PAPEES ON MECHANICAL AND PHYSICAL SUBJECTS. HonDon: C. J. CLAY AND SONS, CAMBRIDGE UNIVERSITY PRESS WAREHOUSE, AVE MARIA LANE. lassflofo: 50, WELLINGTON STREET. H,etp?ig: F. A. BROCKHAU8. Hotft: THE MACMILLAN COMPANY. Bombag: E. SEYMOUR HALE. PAPERS ON MECHANICAL AND PHYSICAL SUBJECTS BY OSBORNE REYNOLDS, F.R.S., MEM. INST. C.E., LL.D., PROFESSOR OP ENGINEERING IN THE OWENS COLLEGE, AND HONORARY FELLOW OF QUEENS' COLLEGE, CAMBRIDGE. REPRINTED FROM VARIOUS TRANSACTIONS AND JOURNALS. VOLUME II 18811900 OF IFO| CAMBRIDGE: AT THE UNIVERSITY PRESS. 1901 [All Eights reserved.] amirtltge : PRINTED BY J. AND C. F. CLAY, AT THE UNIVERSITY PRESS. PREFACE TO VOLUME II. THIS Volume includes the Reprint of my papers on mechanical subjects, following on to those printed in Volume I., from the year 1881 up to date. At the expressed wishes of the authors this Volume also includes the Reprint of the second parts of two papers, of which I contributed the first parts only. One of these is the paper " On the Theory of the (Steam-Engine) Indicator and the Errors in Indicator Diagrams." Of this the second part was contributed by Professor A. W. Brightmore, D.Sc. The other being the paper " On the Mechanical Equivalent of Heat," and of this the second part was contributed by Mr W. H. Moorby, M.Sc. OSBORNE REYNOLDS. 19, LADYBABN ROAD, MANCHESTER. 95916 CONTENTS OF VOL. II. 41. On the Fundamental Limits to Speed .... 1 24 The different limits imposed by the several properties of material the limits determined by the ratio of strength to heaviness stresses due to acceleration in the coupling rods of locomotives the de- structive effect of periodic forces synchronising with the natural period of the structure the extent to which balancing of machines is possible. 42. On an Elementary Solution of the Dynamical Problem of Isochronous Vibration 25 27 43. The Comparative Resistances and Stresses in the Cases of Oscillation and Rotation with Reference to the Steam- Engine and Dynamo 28 50 The friction in the two cases the dynamics of oscillations controlled by a crank effect of reservoirs of energy loss of energy by friction resulting from pressures caused by inertia application to the steam- engine and dynamo. 44. An Experimental Investigation of the circumstances which determine whether the Motion of Water shall be direct or sinuous, and of the Law of Resistance in Parallel Channels 51 105 Section I. Introduction. The failure of the theory of hydrodynamics to explain why the resist- ance is in some cases proportional to the velocity, and in others to the square of the velocity direct and sinuous motion the effect of viscosity character of motion dependent on dimensional properties the evidence of these in the equations of motion experiments by means of coloured bands in glass tubes prove the existence of a critical velocity at which eddying motion begins two streams in opposite directions experiments showing the resistance is constant if v a p/p . c results shown to agree with both Darcy's and Poiseuille's experi- ments 51-67 Section II. Description of the experiments in glass tubes by means of colour bands relations between critical velocity, size of tube and viscosity the sudden appearance of the eddies effect of initial disturbances effect of size of disturbance on instability ...... 68 77 Section III. Experiments to determine critical velocity by measuring the resistance the apparatus methods of measuring the discharge and the pressures effect of temperature results general law of resistance brought out by the method of logarithmic homologues 78 98 Vlll CONTENTS. PAGES Section IV. Application to Darcy's experiments their reduction by means of logarithmic homologues effect of temperature when the critical velocity is passed effect of roughness of surface of pipes . . 99105 45. The Transmission of Energy 106 131 Directed and undirected energy the sources of energy transmission by means of coal and corn transmission of stored energy in the directed form distribution by means of compressed water, com- pressed air, ropes and shafts. On the Equations of Motion and the Boundary Conditions for Viscous Fluids 132 137 The equations of motion shown to be at variance with the boundary conditions modification of the equations to satisfy the boundary conditions. 47. On the General Theory of Thermo- Dynamics . . . 138 1.52 Joule's law and Carnot's ideal engine experimental illustration of the second law of thermodynamics by mechanisms working by means of undirected energy the limits of the steam-engine possibilities of the gas-engine. 48. On the Two Manners of Motion of Water . . . 153162 The inadequacy of the theory of fluid motion to account for the actual behaviour of fluids internal motions seen by introducing colour bands conditions for steady motion converging channels generally steady, and diverging channels unsteady parallel streams steady below and unsteady above a certain velocity effect of viscosity. 49. On the Theory of the Steam-Engine Indicator and the Errors in Indicator- Diagrams ..... 163 180 Requirements that the diagram may be exact. The disturbances on the pencil (1) disfigurement of the diagram caused by the inertia of the mechanism (2) the friction of the pencil the general effect to increase size of diagram. The disturbances on the drum (1) inertia of the drum (2) varying stiffness of the spring (3) the friction of the drum shortening of diagram and reduction of mean pressure. 49A. Experiments on the Steam-Engine Indicator . . . 181 202 Description of the apparatus testing of the springs effect of oscilla- tions of spring stretching of indicator-cord. 50. On the Dilatancy of Media composed of Rigid Particles in contact. With Experimental Illustrations . . 203 216 Any change of shape causes a change of volumes in a granular medium equal spheres arranged as a pile of shot have a density \/2 times as much as when arranged in a cubical formation condition of maximum density very stable friction tends to increase stability experiments with sand and shot contained in bags dilatancy of media a possible explanation of the force of attraction also of cohesion and chemical combination. CONTENTS. IX 51. Experiments showing Dilatancy, a Property of Granular^ Material, possibly connected with Gravitation 52. On the Theory of Lubrication and its Application to Mr Beauchamp Tower s Experiments, including an Experimental Determination of the Viscosity of Olive Oil 217227 228310 Section I. Introduction. Discordance of experimental results Mr Tower's discovery of the sepa- rating film of oil, &c. the idea of a hydrodynamical theory of lubrication the equation of lubrication mentioned before Section A of the British Association at Montreal, and subsequently integrated the comparison of the theoretical results with experimental shows a temperature effect determination of the variation of viscosity of olive oil brings the theory into complete accordance with experiments, and shows how various circumstances affect the results the difference in the radii of brass and journal, and the point of nearest approach of brass to journal, and explanation of increased heating on first reversal the limits of complete lubrication, incomplete lubrication, and necking the general arrangement of the paper . Section II. The Properties of Lubricants. Definition of viscosity the character of viscosity the two viscosities ' experimental determination of the value of /i for olive oil, &c., Figs. 2 and 3, Table I. the comparative values of /* for different fluids and different units 234242 faction 111. General View of the Action of Lubincation. The case of two nearly parallel surfaces separated by a viscous fluid the case of revolving cylindrical surface the effect of a limited supply of lubricating material the relation between resistance, load, and speed for limited lubrication the conditions of equilibrium with cylindrical surfaces the wear and heating of bearings . . . 242 258 Section IV. The Equations of Hydrodynamics as Applied to Lubrication. The complete equations for interior of viscous fluid simplified the boundary conditions the first integration of the resulting equations, equations of lubrication the conditions under which further inte- gration has been undertaken ........ 258 262 Section V. Cases in which the Equations are Completely Integrated. Parallel plane surfaces approaching each other, the surfaces having elliptical boundaries plane surfaces of unlimited length . . . 262 265 Section VI. The Integration of the Equations for Cylindrical Surfaces. General adaptation of the equations the method of approximate inte- gration integration of the equations 266 273 Section VII. Solution of the Equations for Cylindrical Surfaces. 273282 282289 c and /v/ -p small compared with unity further approximation to the solution of the equations for particular values of c, Figs. 18, 19 and 20 c = '5 the limit to this method of integrating .... Section VIII. The Effects of Heat and Elasticity. H and a are only to be inferred from experiments the effect of the load and the velocity to alter a the effect of speed on the tempera- ture the formulae for temperature and friction, and interpretation of constants the maximum load at any speed ..... CONTENTS. Section IX. Application of the Equations to Mr Tower's Experiments. References to Mr Tower's reports, Tables L, IX., and XII. the effect of necking the journal first approximation to the difference in the radii of the journal and brass No. 1 the rise in temperature of the film, owing to friction the actual temperature of the film the variation of a with the load application of the equations to the- v circumstances of Mr Tower's experiments, Table IV. the velocity of maximum carrying power application of the equations to deter- mine the oil pressure with brass No. 2 conclusions . . . 289 311 V/ 53. On the Flow of Oases . 311320 Experiments show that the flow of gas from one vessel into another is independent of the pressures when their ratio is greater than two, whilst according to the theory the flow diminishes and finally ceases as that ratio is increased this anomaly due to the assumption made in the theory that the pressure at the orifice is the pressure in the receiving vessel at a distance from the orifice the assumption avoided by the integration of the fundamental equations of fluid motion. 54. On Methods of Investigating the Qualities of Lifeboats . 321 325 By using models made to scale according to the laws of dynamic similarity, experiments could be made in ordinary weather to correspond to the severest storms with full-sized lifeboats. 55. On certain Laws relating to the Regime of Rivers and Estuaries, and on the possibility of Experiments on a small Scale 326335 The action of water to raise or lower the beds of rivers and estuaries shown to depend on the character of the motion, and to be inde- pendent of the magnitude or velocity of the stream tidal rivers experiments on a model of the Mersey Estuary resemblance of the contours to the charts of the Mersey. 56. On the Triple- Expansion Engines and Engine-Trials at the Whitworth Engineering Laboratory, Owens College, Manchester 336379 Description of the engines, boilers, and connections the appliances for measuring the condensing water, and hot-well discharge arrangement of indicators the specially designed hydraulic brakes their advan- tages over friction brakes objects and methods of conducting the trials the checks on the heat and water afforded by the surface condenser the radiation method of combining the diagrams to show the proportion of steam condensed at all points in the expansion the missing quantity and the effect of the steam jackets in reducing it the relative effect of the jackets in the three cylinders mean results of the trials. 57. Report of the Committee appointed to investigate the Action of Waves and Currents on the Beds and Foreshores of Estuaries by means of Working Models . . . 380 409 Experiments to determine how the distribution of the sand is affected by the horizontal and vertical dimensions of the models, and the tide period description of the models and the tide generators method of surveying conditions of experiments results conclusions plans. CONTENTS. XI 58. Second Report of the Committee appointed to investigate the Action of Waves and Currents on the Beds and Fore- shores of Estuaries by means of Working Models . 410 481 Experiments to ascertain the law of the limits for dynamic similarity critical values of the criterion of similarity general distribution of sand in V-shaped estuaries effects of land water the automatic tide gauges description of the experiments plans. 59. Third Report of the Committee appointed to investigate the Action of Waves and Currents on the Beds and Fore- shores of Estuaries by means of Working Models . 482 518 V-shaped estuary with a long tidal river possible condition of in- stability tides varying from spring to neap effect of training walls effect of groins general results of the investigation descrip- tion of the experiments plans. 60. On Two Harmonic Analyzers 519 523 An instrument for detecting and identifying periodic vibrations in structures an appliance for setting up vibrations in a structure so as to find its natural period. 61. Study of Fluid Motion by means of Coloured Bands . 524-=-534 Experiments showing how the internal motions, otherwise invisible, are shown by the use of colour bands the small resistance to wave motions internal fluid motion generally a process of mixing an experiment showing a straight vortex illustrating how internal waves can exist without motion on the outside boundary. 62. On the Dynamical Theory of Incompressible Viscous Fluids and the Determination of the Criterion . . . 535 577 Section I. Introduction. Stokes' dissipation function and the author's determination of the critical velocity of water considerations which show that the criterion follows from the equations of motion basis of the method of analysis summary of conclusions of the investigation ..... 535 544 Section II. The mean-motion and heat-motions as distinguished by periods mean-mean-motion and relative-mean-motion discriminative cause and action of transformation two systems of equations a dis- criminating equation 544 563 Section III. The criterion of the conditions under which relative-mean-motion cannot be maintained in the case of incompressible fluid in uniform sym- metrical mean-flow between parallel solid surfaces expression for the resistance ........... 563 577 63. Experiments shotving the Boiling of Water in -an Open Tube at Ordinary Temperatures ..... 578 587 Explanation of the hissing noise in the kettle reduction of pressure in a contracting channel sufficient to cause boiling conditions necessary as explained by the author's determination of the critical velocity of water. Xll CONTENTS. PAGES 64. On the Behaviour of the Surface of Separation of Two Liquids of different Densities 588 590 65. On Methods of Determining the Dryness of Saturated Steam and the Condition of Steam Gas . . . 591 600 The conditions of steam in Regnault's experiments wet steam wire- drawing calorimeters theory of the reductions the erroneous as- sumption that the specific heat of steam above the temperature of saturation is constant and equal to that of the steam gas in Regnault's experiments the possibility of obtaining an accurate estimate means of assuring the final condition, that of steam gas. 66. On the Method, Appliances and Limits of Error in the direct Determination of the Work Expended in Raising the Temperature of Ice-Gold Water to that of Water Boiling under a pressure of 29'899 inches of Ice-Cold Mercury in Manchester ...... 601 733 PART I. The standard of temperature in Joule's determination description of the experimental steam-engine and other appliances used in this investigation the brake and the possible errors due to fluctuations of the speed and the turning moment the cyclic variations of speed thermometer scales avoided by working between the standard tem- peratures 32 and 212 the additional appliances required con- duction and radiation of heat the standards of length, mass, and temperature air in the water the specific heat of the water complete table of the corrections for all circumstances affecting the accuracy of the results 601 657 PART II. Details of the several parts of the apparatus and measuring appliances the system of conducting the trials comparison of the thermometers adjustments of the brake leakages details of the different series of trials the correction ......... 658 733 67. On the Slipperiness of Ice 734 738 An explanation afforded by the author's theory of lubrication which shows that continuous rectilinear motion is one of the only two cases in which complete lubrication is possible. 41. ON THE FUNDAMENTAL LIMITS TO SPEED. I. [From "The Engineer," Oct. 28, 1881.] AMONG the facts which are so familiar to us as not to command our attention are the limits to the rates at which we can move over the surface of this earth, or, to put it more generally, the limits to the rate at which terrestrial objects can move. Everyone is now familiar with the fact that railway trains do not exceed sixty or seventy miles an hour; that steamboats do not exceed twenty-five miles an hour ; carriages on ordinary roads, ten or twelve. The fastest running animals rarely exceed a mile in two minutes, or the fastest bird a mile a minute. That there are circumstances on which these limits depend must be generally recognised ; but, while speed is the highest of our mechanical ambitions, how many of those who find themselves confined for nine hours between London and Edinburgh have ever asked themselves, why should there be a limit to speed at all ? In the early days of railroads the question as to the possibility of exceed- ing the speed of animals was very prominent ; and many of the immediate circumstances on which possible speed depends such as the strength and elasticity of the machine, and the smoothness of the road have since received due attention. This was a matter of necessity, just as, in attempt- ing to gain a higher standpoint on the side of a hill, account must be taken of the difficulties of the ground immediately above one. But such notice is a very different thing from a general survey of the limit imposed by the height of the hill itself. While we were still in the valley, and the immediate difficulties of ascent were great, our aspirations might well fall short of the top of the hill, which would not then become an object of attention. But having toiled up a great way, and having apparently reached a flat, or . 2 ON THE FUNDAMENTAL LIMITS TO SPEED. [41 nearly flat, plane on which we are wandering without making any consider- able ascent, it cannot but be a matter of interest and importance to make a more general exploration, and endeavour to ascertain what is the nature of the country behind and above the clouds which surround us. The greatest speeds attained have not increased now for many years. It is probable that the run from Holyhead to London is still the fastest journey ever accomplished over so long a distance, although the number of instances in which this speed is approximately reached are now numerous, and continually increasing. With animals there is no great alteration why should there be ? And with machines, locomotives or steamboats, the improvement is that the average speed more nearly reaches the maximum, rather than any extension of the maximum. Noticing this, we cannot avoid the surmise that the obstruction to further advance arises from something more fundamental than mere economy or imperfection of mechanical con- trivance. The question as to how far this is the case must admit of an answer if the circumstances can be subjected to a complete theoretical examination. The problem is very complicated, and it may well be doubted whether our knowledge of the circumstances and possibilities of art is sufficient to enable us to arrive at a definite conclusion. But what we may do is to look, in the first instance, for any circumstance which imposes a definite limit to possible speeds, and having investigated the law of this limit, look for other limits, and having examined each separately, endeavour to arrive at the result when they are taken in conjunction. To begin with, it will be well to try and catch sight of the top of the hill from a distance. Going far away from the complexity of our immediate problem, we may ask whence there can be any limit to possible speeds ? Any limitations in the circumstances on which speed depends would cause a limit to speed and, although perhaps not very obvious, consideration will show that speed depends on certain physical and mechanical properties of material, and that these are essentially limited. Thus the strength of material is limited. Some materials are stronger than others, but the strength of the strongest is easily reached, and although improvement in art brings the stronger and more appropriate materials within reach, still by no tittle have we been able to extend the strength of the strongest beyond what it has been, so to speak, fixed by nature. When compared by heaviness, natural tissues are the strongest materials. A silk cord will sustain more than a steel wire of the same weight, and such a wire is the strongest form of any manufactured material. To the limited strength, as compared with the weight of material, then, we may look for a limit to possible speeds; and this is not all. There are other limits for instance, the limited temperature at which material retains its strength ; in fact, the properties and powers of material are essentially limited in all directions, and, inasmuch as speed 41] ON THE FUNDAMENTAL LIMITS TO SPEED. 3 depends on these properties, it must be limited. If we take a somewhat closer view, the immediate conclusion is that there are at least two distinct sources of a limit to speed. The first and most obvious of these is that the resistance to motion requires that the moving object should be continually urged forward by a force, and the maintenance of this force requires additions to the weight of the moving object, which additions increase the resistance; so that at a certain speed there will be a balance between the resistance and the force, any increase in the force causing a still greater increase in the resistance. This may be illustrated by reference to a railway. The resistance of the engine is the addition necessary to maintain the motion. Taking the best results, the resistance of an engine at high speeds is about 45 Ib. per ton of its weight. If, then, the locomotive weighs 20 tons it would require a steady pull of 900 Ib. to balance its resistance. To maintain this force a certain pressure of steam must act on the pistons. To keep up this pressure the cylinders must be filled and emptied every revolution of the driving wheels say, every 2(>'4 ft., or 200 times per mile. To maintain the speed then the boiler must supply steam enough to fill the cylinders 200 times per mile, i.e., in whatever time the mile is run. Now the power of supplying steam by the boiler is limited. A boiler of a certain weight cannot be made to supply more than a certain amount of steam, and if we know the shortest time in which the boiler will produce 200 cylinders full of steam at the pressure required to move the engine, we know the shortest time in which it could run a mile, or the limit of speed arising from this source. To increase the size of the boilers would be to increase the weight and consequent resistance of the engine, so that the only chance of extending the limit is to increase the steam -producing power of the same weight of boiler ; and the question whether this actual limit has been reached is a question as to whether there still remains, after all these years, room for improvement in the best boilers whether, in fact, the steam-producing power of boilers has reached the limit imposed by the limit to the strength and other properties of material of which they may be constructed. The case of the locomotive has been introduced here merely for the sake of illustrating the fact that, however distant, there is a limit to possible speeds arising from this source. As a matter of fact this limit is not actually reached, for, as will be subsequently shown, there are other and inferior limits which come in; that is when the engine is running without a train, but when the train is added, as it must be from an economical point of view, then the steam -producing power of the boiler does impose an economical limit on the speed of the train. The case of steamboats is somewhat different. With these the resistance 12 4 ON THE FUNDAMENTAL LIMITS TO SPEED. [41 increases in a high ratio with the speed, as the square of the speed, so that not only have the cylinders to be filled at a rate proportional to the speed of the boat, but to maintain the requisite force the size of the pistons or the pressure of the steam must increase as the square of the speed ; so that instead of being, as with the locomotive, nearly in the simple ratio to the speed, the quantity of steam required in a given time varies as the cube of the speed. Thus, in the case of steamboats, the steam-producing capacity of a certain weight of boiler is the source of the actual as well as the economical limit to the speed. This limit has been reached with the modern steam launch and torpedo boat, in which as much as two-thirds of the whole weight of the ship are given up to the engines and boilers ; the highest speeds so attained being about twenty-five miles an hour. The action of this, which may be called the physical limit to speed, may be traced in animals, but the requisite data for its discussion are wanting. The second funda- mental source of limits to speed is the strength of the parts, and the forces holding these parts, necessary to withstand the forces to which the motion gives rise. This may be called the dynamical limit to speed. This source of limit has received less general notice than the preceding. That the motion of machines and animals necessarily gives rise to forces in and between their parts is not perhaps very obvious, on account of its being so well known that motion itself does not give rise to force between the parts of a moving object. But this is only when the motion is rectilinear and uniform. To stop and start a body or to change its direction requires force proportional to the weight of the body and the rate at which the change is made. In order to realise how all possible motions on the earth are limited, it must be noticed that uniform rectilinear motion is impossible. Objects on the earth have to maintain their motion against such resistances as they encounter by the relative and limited motions of their parts ; with animals by the motion of their legs, wings, or fins ; in machines by the motions of their pistons, cranks, and wheels ; and, even apart from this, uniform motion is impossible owing to the impossibility of maintaining a direct course for instance, a perfectly even road. The limit to the speed of any complex body, such as an animal, an engine, or even a revolving wheel, will depend primarily on the manner in which the general motion depends on or involves change in the speed or direction of motion of any or all of the parts. For example, in the case of all carriages the limit to the strength of the tires of the wheels would limit the speed if there were no inferior limit. That what is called centrifugal force tends to burst the tires must be universally known ; but there is a simplicity about the law of this limit which marks it out as the best illustration of the class of limits which arise from acceleration. 41] ON THE FUNDAMENTAL LIMITS TO SPEED. 5 The bursting tension of the tire caused by the revolution of the wheel is the result of the centrifugal force acting on each elementary portion of the tire, and is the same as if the tire were subject to an outward pressure equal to the centrifugal force all over its inner surface. The dynamical problem of estimating the centrifugal tension from the weight, diameter, and speed of revolution of the tire is not difficult, but it will be sufficient here to state the result. The tension per square inch of section of the tire is '37 multiplied by the weight of a cubic inch of the material and the square of the velocity in feet per second. The limit of speed is that which causes a centrifugal tension equal to the greatest stress the material will safely bear. With iron this is about 15,000 Ib. per square inch. A cubic inch of iron weighs '24 lb., so that the velocity squared is equal to 11x15,000 or 165,000, or, roughly, the velocity equals 400 ft. per second. This, which is 270 miles an hour, is the limit arising from centrifugal force to the safe velocity ; for steel tires, the strength of which is about double that of iron, the limit becomes 380 miles an hour. It should be noticed that neither the diameter of the wheel nor the thickness of the tire makes any difference to this limit, which depends solely on the ratio between the strength and heaviness of the material. If we could get a stronger material, then we might extend the limit, but as natural fibres are the only materials stronger than steel, and these do not possess the hardness necessary for tires, there is absolutely no prospect of any extension in this direction. The velocity of the train is the same as the velocity of the tire, so that the figures given above show the limit to the velocity of the train arising from the centrifugal force on the tire that is, supposing the tire subject to no forces but those considered. Looked at in this way, the limit appears well away from any speeds already realised. But as the tire is subject to forces arising from its contact with the rail and from the load on the wheel, the margin left for centrifugal force is much less than what has been stated, so that the actual limit, which involves complex considerations, is really much lower. Wheels have been here considered as affording the simplest example of how changes in the direction, or speed of motion in the parts, of a moving object must cause a limit to the speed at which the object can move, and not 1> -cause the wheels are the parts which would give way first were the speed to be increased. In the locomotive, as at present constructed, there are parts the coupling and connecting rods, for instance which would give way under these accelerations before the tires ; and it will be the object in a subsequent article to discuss somewhat fully the limit to speed imposed by these, as well as by othor parts of the machinery. In the case of animals there are no wheels, but the problem does not 6 ON THE FUNDAMENTAL LIMITS TO SPEED. [41 differ greatly ; for the forces required to stop and start the limbs tax the strength of these in much the same way as the strength of the tires is taxed by centrifugal force. So that the conclusion is the same, that the strength, as compared with the heaviness, of the material of the bones and tissues of animals determines a limit to the possible speed ; which conclusion is borne out by the fact that the strength, as compared with the heaviness of these materials, is as high, or higher, than that of any other materials the strength being that required to resist the particular forces which the parts are generally called upon to sustain, i.e., bone to resist crushing, and sinews to resist tension. Before closing this article, which is intended as an introduction to the more definite discussion of certain particular cases where these limits come in, it should be pointed out that besides the two sources of limits to speed which have been particularly noticed, viz., those which arise from the strength of the material, and those from the limited capacity of producing energy, there are other sources of limits. One of these, of a physical kind, is the inability to get rid of the heat produced at the joints by friction. The heating of bearings, which is a very common source of the actual limit to speed, although it has not apparently received much attention except in a practical way, admits of theoretical consideration as being subject to definite laws. Another source of the limit to speed, of the greatest practical importance, although more complex than the preceding, is the effect of the moving pieces and the forces between these to cause unsteadiness to the motion of the whole structure. The difficulty of keeping a railway train steady has perhaps as much to do with the actual speed attained as any other cause. In so far as this unsteadiness arises from the unevenness of the road, and the mere disturbing forces caused on the frame by the moving pieces, it belongs to the class of dynamical limits, but it depends on a particular property of matter not involved in other cases of this class of limits. The rocking of a structure depends on the character of its elasticity, and on the period as well as the magnitude of the disturbing forces ; and, as a matter of fact, the tendency to vibrate would impose a limit on the speed of most machines, so that it is entitled to a place amongst the sources of limit, and may be called the elastic limit. So far, then, we see that there are four distinct sources of limits to speed. The limited capacity of producing energy, the limited strength of the material, the limited power of discharging the heat produced by friction, and the elastic limit. In pointing out the general nature of these limits, attention has been directed to objects with powers of locomotion as being more familiar ; there are, however, the same sources of limits to the speed of stationary machinery, such as steam-engines and tools. 41] ON THE FUNDAMENTAL LIMITS TO SPEED. II. [From "The Engineer," Nov. 18, 1881.] To obtain an idea of the effect of accelerations, we may take an instance of a moving machine, and supposing its speed to increase, consider which of its parts would give way first. The locomotive seems to afford the best example. Imagine, then, a locomotive to be started down a long incline with the steam fully on ; what part of the machine would give way first ? In the case of an engine with its wheels coupled, the question may be answered with certainty. The coupling rods would be thrown off. Although perhaps not generally known, this has been shown both theoretically and practically. Anyone with the smallest mechanical insight, observing from a distance a coupled engine in motion, cannot fail to perceive that the rapid up-and-down motion of these rods, which are held only at the ends, must call for great strength to prevent them breaking in the middle. That the strength so called for approaches the actual strength of the rods can, of course, only be ascertained by definite calculation. Six years ago the case of one of these rods was taken as an example, to illustrate to the engineering class at Owens College the effect of accelerations, and the result of the calculation then made was to show that the strength called for when the engine was running at 70 miles an hour was nearer the limit imposed by the actual strength of the material than is usually considered safe in estimating the size of such structures. Thus, instead of 10,000 lb., the stress amounted in this example to 15,000 lb. The fact was surprising enough to arrest attention, and raise a question as to the considerations which had led to the proportions of these rods. On reference to the text-books and manuals it was found that the effects of accelerations had no place in them, so that it would appear that engineers have had no rule to go by but that of experience; or, in other words, that the dimensions of these rods have been arrived at by the process of trial and failure. All these facts considered, the matter seemed one of no small mechanical interest. For apart from the importance of these rods and the desirability of supplying a theoretically derived formula in place of empirical rules, the experience of the fitness of these rods has been so ample, that as soon as we are in a position to calculate the stresses in their material, they furnish a very important test as to the factor of safety for such parts of machinery. Thus, it appears that while a rule has been laid down that a certain stress is the greatest which the iron in any important part of a machine should bear, these very important parts have been unwittingly 8 ON THE FUNDAMENTAL LIMITS TO SPEED. [41 allowed to bear, and have borne safely, half as much again as that given by the rule. That the stress in these rods may be as great as appeared from theoretical consideration, or, at least, that they are the parts of the engine which first give way when an undue speed is attained, has been confirmed by the records of railway accidents. Shortly after the first investigations were made, a train having on it three similar coupled engines ran away down an incline, the brakes being overpowered, and eye-witnesses described how the first symptom of disaster was the flying off of the coupling rods from one of the engines, those from the others following immediately after. In 1878 attention was called to these facts at a meeting of the Manchester Literary and Philosophical Society, and they excited the interest of Dr Joule, who has kindly sent the author published accounts of several instances of the failure of these rods in cases of high speeds. Amongst these was the following extract from a letter published in the Manchester Courier. The accident occurred on the Cheshire line from Manchester to Liverpool, on which the speeds are very high. The author of the letter has clearly used the term connecting rod in the sense of coupling rod. " Shortly after we had passed one of the small stations on the way, and before reaching Warrington, the connecting rod of the engine, or some other material -portion of that part of the mechanism, became broken, and flew off with such force as to strike the embankment on the near side, and thence rebound with terrible power into the window of one of the third-class carriages immediately behind, completely smashing in the woodwork, as well as all the glass, to the great danger of one or more passengers within, but who escaped uninjured. I was a passenger on another occasion, on the same journey, when the connecting rod snapped in two, and the two pieces continued to whirl round until the train could be stopped, to the great risk of driving the engine and carriages from the metals. And I have heard it said that accidents of a similar kind have occurred on other occasions." The theory of these rods has been taught in the engineering classes at Owens College for several years, but its first appearance in print seems to have been in a letter in The Engineer of May 27th, 1881, signed " S. R.," dated Manchester, May llth; and more fully in an article which appeared in The Engineer, of Sept. 9th, 1881. Leaving what we may call the swinging forces out of consideration, the coupling rods are designed to withstand certain forces which cannot exceed a definite amount. This amount may be estimated for each particular case. The utmost one rod can be called upon to do is to turn one pair of wheels against the whole friction between the wheels and the rail, which latter may be sanded. In such a case, F, the coefficient of friction, would be about - 3. Let R be the radius of the wheels in inches, L the length of the cranks, P the pressure between the wheel and the rail in pounds ; then taking T for the force in pounds, tension or compression, in the 41] ON THE FUNDAMENTAL LIMITS TO SPEED. 9 rod necessary to cause the pair of wheels to slide when the other rod is in the line of centres, LT=FRP, . Li T may be either tension or compression, but it is the latter that is the most important for the present consideration. If now we take the swinging action into account, we have to add the effect of the vertical force which must act on each point of the rod in order to change its vertical motion. Relatively to the engine each point of the rod will describe a circle exactly similar to that described by either crank pin. In describing its circle each portion of the rod will be subject to centrifugal force. Consider a cubic inch of material of weight w, the centrifugal force of this by the well-known formula is wv*L*_ wv*L - Where v is the velocity of the engine in feet per second, and * c " 47700000 D 3 ' 44] THE LAW OF RESISTANCE IN PARALLEL CHANNELS. 67 the units being metres and degrees centigrade. It will be observed that this value is much less than the critical velocity at which steady motion broke down ; the ratio being 437 to 278. 17. The general law of resistance. The logarithmic homologues all consist of two straight branches, the lower branch inclined at 45 degrees and the upper one at n horizontal to 1 vertical. Except for the small distance beyond the critical velocity these branches constitute the curves. These two branches meet in a point on the curve at a definite distance below the critical pressure, so that, ignoring the small portion of the curve above the point before it again coincides with the tipper branch, the logarithmic homologue gives for the law of resistance for all pipes and all velocities . D' . / R D \ A i = (B-v\ , P" V p ) where n has the value unity as long as either number is below unity, and then takes the value of the slope n to 1 for the particular surface of the pipe. If the units are metres and degrees centigrade, 4 = 67,700,000, B = 396, P = (1 + 0-0336 T+ 0-000221 T*)~\ This equation then, excluding the region immediately about the critical velocity, gives the law of resistance in Poiseuille's tubes, those of the present investigation and Darcy's, the range of diameters being from 0-000013 (Poiseuille, 1845) to 0-5 (Darcy, 1857), and the range of velocities from 0-0026) 7 [ metres per sec., 188,3. TiO / Uv J This algebraical formula shows that the experiments entirely accord with the theoretical conclusions. The empirical constants are A, B, P, and n ; the first three relate solely to the dimensional properties of the fluid summed up in the viscosity, and it seems probable that the last relates to the properties of the surface of the pipe. Much of the success of the experiments is due to the care and skill of Mr Foster, of Owens College, who has constructed the apparatus and assisted me in making the experiments. 52 68 ON THE MOTION OF WATER, AND OF [44 SECTION II. Experiments in glass tubes by means of colour bands. 18. In commencing these experiments it was impossible to form any very definite idea of the velocity at which eddies might make their appearance with a particular tube. The experiments of Poiseuille showed that the law of resistance varying as the velocity broke down in a pipe of say 0'6 millim. diameter ; and the experiments of Darcy showed this law did not hold in a half-inch pipe with a velocity of 6 inches per second. These considerations, together with the comparative ease with which experiments on a small scale can be made, led me to commence with the smallest tube in which I could hope to perceive what was going on with the naked eye, expecting confidently that eddies would make their appearance at an easily attained velocity. 19. The first apparatus. This consisted of a tube about inch or 6 millims. in diameter. This was bent into the siphon form having one straight limb about 2 feet long and the other about 5 feet (Fig. 10). Fig. 10. Fig. KX. The end of the shorter limb was expanded to a bell mouth, while the 44] THE LAW OV RESISTANCE IN PARALLEL CHANNELS. 69 longer end was provided with an indiarubber extension on which was a screw clip. The bell-mouthed limb was held vertically in the middle of a beaker, with the mouth several inches from the bottom as shown in Figs. 10 and 10'. A colour tube about 6 millims. in diameter, also of siphon form, was placed as shown in the figure, with the open end of the shorter limb just under the bell mouth, the longer limb communicating through a controlling clip with a reservoir of highly coloured water placed at a sufficient height. A supply-pipe was led into the beaker for the purpose of filling it ; but not with the idea of maintaining it full, as it seemed probable that the inflowing water would create too much disturbance, experience having shown how important perfect internal rest is to successful experiments with coloured water. 20. The first experiment The vessels and the siphons having been filled and allowed to stand for some hours so as to allow all internal motion to cease, the colour clip was opened so as to allow the colour to emerge slowly below the bell (Fig. 11). Fig. 11. Fig. 12. Then the clip on the running pipe was opened very gradually. The water was drawn in at the bell mouth, and the coloured water entered, at first taking the form of a candle flame (Fig. 12), which continually elongated until it became a very fine streak, contracting immediately on leaving the colour-tube, and extending all along the tube from the bell mouth to the outlet (Fig. 10). On further opening the regulating clip so as to increase 70 ON THE MOTION OF WATER, AND OF [44 the velocity of flow, the supply of colour remaining unaltered, the only effect was to diminish the thickness of the colour band. This was again increased by increasing the supply of colour, and so on until the velocity was the greatest that circumstances would allow until the clip was fully open. Still the colour band was perfectly clear and definite throughout the tube. It was apparent that if there were to be eddies it must be at a higher velocity. To obtain this about 2 feet more were added to the longer leg of the siphon, which brought it down to the floor. On trying the experiment with this addition, the colour band was still clear and undisturbed. So that for want of power to obtain greater velocity this experiment failed to show eddies. When the supply pipe which filled the beaker was kept running during the experiment, it kept the water in the beaker in a certain state of disturb- ance. The effect of this disturbance was to disturb the colour band in the tube, but it was extremely difficult to say whether this \yas due to the wavering of the colour band or to genuine eddies. 21. The final apparatus. This was on a much larger scale than the first. A straight tube, nearly 5 feet long and about an inch in diameter, was selected from a large number as being the most nearly uniform, the variation of the diameter being less than l-32nd of an inch. The ends of this tube were ground off plane, and on the end which appeared slightly the larger was fitted a trumpet mouth of varnished wood, great care being taken to make the surface of the wood continuous with that of the glass (Fig. 13). The other end of the glass pipe was connected by means of an indiarubber washer with an iron pipe nearly 2 inches in diameter. The iron pipe passed horizontally through the end of a tank, 6 feet long, 18 inches broad and 18 inches deep, and then bent through a quadrant so that it became vertical, and reached 7 feet below the glass tube. It then terminated in a large cock, having, when open, a clear way of nearly a square inch. This cock was controlled by a long lever reaching up to the level of the tank. The tank was raised upon trestles about 7 feet above the floor, and on each side of it, at 4 feet from the ground, was a platform for the observers. The glass tube thus extended in a horizontal direction along the middle of the tank, and the trumpet mouth was something less than a foot from the end. Through this end, just opposite the trumpet, was a straight colour 44] THE LAW OF RESISTANCE IN PARALLEL CHANNELS. 71 tube three-quarters of an inch in diameter, and this tube was connected, by means of an indiarubber tube with a clip upon it, with a reservoir of colour, which for good reasons subsequently took the form of a common water bottle. Fig. 13. With a view to determining the velocity of flow, an instrument was fitted for showing the changes of level of the water in the tank to the 100th of an inch (Fig. 14). Thermometers were hung at various levels in the tank. '22. The final experiments. The first experiment with this apparatus was made on 22nd February, 1880. By means of a hose the tank was filled from the water main, and having been allowed to stand for several hours, from 10A.M. to 2 P.M., it was then found that the water had a temperature of 46 F. at the bottom of the tank, and 47 F. at the top. The experiment was then commenced in the same ON THE MOTION OF WATER, AND OF [44 manner as in the first trials. The colour was allowed to flow very slowly, and the cock slightly opened. The colour band established itself much as before, and remained beautifully steady as the velocity was increased until, Fig. 14. all at once, on a slight further opening of the valve, at a point about two feet from the iron pipe, the colour band appeared to expand and mix with the water so as to fill the remainder of the pipe with a coloured cloud, of what appeared at first sight to be of a uniform tint (Fig. 4, p. 59). Closer inspection, however, showed the nature of this cloud. By moving the eye so as to follow the motion of the water, the expansion of the colour band resolved itself into a well-defined waving motion of the band, at first without other disturbance, but after two or three waves came a succession of well-defined and distinct eddies. These were sufficiently recognisable by following them with the eye, but more distinctly seen by a flash from a spark, when they appeared as in Fig. 5, p. 60. The first time these were seen the velocity of the water was such that the tank fell 1 inch in 1 minute, which gave a velocity of O m- 627, or 2 feet per second. On slightly closing the valve the eddies disappeared, and the straight colour band established itself. Having thus proved the existence of eddies, and that they came into existence at a certain definite velocity, attention was directed to the relations between this critical velocity, the size of the tube, and the viscosity. Two more tubes (2 and 3) were prepared similar in length and mounting to the first, but having diameters of about one-half and one-quarter inch respectively. 44] THE LAW OF RESISTANCE IN PARALLEL CHANNELS. 73 In the meantime an attempt was made to ascertain the effect of viscosity by using water at different temperatures. The temperature of the water from the main was about 45, the temperature of the room about 54 ; to obtain a still higher temperature, the tank was heated to 70 by a jet of steam. Then taking, as nearly as we could tell, similar disturbances, the experiments which are numbered 1 and 2 in Table I., page 74, were made. To compare these for the viscosity, Poiseuille's experiments were available, but to prevent any accidental peculiarity of the water being overlooked, experiments after the same manner as Poiseuille's were made with the water in the tank. The results of these however agreed so exactly with those of Poiseuille that the comparative effect of viscosity was taken from Poiseuille's formula P-I = 1 + 0-03368T + 0-000221 T\ where P x. fj, with the temperature and T is temperature centigrade. The relative values of P at 47 and 70 Fah. are as 1-3936 to 1, while the relative critical velocities at these temperatures were as 1-45 to 1, which agreement is very close considering the nature of the experiments. But whatever might have been the cause of the previous anomalies, these were greatly augmented in the heated tank. After being heated the tank had been allowed to stand for an hour or two, in order to become steady. On opening the valve it was thought that the eddies presented a different appearance from those in the colder water, and the thought at once suggested itself that this was due to some source of initial disturbance. Several sources of such disturbance suggested themselves the temperature of the tank was 11 C. above that of the room, and tne cooling arising from the top and sides of the tank must cause circulation in the tank. A few streaks of colour added to the water soon showed that such a circulation existed, although it was very slow. Another source of possible disturbance was the difference in the temperature at the top and bottom of the tank, this had been as much as 5. In order to get rid of these sources of disturbance it was necessary to have the tank at the same temperature as the room, about 54 or 55. Then it was found by several trials that the eddies came on at a fall of about 1 inch in 64 seconds, which, taking the viscosity into account, was higher than in the previous case, and this was taken to indicate that there was less disturbance in the water. As it was difficult to alter the temperatures of the building so as to obtain experiments under like conditions at a higher temperature, and it appeared 74 ON THE MOTION OF WATER, AND OF [44 that the same object would be accomplished by cooling the water to its maximum density, 40, this plan was adopted and answered well, ice being used to cool the water. Experiments were then made with three tubes 1, 2, 3, at temperatures of about 51 and 40. All are given in Table I. TABLE I. Experiments with Colour Bands Critical Velocities at which Steady Motion breaks down. Pipe No, 1, glass. Diameter 0'0268 metre; log diameter 2*42828. No. 2, 0-01527 2-18400. No. 3, 0-007886 3'89783. Discharge, cub. metre = '021237 ; log = 2-32709. Date, 1880 Kefer- ence num- ber Pipe Tem- pera- ture, centi- grade Time of dis- charge Velocity, metres log time -logP log V log B e 1 March 1 No. 1 8-3 60 0-6270 1-77815 0-11242 T-79729 1-66200 3 2 11 21 87 0-4325 1-93959 0-25654 1 -63593 1-67930 25 3 I.') 70 0-5374 1-84500 0-19198 1-73035 1-64936 21 April 4 15 12 60 0-6270 1-77815 0-15712 1 -79729 1-61730 11 ."> 11 13 64 0-5878 1-80618 0-16882 1-76926 1 -64464 11 6 11 13 67 0-5614 1-82617 0-16882 1-74927 1-65363 V 7 11 13 64 0-5878 1-80618 0-16882 1-76926 1-64464 8 11 5 54 0-6967 1-73239 0-06963 1 -84305 1 -65898 11 9 * 5 52 0-7235 1-71600 0-06963 1 -85940 1-64269 22 10 11 10 62 0-6068 1-79239 0-13319 1-78305 1-65546 11 11 M 11 64 0-5870 1-80613 0-14525 1-76931 1-65716 25 March 12 No. 2 * 22 1.-).-, 0-7476 2-19033 0-26710 1-87367 1-67523 23 April 13 11 11 110 1-052 2-04139 0-14525 0-02261 1-64814 14 11 11 108 1-072 2-03342 0-14525 0-03058 1-64017 11 15 11 4 83 1-396 1-91907 0-05621 0-14493 1-61486 16 11 4 83 1-396 1-91907 0-05621 0-14493 1-61486 11 17 11 4 83 1-396 1-91907 0-05621 0-14493 1-61486 11 18 55 6 86 1-348 1-93449 0-08278 0-12951 1-59371 > 19 11 6 85 1-362 1-92941 0-08278 0-13459 1-59863 24 20 No. 3 11 220 1-967 2-34242 0-14525 0-29392 1-66300 11 21 5> 10-5 224 1-932 j 2-35024 0-13920 0-28610 1-67687 11 22 55 11 218 1-982 2-33845 0-14525 i 0-29789 1-65903 11 23 55 11 116 2-004 2-33445 0-14525 0-30189 1-65503 2:, 24 1) 4 164 2-637 2-21484 0-05621 0-42150 1-62446 11 25 11 4 172 2-517 2-23552 0-05621 0-40082 1 -64514 11 26 11 6 176 2-460 2-24551 0-08278 0-39083 1-62856 11 27 11 6 176 2-460 2-24551 0-08278 0-39083 : 1-62856 11 28 6 174 2-488 2-24054 0-08278 0-39580 ! 1-62359 11 29 6 177 2-446 2-24791 0-08278 0-38837 1-63102 This gives the mean value for log, TG4139 ; and U a = 44] THK LAW OF RESISTANCE IN PARALLEL CHANNELS. 75 In reducing the results the unit taken has been the metre and the tem- perature is given in degrees centigrade. The diameters of the three tubes were found by filling them with water. The time measured was the time in which the tank fell 1 inch, which in cubic metres is given by Q = -021237. In the table the logarithms of P, v, and B s are given, as well as the natural numbers for the sake of reference. The velocities v have been obtained by the formula _ B s being obtained from the formula The filial value of B s is obtained from the mean value of the logarithm of B t . 23. . The results. The values of log B s show a considerable amount of regularity, and prove, I think conclusively, not only the existence of a critical velocity at which eddies come in, but that it is proportional to the viscosity and inversely proportional to the diameter of the tube. The fact, however, that this relation has only been obtained by the utmost care to reduce the internal disturbances in the water to a minimum must not be lost sight of. The fact that the steady motion breaks down suddenly shows that the rluid is in a state of instability for disturbances of the magnitude which cause it to break down. But the fact that in some conditions it will break down for a large disturbance, while it is stable for a smaller disturbance shows that there is a certain residual stability so long as the disturbances do not exceed a given amount. The only idea that I had formed before commencing the experiments was that at some critical velocity the motion must become unstable, so that any disturbance from perfectly steady motion would result in eddies. I had not been able to form any idea as to any particular form of dis- turbance being necessary. But experience having shown the impossibility of obtaining absolutely steady motion, I had not doubted but that appearance of eddies would be almost simultaneous with the condition of instability. 76 ON THE MOTION OF WATER, AND OF [44 I had not, therefore, considered the disturbances except to try and diminish them as much as possible. I had expected to see the eddies make their appearance as the velocity increased, at first in a slow or feeble manner, indicating that the water was but slightly unstable. And it was a matter of surprise to me to see the sudden force with which the eddies sprang into existence, showing a highly unstable condition to have existed at the time the steady motion broke down. This at once suggested the idea that the condition might be one of instability for disturbance of a certain magnitude and stable for smaller disturbances. In order to test this, an open coil of wire, as in Fig. 15, was placed in the tube so as to create a definite disturbance. Fig. 15. Eddies now showed themselves at a velocity of less than half the previous critical velocity, and these eddies broke up the colour band, but it was difficult to say whether the motion was really unstable or whether the eddies were the result of the initial disturbance, for the colour band having once broken up and become mixed with the water, it was impossible to say whether the motion did not tend to become steady again later on in the tube. Subsequent observation however tended to show that the critical value of the velocity depended to some extent on the initial steadiness of the water. One phenomenon in particular was very marked. Where there was any considerable disturbance in the water of the tank and the cock was opened very gradually, the state of disturbance would first show itself by the wavering about of the colour band in the tube ; sometimes it would be driven against the glass and would spread out, and all without a symptom of eddies. Then, as the velocity increased but was still com- paratively small, eddies, and often very regular eddies, would show themselves along the latter part of the tube. On further opening the cock these eddies would disappear and the colour band would become fixed and steady right through the tube, which condition it would maintain until the velocity reached its normal critical value, and then the eddies would appear suddenly as before. Another phenomenon very marked in the smaller tubes, was the inter- 44] THE LAW OF RESISTANCE IN PARALLEL CHANNELS. 77 mittent character of the disturbance. The disturbance would suddenly come on through a certain length of the tube and pass away and then come on again, giving the appearance of flashes, and these flashes would often commence successively at one point in the pipe. The appearance when the flashes succeeded each other rapidly was as shown in Fig. 16. Fig. 16. This condition of flashing was quite as marked when the water in the tank was very steady, as when somewhat disturbed. Under no circumstances would the disturbance occur nearer to the trumpet than about 30 diameters in any of the pipes, and the flashes generally, but not always, commenced at about this distance. In the smaller tubes generally, and with the larger tube in the case of the ice-cold water at 40, the first evidence of instability was an occasional flash beginning at the usual place and passing out as a disturbed patch two or three inches long. As the velocity was further increased these flashes became more frequent until the disturbance became general. I did not see a way to any very crucial test as to whether the steady motion became unstable for a large disturbance before it did so for a small one ; but the general impression left on my mind was that it did in some way as though disturbances in the tank, or arising from irregularities in the tube, were necessary to the existence of a state of instability. But whatever these peculiarities may mean as to the way in which eddies present themselves, the broad fact of there being a critical value for the velocity at which the steady motion becomes unstable, which critical value is proportional to pc' where c is the diameter of the pipe and fi/p the viscosity by the density, is abundantly established. And cylindrical glass pipes for approximately steady water have for the critical value V = where in metres B s = 4370 about. 78 ON THE MOTION OF WATER, AND OF [44 SECTION III. Experiments to determine the critical velocity by means of resistance in the pipes. 24. Although at first sight such experiments may appear to be simple enough, yet when one began to consider actual ways and means, so many uncertainties and difficulties presented themselves, that the necessary courage for undertaking them was only acquired after two years' further study of the hydrodynamical aspect of the subject, by the light thrown upon it by the previous experiment with the colour bands. This has been already explained in Art. 13. Those experiments had shown definitely that there was a critical value of the velocity at which eddies began, if the water were approximately steady when drawn into the tube, but they had also shown definitely, that at such critical velocity, the water in the tube was in a highly unstable condition ; any considerable disturbance in the water causing the break down to occur at velocities much below the highest that could be attained when the water was at its steadiest ; suggesting that if there were a critical velocity at which, for any disturbance whatever, the water became stable, this velocity was much less than that at which it would become unstable for infinitely small disturbances ; or, in other words, suggesting that there were two critical values for the velocity in the tube, the one at which steady motion changed into eddies, the other at which eddies changed into steady motion. Although the law for the critical value of the velocity had been suggested by the equations of motion, it was, as already explained, only at the beginning of this year that I succeeded in dealing with these equations so as to obtain any theoretical explanation of the dual criteria ; but having at last found this, it became clear to me that, if in a tube of sufficient length the water were at first admitted in a high state of disturbance, then as the water proceeded along the tube, the disturbance would settle down into a steady condition, which condition would be one of eddies or steady motion, according to whether the velocity was above or below what may be called the real critical value. The necessity of initial disturbance precluded the method of colour bands, so that the only method left was to measure the resistance at the latter portion of the tube in conjunction with the discharge. The necessary condition was somewhat difficult to obtain. The change in the law of resistance could only be ascertained by a series of experiments 44] THK LAW OF RESISTANCE IN PARALLEL CHANNELS. 79 which had to be carried out under similar conditions as regards temperature, kind of water, and condition of the pipe; and in order that the experiments might be satisfactory, it seemed necessary that the range of velocities should extend far on each side of the critical velocity. In order to best ensure these conditions, it was resolved to draw the water direct from the Manchester main, using the pressure in the main for forcing the water through the pipes. The experiments were conducted in the workshop in Owens College, which offered considerable facilities owing to arrangements for supplying and measuring the water used in experimental turbines. Fig. 17. 25. The apparatus is shown in Fig. 17. 80 ON THE MOTION OF WATER, AND OF [44 As the critical value under consideration would be considerably below that found for the change for steady motion into eddies, a diameter of about half an inch (12 millims.) was chosen for the larger pipe, and one quarter of an inch for the smaller, such pipes being the smallest used in the previous experiments. The pipes (4 and 5) were ordinary lead gas, or water pipes. These, which, owing to their construction, are very uniform in diameter, and when new, present a bright metal surface inside, seemed well adapted for the purpose. Pipes 4 (which was a quarter-inch pipe) and 5 (which was a half-inch) were 16 feet long, straightened by laying them in a trough formed by two inch boards at right angles. This trough was then fixed so that one side of the trough was vertical and the other horizontal, forming a horizontal ledge on which the pipes could rest at a distance of 7 feet from the floor ; on the outflow ends of the pipes cocks were fitted to control the discharge, and at the inlet end the pipes were connected, by means of a T branch, with an indiarubber hose from the main ; this connexion was purposely made in such a manner as to necessitate considerable disturbance in the water entering the pipes from the hose. The hose was connected, by means of a quarter- inch cock, with a four-inch branch from the main. With this arrangement the pressure on the inlet to the pipes was under control of the cock from the main, and at the same time the discharge from the pipes was under control from the cocks on their ends. This double control was necessary owing to the varying pressure in the main, and after a few preliminary experiments a third and more delicate control, together with a pressure gauge, were added, so as to enable the observer to keep the pressure in the hose, i.e., on the inlets to the pipes, constant during the experiments. This arrangement was accomplished by two short branches between the hose and the control cock from the main, one of these being furnished with an indiarubber mouthpiece with a screw clip upon it, so that part of the water which passed the cock might be allowed to run to waste, the other branch being connected with the lower end of a vertical glass tube, about 6 millims. in diameter and 30 inches long, having a bulb about 2 inches diameter near its lower extremity, and being closed by a similar bulb at its top. This arrangement served as a delicate pressure gauge. The water entering at the lower end forced the air from the lower bulb into the upper, causing a pressure of about 30 inches of mercury. Any further rise increased this pressure by forcing the air in the tubes into the upper bulb, and by the THE LAW OF RESISTANCE IN PARALLEL CHANNELS. 81 weight, of water in the tube. During an experiment the screw clip was con- tinually adjusted, so as to keep the level of the water in the glass tube between the bulbs constant. 26. The resistance gauges. Only the last 5 feet of the tube was used for measuring the resistance, the first 10 or 11 feet being allowed for the acquirement of a regular condition of flow. It was a matter of guessing that 10 feet would be sufficient for this, but since, compared with the diameter, this length was double as great for the smaller tube, it was expected that any insufficiency would show itself in a greater irregularity of the results obtained with the larger tube, and as no such irregularity was noticed it appears to have been sufficient. At distances of 5 feet near the ends of the pipe, two holes of about 1 inillim. were pierced into each of the pipes for the purpose of gauging the pressures at these points of the pipes. As owing to the rapid motion of the water in the pipes past these holes, any burr or roughness caused in the inside of the pipe in piercing these holes would be apt to cause a disturbance in the pressure, it was very important that this should be avoided. This at first seemed difficult, as owing to the distance 5 feet of one of the holes from the end of pipes of such small diameter, the removal of a bun-, which would be certain to ensue on drilling the holes from the outside, was difficult. This was overcome by the simple expedient suggested by Mr Foster of drilling holes completely through the pipes and then plugging the side on which the drill entered. Trials were made, and it was found that the burr thus caused was very slight. Before drilling the holes short tubes had been soldered to the pipes, so that the holes communicated with these tubes ; these tubes were then con- nected with the limbs of a siphon gauge by indiarubber pipes. These gauges were about 30 inches long ; two were used, the one con- taining mercury, the other bisulphide of carbon. These gauges were constructed by bending a piece of glass tube into a U form, so that the two limbs were parallel and at about one inch apart. Glass tubes are seldom quite uniform in diameter, and there was a difference in the size of the limbs of both gauges, the difference being con- siderable in the case of the bisulphide of carbon. The tubes were fixed to stands with carefully graduated scales behind them, so that the height of the mercury or carbon in each limb could be read. It had been anticipated that readings taken in this way would be o. 11. ii. 6 82 ON THE MOTION OF WATER, AND OF [44 sufficient. But it turned out to be desirable to read variations of level of the smallness of j^ooth of an inch or J^th of a millimetre. A species of cathetometer was used. This had been constructed for my experiments on Thermal Transpiration, and would read the position of the division surface of two fluids to T oio^ n mcn (P a e 258, Vol. I.). The water was carefully brought into direct connexion with the fluid in the gauge, the indiarubber connexions facilitating the removal of all air. 27. Means adopted in measuring the discharge. For many reasons it was very desirable to measure the rate of discharge in as short a time as possible. For this purpose a species of orifice or weir gauge was constructed, consisting of a vertical tin cylinder two feet deep, having a flat bottom, being open at the top, with a diaphragm consisting of many thicknesses of fine wire gauze about two inches from the bottom ; a tube connected the bottom with a vertical glass tube, the height of water in which showed the pressure of water on the bottom of the gauze ; behind this tube was a scale divided so that the divisions were as the square roots of the height. Through the thin tin bottom were drilled six holes, one an eighth of an inch diameter, one a quarter of an inch, and four of half an inch. These holes were closed by corks so that any one or any combination could be used. The combinations used were : Gauge No. 1. The ^ inch hole alone. No. 2. The inch hole alone. No. 3. A inch hole alone. No. 4. Two \ inch holes. No. 5. Four \ inch holes. According to experience, the velocity with which water flows from a still vessel through a round hole in a thin horizontal plate is very nearly propor- tional to the area of the hole and the square root of the pressure, so that with any particular hole the relative quantities of water discharged would be read off at the variable height gauge. The accuracy of the gauge, as well as the absolute values of the readings, was checked by comparing the readings on the gauge with the time taken to fill vessels of known capacity. In this way coefficients for each one of the combinations 1, 2, 3, 4, 5 were obtained as follows : 44] THE LAW OF RESISTANCE IN PARALLEL CHANNELS. TABLE II. 83 No. of Gauge Readings on Time Gauge Quantity Coefficient Logarithmic coefficient Seconds c.c. Gauge No. 1 ib. 19-55 61 59 1160 1160 j -966 T-985 No. 2 5-3 54 1160 4-055 608 ib. 15-3 full A 4-055 No. 3 15 360 A 16-220 1-210 No. 4 15 178 A 32-440 1-511 No. 5 15 90 A 64-880 1-812 From this table it will be seen that the absolute values of the coefficients were obtained from experiments on the gauges No. 1 and No. 2, the co- efficients for the gauges 3, 4, and 5 being determined by comparison of the times taken to fill a vessel of unknown capacity, which stands in the Table as A. The relative value of these coefficients came out sensibly proportional to the squares of the diameters of the apertures. For the smaller velocities it was found that the gauge No. 1 was too large, and in order not to delay the experiment in progress, two glass flasks were used : these are distinguished as flasks (1) and (2) ; their capacities, as subsequently determined with care, were 303 and 1160 c.c. The dis- charge as measured by the times taken to fill these flasks are reduced to c.c. per second by dividing the capacities of the flasks by the times. 28. The method of carrying out the experiments was generally as follows : My assistant, Mr Foster, had charge of the supply of water from the main, keeping the water in the pressure gauge at a fixed level. The tap at the end of the tube to be experimented upon being closed, the zero reading of the gauge was carefully marked, and the micrometer adjusted so that the spider line was on the division of water and fluid in the left-hand limb of the gauge. The screw was then turned through one entire revolution, which lowered the spider line one-fiftieth of an inch ; the tap at the end of the pipe was then adjusted until the fluid in the gauge came down to the spider line; having found that it was steady there, the discharge was measured. This having been done, the spider line was lowered by another complete revolution of the screw, the tap again adjusted, and so on, for about 20 midingH, which meant about half an inch difference in the gauge. Then the readings were taken for every five turns of the screw until the limit of the range, about 2 inches, was reached. After this, readings were taken by 62 84 ON THE MOTION OF WATER, AND OF [44 simple observation of the scale attached to the gauge. In taking these readings the best plan was to read the position of the mercury or carbon in both limbs of the gauge, but this was not always done, some of the readings entered in the notes referred to one or other limb of the gauge, care having been taken to indicate which. In the Tables III., IV., and V. of results appended, the noted readings are given and the letters r, I, and b signify whether the reading was on the right or left limb, or the sum of the readings on both limbs. The readings marked I and r are reduced by the correction for the difference in the size of the limbs as well as the coefficient for the particular fluid in the gauge. Thus it was found with the mercury tube that when the left limb had moved through 39 divisions on the scale the right had moved through 41, so that to obtain the sum of these readings, the readings on the left, or those marked I, had to be multiplied by 2'05, and those on the right by 1-95. With the bisulphide of carbon gauge, 11 divisions on the left caused 9 on the right, so that the correction for the reading on the left was T8 and on the right 2*2. 29. Comparison of the pressure gauges. The pressures as marked by the gauges were reduced to the same standard by comparing the gauges ; thus '25 of the left limb of the mercury corresponded with 24 inches on both limbs of the bisulphide. Therefore to reduce the readings of the bisulphide of carbon to the same scale as those of the mercury they were multiplied by 25 x 20-5 24 = 0-0213. This brought the readings of pressure to the same standard, i.e., an inch of mercury, but these were further reduced by the factor 0'00032 to bring them to metres of water. As it was convenient for the sake of comparison to obtain the differences of pressure per unit length of the pipe, the pressures in metres of water have been divided by 1'524, the length in metres between the gauge holes, and these reductions are included in the tables of results in the column headed i. From the discharges, as measured by the various gauges, reduced to cubic centimetres, the mean velocity of the water was found by dividing by the area of the section of the pipe. 44] THE LAW OF RESISTANCE IN PARALLEL CHANNELS. 85 30. Sections and diameters of the pipes. The areas were obtained by carefully measuring the diameters by means of fitting brass plugs into the pipes, and then measuring the plugs. In this way the diameters were found to be Diameter, No. 4 pipe, '242 inch, 6'15 millims. No. 5 pipe, -498 inch, 127 millims. These gave the areas of the sections Section, No. 4 pipe, 297 square millims. No. 5 pipe, 125 square millims. The discharge in cubic centimetres, divided by the area of section in square millimetres, gave the mean velocity in metres per second, as given in the Tables III., IV., and V. The logarithms of i and v are given for the sake of comparison. 31. The temperature. The chief reason why the water from che main had been used, was from the necessity of having constant temperature throughout the experiments, and my previous experience of the great constancy of the temperature of the water in the mains, even over a period of some weeks. At the commencement of the experiments the temperature of the water when flowing freely was found to be 5 C. or 41 F., and it remained the same throughout the experiments. Nevertheless, a fact which had been overlooked caused the temperature in the pipes to vary somewhat and in a manner somewhat difficult to determine. This fact, which was not discovered until after the experiments had been reduced, was that the temperature of the workshop being above that of the main, the water would be warmed in flowing through the pipes to an extent depending on its flow. The possibility of this had not been altogether overlooked, and an early observation was made to see if any such warming occurred, but as it was found to be less than half a degree no further notice was taken until on reducing the results it was found that the velocities obtained with the very smallest discharges presented considerable discre- pancies in various experiments; this suggested the cause. The discrepancies were not serious if explained, so that all that was necessary was to carefully repeat the experiments at the lower velocities observing the temperatures of the effluent water. This was done, and further experiments were made (see Art. 33). iQ O >O t-~ i I CO Oi !> iO t~ ",O -* op ryi co -3* >p cp IN I^H I.IH M lAi I^H >O CO N r- * CO cc co r-i cc cc i i OS CN ^ >O CD !> ICC IN IN IN KM IN IN IN IN IN IN IN lr icooo OC>r7it;qf;p O O O O O O O O O O O - iWQOCOOOOCCONl-i OOOOOO CDCNQOOOCO OOOOOO OOOOOO a II ip ip AH O COCDOJfNOQO r-iNCOOCDl^ POOPPP OOOOOO OOOOO' i i IF i laqiunn aoua i i N CO -O>CQOi 1 T* I"- rH oooooooo oooooooo Tt CO 00 OOOt COiO (M i ( O OO iO (J^ CO CO CD iO O O g oooooop'7* r'r'T'T'T'? 4 - 66666666 66666000 jaqtutm aouaaajag w - cp O5 O ^3

00 OJ ^ o C- 'TJ co CD O *H 'T? g^ CP S4J a> O CD CD !> OO CDCMOOi ii 'iOOCO iO O -* >O 00 i i-fOO OOOSO^CiClOOO l-t loo ioi ice loc loo 100 loo 100 100 100 100 IOO I(M I ' S 2 g I'" s^s ij, B CO >O 00 CO i il^OOO oooooooo OOOOOOOO - 3) -* O S 0) "* HjN O 0) OQ&.S OOi<35O10000000C i IflSt-OJOeOt IQ 1-- 1-- OO 00 Ci O r-i (N OOOOOOOO oooooooo OOOOOOOO Temperature / m^M OPW8HTO3 OOOOOOOOCOO5 0) . tc ^^(M^ t1 1 1 i * ai ^4 * .6 $ 3 H Pressures , rr) ^ In P^ O Q 66666666 O O1 00 00 "* O 1- 00 oooooooo 66666666 ^ o d 53 * - r o a -2 I G >,nmao M a pa G^I co ^f *o co t** oo o^ O fN CO CO CO i ; r- l- 00 I- O OC OO O5 Oi 35 O> O5 CJ O5 O -1 -1 1 <> CO CO > CD r- I :iOO5iOfNO5CDCOOCDCOfOD(MX Tf^iQootNtN'f^ii^ooasoi itMcorf OOOiOi '5 Dianiitcr O'"OI2J. 0'" 1 00100200300400900600700800901 Oil 012 13 14 15 16 17 S/u/wu//Vsiiri- III KVi/ir According to my assumption, the straightness of the curves between the origin and the critical points would depend on the constancy of temperature, 44] THE LAW OF RESISTANCE IN PARALLEL CHANNELS. 91 and it was the small divergences observed that suggested a variation of temperature which had been overlooked. This variation was confirmed by further experiments, amongst which are those contained in Table IV. These showed that the probable variation of the temperature was in Table III. from 12 C. to 9'C. at the critical point, and from 12 C. to 8 C. in Table V., which variations would account for the small deviation from the straight. It only remained, then, to ascertain how far the actual values of v c , the velocity at the critical points, corresponded with the ratio -~ or -~ . For tube 4 from the Table III. D = 0-006 15 metres, v c = (V4426 metres per second at 9 C., at this temperature P = '757 (see p. 73). p Hence putting B c = ^ , VglJ we have B c = 279 7. Again, for tube 5, Table V., at 8 C. = 0127, v c = -2260, P = -7796, whence B c = 272'0. The differences in the values of B c thus obtained, would be accounted for by a variation of a quarter of a degree in temperature, and hence the results are well within the accuracy of the experiments. To each critical velocity, of course, there corresponds a critical value of the pressure. These are determined as follows. The theoretical law of resistance for steady motion may be expressed by A c Pi = B c Pv. And multiplying both sides by ^, This law holds up to the critical velocity, and then the right-hand number is unity, and, if B c has the values just determined : c ~ 92 ON THE MOTION OF WATER, AND OF [44 by Table III. i e = -0516, P 2 =-573, D 3 = -000,000,232, which give A c = 47,750,000. By Table V. i= -00638, P 2 = -607, D* = -00000205, which give A e = 46,460,000, which values of A c differ by less than by what would be caused by half a degree of temperature. The conclusion, therefore, that the critical velocity would vary as j. is abundantly verified. 34. Comparison with the discharges calculated by Poiseuille's formula. Poiseuille experimented on capillary tubes of glass between '02 and '1 millim. in diameter, and it is a matter of no small interest to find that the formula of discharges which he obtained from these experiments is numerically exact for the bright metal tubes 100 times as large. Poiseuille's formula is 777)4 Q = 1836-724 (1 + 0-0336793 T + 0-000220992 T 8 ) ^- , T = temperature in degrees centigrade. H = pressure in millims. mercury. D diameter in millims. L = length in millims. Q = discharge in millims. cubed. Putting i= - P = (1 + 0-336793 T + 0-000220992 T*)~\ 44] THE LAW OF RESISTANCE IN PARALLEL CHANNELS. 93 and changing the units to metres and cubic metres this formula may be written 47 700000 ^t = 278pv, the coefficients corresponding to A c and B e . The agreement of this formula with the experimental results from tubes 4 and 5 is at once evident. The actual and calculated discharges differ by less than 2 per cent., a difference which would be more than accounted for by an error of half a degree in the temperature. 35. Beyond the critical point. The tables show that, beyond the critical point, the relation between i and v differs greatly from that of a constant ratio ; but what the exact relation is, and how far it corresponds in the two tubes, is not to be directly seen from the tables. In the curves (diagram I. page 90) which result from plotting i and v, it appears that after a period of flatness the curves round off into a parabolic form ; but whether they are exact parabolae, or how far the two curves are similar with different parameters, is difficult to ascertain by any actual comparison of the curves themselves, which, if plotted to a scale which will render the small differences of pressure visible, must extend 10 feet at least. 36. The logarithmic method. So far the comparison of the results has been effected by the natural numbers, but a far more general and clearer com- parison is effected by treating the logarithms of I and v. This method of treating such experimental results was introduced in my paper on Thermal Transpiration, page 283, Vol. I. Instead of curves, of which i and v are the absciss* and ordinates, logt and log v are taken for the absciss* and ordinates, and the curve so obtained is the logarithmic homologue of the natural curve. The advantage of the logarithmic homologues is that the shape of the curve is made independent of any constant parameters, such parameters affecting the position of all points on the logarithmic homologue similarly. Any similarities in shape in the natural curves become identities in shape in the logarithmic homologues. How admirably adapted these logarithmic homologues are for the purpose in hand is at once seen from diagram II., which contains the logarithmic homologues of the curves for both pipes 4 and 5. A glance shows the similarity of these curves, and also their general character. But it is by tracing one of the curves, and shifting the paper 94 ON THE MOTION OF WATER, AND OF [44 rectangularly until the traced curve is superimposed on the other, that the exact similarity is brought out. It appears that, without turning the paper at all, the two curves almost absolutely fit. It also appears that the horizontal and vertical components of the shift are Horizontal shift '913 Vertical shift '294 which are, within the accuracy of the work, respectively identical with the J)3 J) differences of the logarithms of -^ and -,j for the two tubes. 37. The general law of resistance in pipes. The agreement of the logarithmic homologues shows that not only at the critical velocities, but for all velocities in these two pipes, pressure which renders (Z) 3 //* 2 ) i the same in both pipes corresponds to velocities which render (D/fi) v the same in both pipes. This may be expressed in several ways. Thus if the tabular value 44] THE LAW OF RESISTANCE IN PARALLEL CHANNELS. 95 of i for each pipe, plotted in a scale, be multiplied by a number propor- tional to D 3 /P- tor that particular pipe, and the values of v by a number proportional to D/P, then the curves which have these reduced values of i and v for abscissae and ordinates will be identical. A still more general expression is that if expresses the relation between i and v for a pipe in which D=l, T= 0, P= 1, P expresses the relation for every pipe and every condition of the water. The determination of the relation between circumstances of motion and the physical condition of the water in such a general form was not contem- plated when the experiments were undertaken, and must be considered as a result of the method of logarithmic homologues, which brought out the relation in such a marked manner that it could not be overlooked. Nor is this all. It had formed no part of my original intention to re-investigate the law of resistance in pipes for velocities above the critical value, as this is ground which had been very much experimented upon, and experiments seemed to show that the law was either indefinite or very complex a conclusion which did not seem inconsistent with the supposition that above this point the resistance depended upon eddies which might be somewhat uncertain in their action. But although it was not my intention to investigate laws, I had made a point of continuing the experiments through a range of pressures and velocities very much greater I think than had ever been attempted in the same pipe. Thus it will be noticed that in the larger tube the pressure in the last experiment is four thousand times as large as in the first. In choosing the great range of pressures I wished to bring out what previous experiments had led me to expect, namely, that in the same tube for sufficiently small pressures the pressure is proportional to the velocity, and for sufficiently great pressures, the pressure was proportional to the square of the velocity. Had this been the case not only would the lowest portion of the logarithmic homologues up to the critical point have come out straight lines inclined at 45 degrees, but the final portion of the curve would have come out a straight line at half this inclination, or with a slope of two horizontal to one vertical. 96 ON THE MOTION OF WATER, AND OF [44 The near approach of the lower portions of the curve to the line at 45 led me, as I have already explained, to discover that the temperatures had risen at the lower velocities, and to make a fresh set of experiments, some of which are given in Table IV., in which, although the temperatures were not constant, they were sufficiently different from the previous ones to show that the discrepancy in the lower portions of the curves might be attributed to variations of temperature, arid the agreement with the line of 45 con- sidered as within the limits of accuracy of experiment. When the logarithms of the upper portions of the curve came to be plotted, the straightness and parallelism of the two lines was very striking. There are a few discrepancies which could not be in any way attributed to temperature, as with so much water moving this was very constant, but on examination it was seen that these discrepancies marked the changes of the discharge gauges. The law of flow through the orifices not having been strictly as the square roots of the heights, the manner in which the gauges had been compared forbade the possibility of there being a general error from this cause ; the middle readings on the gauge were correct, so that the discrepancies, which are small, are mere local errors. This left it clear that whatever might be their inclination the lines expressed the laws of pressures and velocities in both tubes, and since the lines are strictly parallel, this law was independent of the diameter of the tube. This point has been very carefully examined, for it is found that the inclination of these lines differs decidedly from that of 2 to 1, being T723 to 1, and so giving a law of pressures through a range 1 to 50 of i oc v 1 ' 723 . This is different from the law propounded by any of the previous experi- menters, who have adhered to the laws i = v 2 , or i = Av + Bv 2 . That neither of these laws would answer in case of the present experiments was definitely shown, for the first of these would have a logarithmic homo- logue inclined at 2 to 1, and the second would have a curved line. A straight logarithmic homologue inclined at a slope T723 to 1 means no other law than i oc v 1 ' 733 . I have therefore been at some pains to express the law deduced from my experiments on the uniform pipes so that it may be convenient for application. This law as already expressed is simply Dv\ 44] THE LAW OF RESISTANCE IN PARALLEL CHANNELS. 97 where / is such that is the equation to the curve which would result from plotting the resistance and velocities in a pipe of diameter 1 at a temperature zero. The exact form of / is complex, this complexity is however confined to the region immediately after the critical point is passed. Up to the critical point D 3 . Dv c p2 l ~~ c p ' After the critical point is passed the law is complex until a velocity which is l'325v c is reached. Then as shown in the homologues the curve assumes a simple character again, that is, the logarithmic homologue becomes a straight line inclined at 1723 to 1. Referring to the logarithmic homologues (diagram 2, page 94), it will be seen that although the directions of the two straight extremities of the curve do not meet in the critical point, their intersection is at a constant distance from this point, which in the logarithmic curves is, both for ordinates and abscissa?, 0154. These points o are therefore given by log pV^og^T + 0'154 Dv e Dv log -p- = log -p h Therefore putting P 2 P A = ^, B=f- Dh n Dv n log A = log B = log B c + 0-154 and by the values of A c and B c previously ascertained (Art. 33, p. 92), Iog4 = 7-8311, 4 = 67,700,000 log B = 2-598, B = 396-3 For feet log A = 6'28414, A = l ,935,000 log = 1-56603, 5= 36-9. O. R. II. 7 98 ON THE MOTION OF WATER, AND OF [44 We thus have for the equation to the curves corresponding to the upper straight branches , D 3 . And if n have the value 1 or T722 according as either member of this equation is < or > 1 the equation . D 3 . _ /BDv\ n is the equation to a curve which has for its logarithmic homologue the two straight branches intersecting in o, and hence gives the law of pressures and velocities, except those relating to velocities in the neighbourhood" of the critical point, and these are seldom come across in practice. Dv By expressing n as a discontinuous function of B c -p the equation may be made to fit the curve throughout. 38. The effect of temperature. It should be noticed that although the range is comparatively small, still the displacement of the critical point in Tables III. and IV. is distinctly marked. The temperatures were respectively 9 C., 5 C. At 9 log P-^ 01 2093 At 5 log P- 1 = 0-06963 Difference = '05130 This should be the differences in the values of log v c in Tables III. and IV. The actual difference is '062. Also the differences in log i c should be the differences in P 2 or '10260, whereas the actual difference is 121. The errors correspond to a difference of about 1 C., which is a very probable error. It would be desirable to make experiments at higher temperature, but there were great difficulties about this which caused me, at all events for the time, to defer the attempt. 44] THE LAW OF RESISTANCE IN PARALLEL CHANNELS. 99 SECTION IV. Application to DARCY'S experiments. 39. DARCY'S experiments. The law of resistance came out so definitely from my experiments that, although beyond my original intention, I felt constrained to examine such evidence as could be obtained of the actual experimental results obtained by previous experimenters. The lower velocities, up to the critical value, were found, as has already been shown (Art. 35), to agree exactly with Poiseuille's formula. For velocities above the critical values the most important experiments were those of Darcy approved by the Academy of Sciences and published 1845 on which the formula in general use has been founded. Notwith- standing that the formula as propounded by Darcy himself could not by any possibility fit the results which I have obtained, it seemed possible that the experiments on which he had based his law might fit my law. A com- parison was therefore undertaken. This was comparatively easy, as Darcy's experimental results have been published in detail. Altogether he experimented on some 22 pipes, varying in diameter from about the size of my largest, O m- 0014 up to O m '5. They were treated in several sets, according to the material of which they were composed wrought- iron gas-pipes, lead pipes, varnished iron pipes, glass pipes, new cast-iron and old rusty pipes. The method of experimenting did not differ from mine except in scale, the distance between Darcy's gauge points being 50"' instead of 5 feet in my case. The great length between Darcy's gauge points entailed his having joints in his pipes between these points, and the nature of his pipes was such as to preclude the possibility of a very uniform diameter. His experiments appear to have been made with extreme care and very faithfully recorded, but the irregularity in the diameters, which appears to have been as much as 10 per cent., and the further irregularity of the joints, preclude the possibility of the results of his experiments following very closely the law for uniform pipes. Another important matter to which Darcy appears to have paid but little attention was temperature. It is true that in many instances he has given the temperature, but he does not appear to have taken any account of it in his discussion of his results, although it varied as much as 20 C. in the cases where he has given it, and as his pipes, 300 metres long, were in the open air, the effect of the sun on the pipes would have led to still larger differences. 72 100 ON THE MOTION OF WATER, AND OF [44 The effect of these various causes on his results may be seen, as he took the precaution to use two pressure gauges on separate lengths of 50"' of his pipes, and the records from these two gauges by no means always agree, particularly for the lower velocities. In one case the results are as wide apart as 15 to 7, and often 10 or 15 per cent. In arriving at tabular values for i he has taken the mean of the two gauges. Taking these things into account, I could not possibly expect any close agreement with my results ; still, as experiments on pipes of such large diameters are not likely to be repeated, at any rate with anything like the same care and success, they offered the only chance of proving that my law was general. 40. Reduction of the experimental results. Rejecting all the experi- ments on rusty and rough pipes, i.e., selecting the lead, the varnished, the glass, and new cast-iron pipes, which ranged from half-an-inch to twenty inches diameter, I had the logarithmic homologues drawn. These are shown on diagram 3. In the case of two of the smaller pipes the Lines show calculated remits. Dots iliow experimental result 3\ -2 -1 Diameter Temp. Surface A Omm. 014 10 C. Glass) B C D 6 E 12 F 14 G 27 H 41 I 26 J 82 270 lf }Poiseuille L 285 650 ,, M 81 15 5 L ad No. 4 N 137 70 5 No. 5 O 188 00 X } P 500 00 X Q 243 00 X f Darcy R 244 00 12 C. Varnished S 49 60 21 C. J Diameter Temp. Surface K 196mm. 00 x Varnished 00 21 C. 90 15 C. Cast Iron new 00 15 00 00 20 70 68 C. I. incrusted ib. cleaned Glass Darcy smallest velocity is well below the critical point, and in several of the other pipes the smallest velocity is near the critical velocity. This accounts for the lower ends of the logarithmic curves being somewhat twisted ; for the remainder of the logarithmic homologues are nearly straight; some are slightly bent one way and some another, but they are none of them more bent than may be attributed to experimental inaccuracy. 44] THE LAW OF RESISTANCE IN PARALLEL CHANNELS. 101 The inclinations of the upper ends of the lead and bituminous pipes is 1746, slightly greater than mine ; but in the cases of the glass pipes and the cast-iron pipes the slopes are T82 and T92 respectively. So much appeared from the logarithmic homologues themselves, but the most important question was, would the curves agree with the results calculated from the formula 41. Comparison with the law of resistance. In applying this test I was at first somewhat at a loss on account in some cases of the want of any record of the temperature, and the doubt as to such temperatures as had been recorded being the temperature of the water in the pipes between the gauges. The dates at which the experiments were made to a certain extent supplied the deficiency of temperature, the temperatures given fixing the law of temperature, so that the probable temperature could be assumed where it was not given. Assuming the temperature, the values of . _P_ l ~AD 3 ' P_ v ~ BD' were calculated for each tube, using the values of A and B as already determined, \ogi and v are the co-ordinates of the intersection of the two straight branches of the logarithmic curves, so that the application of the formula to the results was simply tested by continuing the straight upper branches of the logarithmic homologues to see whether they passed through the corresponding point 0. The agreement, which is shown in diagram 3, page 100, is remarkable. There are some discrepancies, but nothing which may not be explained by inaccuracies, particularly inaccuracies of temperature. 42. The effect of the temperature above the critical point. It is a fact of striking significance, physical as well as practical, that while the temperature of the fluid has such an effect at the lower velocities that, ccvteris panbus, the discharge will be double at 45 C. what it is at 5 C., so little is the effect at the higher velocities that neither Darcy nor any other experimenter seems to have perceived any effect at all. In my experiments the temperature was constant, 5 C. at the higher velocities, so that I had no cause to raise this point till I came to Darcy 's result, and then, after perplexing myself considerably to make out what the 102 ON THE MOTION OF WATER, AND OF [44 temperatures were, I noticed the effect of the temperature is to shift the curves 2 horizontal to 1 vertical, which corresponds with a slope of 2 to 1, and so nearly corresponds with the direction of the curves at higher velocities that variations of 5 or 10 C. produce no sensible effect; or, in other words, the law of resistance at the higher velocity is sensibly independent of the temperature, i.e., of the viscosity. Thus not only does the critical velocity at which eddies come in, diminish with the viscosity, but the resistance after the eddies are established is nearly, if not quite, independent of the viscosity. 43. The inclinations of the logarithmic curves. Although the general agreement of the logarithmic homologues completely establishes the relations between the diameters of the pipes, the pressures, and velocities, for each of the four classes of pipes tried, viz., the lead, the varnished pipes, the glass pipes, and the cast-iron, there are certain differences in the laws connecting the pressures and velocity in the pipes of different material. In the logarithmic curves this is very clearly shown as a slight but definite differ- ence between the inclination of the logarithmic homologues for the higher velocities. The variety of the pipes tried reduces the possible causes of this difference to a small compass. It cannot be due to any difference in diameters, as at least three pipes of widely different diameters belong to each slope. It is not due to temperature. This reduces the cause for the different values of n to the irregularity in the pipes owing to joints and other causes, and the nature of the surfaces. The effect of the joints on the values of n seems to be proved by the fact that Darcy's three lead pipes gave slightly different values for n, while my two pipes without joints gave exactly the same value, which is slightly less than that obtained from Darcy's experiments. Darcy's pipes were all of them uneven between the gauge points, the glass and the iron varying as much as 20 per cent, in section. The lead were by far the most uniform, so that it is not impossible that the differences in the values of n may be due to this unevenness. But the number of joints and unevenness of the tarred pipes corresponded very nearly with the new cast-iron, and between these there is a very decided difference in the value of n. This must be attributed to the roughness of the cast-iron surface. 44. Description of Diagram 3. Diagram 3. In this diagram the experiments of Poiseuille and Darcy are brought into comparison with those of the present investigation. 44] THE LAW OF RESISTANCE IN PARALLEL CHANNELS. 103 In consequence of the number of lines, the general aspect of _the diagram is somewhat confused, but such confusion vanishes so soon as it is clearly perceived that each line of dots indicates the logarithmic homologue for some particular pipe as determined by experiment, reduced and plotted in exactly the same manner as for diagram 2, page 94 ; DD and EE being exact repetitions of the logarithmic homologue for pipes 4 and 5, on a somewhat smaller scale. It is at once apparent from diagram 3 how, for the most part, the experiments have been well below or well above the critical values. In the small pipes of Poiseuille the velocities were below the critical values, and hence lie in straight lines inclined at 45. The smallest pipe on which Poiseuille experimented had a diameter of 0'014 million. ; only one experiment, marked A, is shown in the diagram, as the remaining three extended outside the range of the plate. They fall exactly on the dotted line through A, and do not reach the critical value. The same is true of all the rest of Poiseuille's experiments, except those made on a much larger pipe, diameter 0*65 millim., hence it is thought sufficient to plot only one, namely EE. CO shows the experimental results obtained with the pipe 0'65 millim. diameter, and these reach the critical value as given by the formula, and then diverge from the line. It is important to notice, however, that the points are not taken directly from Poiseuille's experiments, which have been subjected to a correction rendered necessary by the fact that Poiseuille did not measure the resistance by ascertaining the pressure at two points in the pipe, but by ascertaining the pressure in the vessels from which and into which the water flowed through the pipe, so that his resistance includes, besides the resistance of the pipe, the pressure necessary to impart the initial velocity to the water. This fact, which appears to have been entirely overlooked, had a very important influence on many of Poiseuille's results. Poiseuille endeavoured to ascertain what was the limit to the application of his law, and, with the exception of his smallest tubes, succeeded in attaining velocities at which the results were no longer in accordance with his law. When I first examined his experiments I expected to find these limiting velocities above the critical velocities as given by my formula. In all cases, however, they were very much below, and it was then I came to see that Poiseuille had taken no account of the pressure necessary to start the fluid. It then became interesting to see how far the deviations were to be explained in this way. 104 ON THE MOTION OF WATER, AND OF [44 In pipes of sensible size the pressure necessary to start the fluid lies between v 2 v 2 - and 1-505 1- , 2 # 2# according to whether the mouthpiece is trumpet-shaped or cylindrical. Poiseuille states that he was careful to keep both ends of his pipe cylindrical, hence according to the law mouthpieces of sensible size, the pressures which v 2 he gives should be corrected by T505 -~- . if This correction was made, and it was then found that with all the smaller tubes Poiseuille's law held throughout his experiments, and with the larger pipe it held up to the critical value and then diverged in exact accordance with my formula, as shown by the line CG. Darcy's experiments in the case of three tubes F, G, I, fall below the critical value, and in all these cases agree very well with the theoretical curve as regards both branches. This, however, must be looked upon as accidental, as at the lower velocities Darcy had clearly reached the limit of sensitiveness of his pressure gauges ; thus, for instance, the experiment close by the letter F is the mean of two readings which are respectively 7 and 15; there is a tendency through- out the entire experiments to irregularity in the lower readings, which may be attributed to the same cause, and this seems to explain the somewhat common deviation of the one or two lower experiments from the line given by the middle dots. A somewhat similar cause will explain cases of deviation in the one or two upper experiments, for the discrepancy in the two gauges here again becomes considerable. For these reasons the intermediate experiments were chiefly considered in determining the slopes of the theoretical lines. These slopes were obtained as the mean of each class of tubes : Lead jointed T79 Varnished 1 -82 Glass 1-79 New cast-iron T88 Incrusted pipe 2' Cleaned pipe T91 44] THE LAW OF RESISTANCE IN PARALLEL CHANNELS. 105 and then in the cases in which the temperature was given, /, /, L, M, N, the points having been determined by the formulae : Log i' = 2 log P - 3 log D - 7-851 Log v = log P - log D - 2-598 the lines having the respective slopes were drawn through these points, and in all cases agreed closely with the experiments. In the cases where the temperature was not given, the values of log i and log v were calculated for 5 C., these are shown along the line marked " line of intersections at 5," through these points lines are shown drawn at an inclination of 2 to 1, which are the lines on which would lie whatever might be the temperature. These with the respective slope lines were drawn so as most nearly to agree with the experiments, these intersect the lines at 2 to 1 in the points which indicate the temperatures, and considering the extremely small effect of the temperature these are all very probable temperatures with the exception of G, H, and S, in which cases is above the line for 5 C. This indicates strongly that in these cases there must have been a small error, 2 or 3 per cent., in determining the effective diameter of the pipes. It seemed very probable that roughness in the pipes, such as might arise from incrustation or badly formed joints, would affect the logarithmic homologues, and for this reason only the smoother classes of pipes were treated ; but with a view to test this idea, the homologues Q and R, which related to the same incrusted pipe before and after being cleaned were drawn, and their agreement is such as to show that for such pipe the effect of incrustation is confined to the effect on the diameter of the pipe, and on the value of n which it raises to 2. This, however, was a large pipe, and the velocities a long way above the critical velocity, so that it is quite possible that the same incrustation in a smaller pipe would have produced a some- what different effect. The general result of this diagram is to show that throughout the entire range from pipes of 0-'"000014 to -m 5 in diameter, and from slopes of pressure ranging from 1 to 700,000 there is not a difference of more than 10 per cent, in the experimental and calculated velocities, and, with very few exceptions, the agreement is within 2 or 3 per cent., and it does not appear that there is any systematic deviation whatever. 45. THE TRANSMISSION OF ENERGY. "Cantor Lectures delivered before the Society of Arts in 1883," I. (Delivered April 23, 1883.) SOME few days ago, during a conversation with a friend, I remarked that I was going to give some lectures at the Society of Arts upon the trans- mission of energy, whereupon my friend inquired, "Is that the transmission of energy by electricity?" To this I replied, "No." The fact is that we have heard so much about electricity that I began to think it was time to recall attention to the fact that there are other means of performing mechanical operations. I am not sure whether, during the various lectures which have been given in this room on electricity, the actual term, transmission of energy, has been used. But whether it has or not, some of the leading ideas connected with it have been before you. I think it may be said that the great interest which the public has mani- fested in the recent advance in the arts relating to electricity has arisen, in a large measure, from the cry of joy with which Faure's battery was received. A cry which said, in so many words, " Here we have at last a means of utilising our waterfalls and natural sources of power in a way that may relieve us of all the anxiety about our coal-fields." To those who had studied the subject it was evident at the time that this cry was premature. And to some of us, at all events, it seems to be a mistake to encourage false hopes, or, rather, knowingly to base hopes on a false foundation, to hold out as a means of replacing our coal what was, in all probability, only another 45] THE TRANSMISSION OF ENERGY. 107 means of increasing its rate of consumption, for every step in art which facilitates the application of power must increase the demand on the acting sources. But this is not all ; the exaggerated claim set up for electricity, diverted, for a time at all events, attention from the true claim, which would have been sufficient in itself had it not thus been put out of sight. It is not our object at present to save our coal, but to turn it to the best advantage, to get the greatest result we can, and if Faure's battery or any subsequent advance in this direction conduces to this, it is no small matter. Now, during the last ten or fifteen years an entirely new aspect has been given to mechanics by the general recognition of the physical entity which we call energy, in different forms. We recognise the one thing under different forms in the raised hammer, the bent spring, the compressed air, the moving shot, the charged jar, the hot water in the boiler, and the separate existence of coal, corn, or metals, and oxygen. We see in the revolution of the shafts and the travel of belts in our mills, the passage of water, steam, and air along pipes, the conveyance of coal, corn, and metals, and the electric currents, the transmission of this same thing energy from one place to another; and in all mechanical actions we perceive but the change of form of the same thing. Taking this general or energy point, of view we may get rid of all the complication arising from special purpose, and recognise nothing but the form of energy in its source, the distance it has to be transmitted, and the special form that must be given to it for its application. And this view, although not the best in which to study the special purpose of mechanics or contrivances, is of great importance, inasmuch as it has revealed many general laws, and many fundamental limits to the possibilities of extension in certain directions. My object in these lectures is to direct your attention to some of the leading mechanical facts and limits revealed by this view. There is one general remark I would wish to make, by way of caution. I hope nothing I may say will be interpreted by any of my hearers into a prediction as to what may happen in the future. I have to deal with facts, and I shall try to deal with nothing but facts. Many of these facts, or the conclusions to be immediately drawn from them, may appear to bear on the possibilities or, rather, the impossibilities of art. But in the Society of Arts I need not point out that art knows no limit ; where one way is found to be closed, it is the function of art to find another. Science teaches us the results that will follow from a known condition of things; but there is always the unknown condition, the future effect of which no science can 108 THE TRANSMISSION OF ENERGY. [45 predict. You must have heard of the statement in 1837, that a steam voyage across the Atlantic was a physical impossibility, which was said to have been made by Dr Lardner. What Dr Lardner really stated, according to his own showing, was that such a voyage exceeded the then present limits of steam-power. In this he was within the mark, as anyone would be if he were to say now that conversation between England and America exceeded the limit of the power of the telephone. But to use such an argument against a proposed enterprise, is to ignore the development of art to which such an enterprise may lead. I wish to do nothing of this kind, and if, in following my subject, I have to point out circumstances which limit the possibilities of present art, and even seek to define the limits thus imposed, it is in the hope of concentrating the efforts of art into what may be possible directions, by pointing out the whereabouts of such barriers as science shows to be impassable. Although the terms energy and power are in continual, we might almost say familiar, use, such use is seldom in strict accordance with their scientific meaning. In many ways the conception of energy has been rendered popular, but a clear idea of the relation of energy to power is difficult. This arises from the extreme generality of the terms ; in any particular case the distinction is easy. I was going to say that it is easiest to express this distinction by an analogy, but, as a matter of fact, everything that seems analogous is really an instance of energy. Power may be considered to be directed energy ; and we may liken many forms of energy to an excited mob, while the directed forms are likened to a disciplined army. Energy in the form of heat is in the mob form ; while energy in the form of a bent spring, or a raised weight, matter moving in one direction, or of electricity, is in the army form. In the one case we can bring the whole effect to bear in any direction, while in the other case we can only bring a certain portion to bear, depending on its concentration. Out of energy in the mob form we may extract a certain portion, depending on its intensity and surrounding circumstances, and it is only this portion which is available for mechanical operations. Now energy in what we may call its natural sources has both these forms. All heat is in the mob form, hence all the energy of chemical separation, which can only be developed by combustion, is in the mob form ; and this includes the energy stored in the medium of coal. The combustion of 1 Ib. of coal yields from ten to twelve million foot-pounds of energy in the mob form of heat ; under no circumstances existing at present can all this be directed, nor have we a right, as is often done, to call this the power of coal. What the exact possible power is we do not know, but probably about four-fifths of this, that is to say, from eight to ten million foot-pounds 45] THE TRANSMISSION OF ENERGY. 109 of energy per pound of coal is the extreme limit it can yield under the present conditions of temperature at the earth's surface. But before this energy becomes power, it must be directed. This direction is at present performed by the steam-engine, which is the best instrument art has yet devised, but the efficiency of which is limited by the fact that before the very intense mob energy of the fire is at all directed, it has to be allowed to pass into the less intense mob energy of hot water or steam. The relative intensities of these energies are something like twenty-five to nine. The very first operation of the steam-engine is to diminish the directable portion of the energy of the pound of coal from nine million to three millions. In addition to this there are necessary wastes of directable energy, and a con- siderable expenditure of already directed energy in the necessary mechanical operations. The result is that, as the limit, in the very highest class engines the pound of coal yields about one and a-half millions of foot-pounds; in what are called "first-class engines," such as the compound engines on steamboats, the pound of coal yields one million, and in the majority of engines, about five or six hundred thousand foot-pounds. These quantities have been largely increased during the last few years; as far as science can predict, they are open to a further increase. In the steam-engine art is limited to its three million foot-pounds per pound of coal ; but gas-engines have already made a new departure, and there seems no reason why art should stop short of a large portion of the nine millions. Other important natural sources of mechanical powers are energy in an already directed or army form, wind and water power. Here the power needs no development, but merely transmission and adaptation, and hence it has one important advantage over the energy of chemical separation. These have both been, and are, good servants to man. But there appears to be what are greater drawbacks in the irregularity of these forces as regards time, and the distribution as regards space. The application of the power of the wind to the propulsion of ships has, doubtless, influenced the economy of the world more than any other mechanical feat ; and, not very long ago, water-power played no relatively unimportant part of the work of the world. But it would seem that both these have had their day, and are now relegated to work of a secondary kind. Some further development of art might however bring them to a foremost place again, by developing their use to a hitherto unprecedented extent. Hitherto both wind and water have only had a local application that is to say, they were used where and when they were wanted. Wind was only used in the sailing of ships on voyages, and for mills, distributed so as to be within range of such corn as was too far from water ; while water- power, though very valuable to a certain limited extent, when near habitable country, was otherwise allowed to run to waste ; and these wastes included 110 THE TRANSMISSION OF ENERGY. [45 by far the larger sources of this power the larger rivers and waterfalls, the tidal estuaries, and last, but not least, the waves of the sea, a source which has never been utilised for good. A modern idea is, that it needs nothing but a possible development of art to render these larger sources not only available for power in their immediate neighbourhood, but available to supply power wherever it is wanted, and so displace the coal, or replace the power as coal becomes exhausted. The desirability of such a result fully explains the entertainment of the pleasant idea ; but, unfortunately, when we come to look closer into the question, the probability of its accomplishment diminishes rapidly. Many of the considerations of which I shall have to speak bear directly on this question ; so that I shall now defer its further consideration, merely pointing out that, to accomplish this result, the power must not only be extracted from the water on the spot and at the same time, but it must be transmitted over hundreds or thousands of miles, and must be stored till it is wanted. It may well be thought that energy in a directed form, or in the army form, may be better transmitted than in the undirected or mob form. As a matter of fact, however, energy has never been and never can be transmitted as mechanical power in large quantities, over more than trifling distances, say, as a limit, twenty or thirty miles. I say never can, because such trans- mission depends on the strength of material ; and unless there is some other material on the earth of Avhich we know nothing, we know the limit of this. This is a part of rny subject into which I shall enter more closely in my second and third lectures. In deprecating the idea that wind and water will ever largely supply the place of coal, I do not for a moment wish it to be thought that I take a gloomy view of the mechanical future of the earth. This, I believe, admits of immense development, and will not for long depend, as it does at present, on the adjacency of coal-fields. This will be explained as I proceed. It must not be forgotten that, after all, the most important source of energy is not coal, but corn and vegetable matter. The power developed in the labour of animals exceeds the power derived from all other sources, including coal, in the ratio of, probably, 20 or 30 to 1 ; so that, after all, if we could find the means of employing such power for the purposes for which coal is specially employed such as driving our ships, and working our locomotives an increase of 10 per cent, in the agricultural yield of the earth would supply the place of all the coal burnt in engines. The energy which may be derived from the oxidisation of corn has as yet only been artificially developed in the form of heat, and this may be the only possible way ; but physiology has not yet advanced to the point of explaining the physical process of the development of energy consequent on the oxidisation 45] THE TRANSMISSION OF ENERGY. Ill of the blood ; and it is at all events an open question whether the energy of corn may not be really a form of directed energy, in which case^corn would yield six or eight times as much energy as coal does at present, consumed in our engines. As consumed in animals, it yields a larger proportion of energy two or three times as much, and may be more whereas by burning it in steam-engines, we cannot get half as much. Should we find an artificial means of developing anything like the full directable power of corn a problem which has not yet been attempted coal would no longer be necessary for power. I do not mention this as a prediction, but as showing that there are, besides wind and water, other, and as yet untried, directions from which mechanical energy may be derived in the future. Electricity is not a natural source of energy, for the simple reason that the metals have mostly been burnt or oxidised during the past history of the earth. But still it is important, at this stage of my lecture, to point out that the energy consequent on the separate existence of metals and oxygen can be developed without combustion, in a totally directed form, i.e., totally available for power. There are many peculiarities which distinguish the group of elementary substances we call metals, but there is no more distinctive feature than this. This is not a primary source of power, but. as it at present appears, it promises to become the most important secondary source. We cannot find metals existing in a separate form but by the use of power ; where and when it exists, we can separate them from the salts, and so store the energy in a form completely available for power. The economical questions relating to such storage of energy will be considered in their place later in the course. It is not, however, only as effecting storage of power that electricity demands our attention, it also affords a means of transmitting power, which has long held an important place in art, and to which all eyes have been recently turned in expectation of something new and startling. Before considering the developments of art, and the circumstances on which their further development depends, I shall turn, for a moment, to the processes of nature. The mechanics of the universe, no less than those relating to human art, depend on the transmission of energy. In nature energy is transmitted in all its forms and under all circumstances, both those which we can imitate in art, and those we can not. The most important point with regard to the artificial transmission of energy is the proportion of power spent in effecting the transmission, and the circumstances on which this proportionate loss depends. Is such loss universal ? So far as we know, it is attendant in a greater or less degree on all artificial means of transmission, and on all transmissions effected by 112 THE TRANSMISSION OF ENERGY. [45 nature on the surface of the earth. If it were not, this earth would be no place to live upon. No motion would ever cease. As it is, the winds and waters are rapidly brought to rest by the friction which they encounter. Currents of wind and currents of water form the principal means by which energy is transmitted over the surface of the earth. But there are other means which experience less resistance. Oscillatory waves, those of sound, are a very efficient means of transmitting energy. Sounds are not trans- mitted to an unlimited distance, chiefly because by the spreading of the wave the sound becomes weaker and weaker as it proceeds. It is also destroyed by the friction of the solid surface of the earth. Hence the sounds which reach us from bodies high up, as the explosion of a meteor, are heard much further than such sounds made at the surface of the earth, although there are two records of artillery having been heard two hundred miles. Owing to such incidental destruction of sound we cannot say from experience that sound waves in air are destroyed, but from the physical properties of gases we know they are. Waves on the sea are another very efficient means of transmitting power, a means which may be called nature's mill. The waves which take up the energy or power from the wind in mid ocean travel onwards, carrying this energy, and experience such slight resistance that they will, after travelling hundreds or thousands of miles, destroy the shores on which they expend the last of their energy. If we could find a means of utilising the energy of waves, we should not only save our coal, but also save our country from the waves ; still, water waves experience resistance which we can better estimate theoretically than practically. These are the principal ways in which energy is transmitted from one part of the earth to another. There are others, such as earthquakes, but they all show the same thing, that power is spent in the transmission of energy. If we look away into interstellar space, the case is altered. Here we see two ways in which energy is transmitted heat, or light, and the motion of the heavenly bodies. In neither of these can we see any direct evidence of resistance or loss of power ; and, as judged by any terrestrial measure, there certainly is none. The distance at which we see stars is a sufficient proof of the freedom with which a wave of light travels ; while the regularity of the motion of the planetary bodies shoAvs that they encounter no sensible resistance. Yet, although not directly perceivable, there are circumstances that strongly suggest that in both these forms, transmission of energy is resisted. If space is unlimited, and there are stars throughout it, why do not we see them at greater distances than we do ? Under these circum- stances there could be no spot in the heavens at which at a sufficiently 45] THE TRANSMISSION OF ENERGY. 113 great distance there was not a star, so that, if the light were not stopped, the whole heavens would be one fiery envelope as bright "as "the sun. This is a question which philosophers have not decided. But one, and the favourite, way out of the difficulty, is to suppose that the light does en- counter resistance, even in interstellar space. This is a subject on which your Chairman of Council has boldly launched ; and whether his hypothesis be right or wrong, it has brought to the front a very interesting subject. With regard to the resistance encountered by the planetary bodies, our evidence is even slighter. A few domesticated comets seem to diminish their speed ; and it is not so long since we were all on the qui vive, by the promise of the spectacle of an old friend, who seemed to have come earlier than he was expected, on purpose to verify a pre- diction of plunging into the sun, but instead of doing so he passed away and was pronounced a stranger, to the joy of the nervous, but some- what to the discomfiture of astronomers. The energy which we derive from the sun comes to us in the form of sunshine, in a highly directed but extremely scattered form, being uniformly distributed all over the illuminated disc of the earth. It reaches the outer atmosphere nearly in the same condition as it left the sun, having traversed ninety odd millions of miles without any sensible ex- penditure of power. In the twenty or thirty miles of the lower atmo- sphere, however, it encounters very great, but variable, resistance. Sometimes half of it, or three-quarters of it, may reach the earth's surface. This is rare in our country, and on the average not more than a very small fraction ever reaches the surface. When the sun does shine, the sunshine is a form of energy which may be, and is, very largely directed so as to yield power. Any such direction which may be accomplished by human art is undertaken at an enormous disadvantage, on account of the scattered manner in which the energy reaches us. The sunshine must be collected before we can make any mechanical use of it. In the abstract, there are two methods. The one would be to accu- mulate the energy of sunshine on a given place, over a long time. This is nature's method. The energy on each portion of the earth's surface, during days, weeks, the whole year, or many years, is accumulated by the growth of vegetables. Corresponding to this, however, art has as yet developed no means whatever. If we don't use the sunshine as it falls, energy is lost for all mechanical purposes. I say if we don't, not that we do use it, but because we can, and have done so in a small way. By means of a lens, or reflectors, the sunshine which falls on a certain o. R. ii. 8 114 THE TRANSMISSION OF ENERGY. [45 place may be concentrated on to a smaller space, and so be sufficient to perform some mechanical operation. In this way small vapour engines have been worked by sunshine. But the cost of the apparatus necessary for such concentration is out of all proportion to the result accomplished, and shows the art difficulties must be got over by a new departure. There is the further consideration that sunshine on land is too valuable for the maintenance of vital energy to allow of its being devoted to mechanical purposes. As regards the perfectness of nature's method, so far as I know, no attempts have even been made to test this. It is probably very wasteful, as are all nature's methods, but it is effective. In the first instance, the energy of sunshine is stored on the spot where it falls, in the tissues, but chiefly in the sap of the grass and vegetation. If this is not re- moved, a large portion of the energy of the year's growth, that which is in the sap, is stored in the seed, and the rest, although apparently again scattered on the decay of the tissues, is to some extent preserved in the ground, and either forwards the next year's crop, or takes the permanent form of peat; and our coal-fields are but evidence of the way in which the directable energy of sunshine has been stored under cir- cumstances where there was no immediate purpose for which to apply it. Under present circumstances, however, this energy is almost everywhere too valuable to admit of secular storage. It is either removed directly by nature's method, the teeth of animals, or allowed to accumulate for a longer period, and then removed by human industry. The further aggregation of this energy involves the transmission of energy in a mechanical sense, and hence involves the expenditure of power. Nature works by means of directly converting this energy into power. The plant accumulates the energy of sunshine, the animal collects and appropriates this energy. This collection is accomplished by the ex- penditure of power, which means a redistribution of that portion of the energy which is capable of direction. The scheme of nature, therefore, is a cycle. The vegetation accumulates the energy, as far as time is con- cerned, leaving it in a scattered form, requiring power to collect it ; this power is in the grass, and only wants direction; this it receives in the animal, which again expends some of the energy in the operation of collecting. If vegetable energy be supplied to the animal in a collected form, then a large portion of the directed energy is available for mechanical purposes. And in this way we may form a rough estimate of the directed energy to be obtained from sunshine in this country. The common agri- cultural rule is one horse or bullock to two acres, such a horse pulling 120 Ibs. at a rate of 3'6 feet per second for eight hours a day. That is a nominal horse. 45] THE TRANSMISSION OF ENEHGY. 115 We thus get something like 3,000,000,000 over and above the energy necessary for the energy spent in eating the corn and moving itself, which we must put down as at least equal in amount. Taking only the available portion, we have the equivalent per acre of nearly three tons of coal burnt in such steam-engines as exist at present. Now the average weight of the vegetable produce from one acre, taking the form of straw and corn, would be about two tons. So that, as far as mechanical power is concerned, coal burnt in our present steam-engines, and corn and straw eaten by horses, yield about the same energy, weight for weight. The energy which we derive from sunshine is scattered all over the earth, and if it is to be utilised at any spot other than that at which the sunshine falls, it must be transmitted by the expenditure of power. The energy required for immediate operations of agriculture absorbs a large proportion of the actual energy grown. The surplus is available for purposes of art, and we may say that the primary object of man has been to render this surplus as large as possible. This is accomplished, in the first instance, by applying the residue of energy to so ameliorate the conditions of agriculture as to increase the yield and diminish the labour. In this way the land is levelled, enclosed, and drained ; buildings are erected, and finally, but most important of all, roads are made. The effect of roads in increasing the surplus energy is probably greater than any other human accomplishment. The only means of transmitting for purposes of collection or other purpose aggregate energy in the shape of corn, without roads, is on the backs of animals. In this way two or three hundred miles was the absolute limit to the distance an animal could proceed, carrying its own food. On a good road a horse will draw a ton of food at twenty miles a day, which would mean that it would proceed 800 miles before it had exhausted its supply, or whatever surplus energy there might be available on one spot, half this would be available at 400 miles distance. The labour of maintaining the roads should, of course, be de- ducted, but this is very small. The labour of constructing canals is very great, but the result is equal ; a horse can move 800 tons twenty miles a day ; or a horse could draw his own food for 80,000 miles on a canal. That is to say, with a canal properly formed, a horse could go five times round the world without consuming more energy than was in the boat behind it. Or corn could be sent round the world with a consumption of one-fifth. On railways, at low speeds, the force required is about ten times greater than on a canal, so that the expenditure in going round the world would be about equal to the total energy drawn. If for a moment we replace the horse by 82 116 THE TRANSMISSION OF ENERGY. [45 the steam-engine, and the corn by coal, we have to add the weight of the engine to the coal, and diminish the efficiency by one-third ; we so get that the consumption of coal for the same load of coal as of corn, would be about double, or an engine would go about one-fourth round the world, consuming in coal the net weight in the train, that is exclusive of carriages and engine. Or for every thousand miles corn is carried by rail, some- thing like 10 per cent, of the energy of the corn is expended in draft. This is exclusive of the expenditure in wear and repairs, which will be certainly equal, if not greater. Taking, then, the mean distance by rail between London and the West of America, as 2,000 miles, the present expenditure in the energy of corn in transit is somewhere about 20 per cent. The expenditure of energy on the ocean varies, but if transported by steam it would be probably 10 per cent, more, so that at the present time we are actually receiving available mechanical energy, transported in the form of corn, over 2,000 miles of land and 3,000 miles of sea, entirely by artificially directed power, with an expenditure of less than 20 per cent. ; a proportion which 200 years ago would have had to have been spent in transmitting it, fifty miles over land ; a result which has been accomplished by the employment in the meantime of the residual energy over and above that necessary for agriculture, together with a further supply drawn from our coal beds. Turning now our consideration to coal, we find that per weight as used at present, this yields rather less power than corn, but not less than two- thirds, and it then appears that coal may be transmitted at the present time, between any two places on the earth which are connected by rail and water, with an expenditure of less than 50 per cent. In instituting this comparison, the standard has been the actual available power, as developed in our present engines and in horses, with which, weight for weight, there is not much difference. But the adaptability of this energy, so developed for particular purposes, renders the one medium much more valuable than the other. Thus for agricultural purposes, weight for weight, horse food is worth in money ten times as much as coal. This shows the extreme difference in the value of energy according to its adaptability ; and the extension, for which there is unlimited scope, of the adaptability of steam power, may render it ten times as valuable as at present ; nor would this be any small proportion compared with the total energy employed in the work of the world. In this country there are said to be between two and three million horses, and we may put the labouring men down at five millions, or the total power derived from corn as over three million horses. From the best information going, the work done by steam in this country does not exceed the labour of two million horses, so that more than half the energy is still derived from corn. A greater proportion of the actual corn used for 45] THE TRANSMISSION OF ENERGY. 117 horse food comes across the Atlantic ; and for many years maize was sold in this country at an average price of 6 or 7 a ton, the cost of transit being a very small matter. Of course the same cost, say 1 per ton, applied to coal would be a serious matter, considering the low price of the latter. But if, in the present state of our art, energy can be transmitted by corn from any part of the world to this country with an insensible rise, there is no reason to suppose but that, with the advance which science shows us, there is every reason to expect coal may be transmitted with a corresponding small increase in its cost, wherever the demand for it is sufficient to recom- pense the outlay necessary for opening the roads or canals. II. (Delivered April 30, 1883.) In my last lecture I dealt with the transmission of energy through the means of coal and corn, showing that by either of these means power may be transmitted by rail ; with an expenditure of 1/1 2,000th per mile, or by water of l/120,OOUth per mile, this either through the agency of horse or steam. This ease of transmission, however, depends entirely on the railroad or water, and is only possible between places so connected. Hence such means are only applicable to what may be called the mains of power. We come to-day to consider other means of transmitting energy in smaller quantities applicable to its distribution for immediate application. Such transmission is not a matter of secondary importance, although the distances over which it is transmitted may be comparatively insignificant. To emphasise this, I may recall what was previously mentioned, namely, that the relative price of corn and coal shows that the power given out by horses is at least ten times as valuable as that of steam, for more than half the purposes for which energy is used ; or that it answers better to burn our coal in bringing corn from America to plough in England, than to use the coal here for ploughing. In fact, for most of the detailed purposes for which power is used, to draw it from a large source (such as a steam-engine), distribute it and adapt it to its purpose, is ten or twenty times more costly than its trans- portation in large quantities over thousands of miles. Now the means of artificially transmitting power may be considered as three. The power may be stored in matter in various ways, and the matter 118 THE TRANSMISSION OF ENERGY. [45 with the energy transported as, for instance, in our watch-springs. The second means is the transmission of power by moving matter, without actually storing the power in the matter as in shafts and belts, hydraulic connection, &c. And the third method, which is distinct from the others, is the transmission of energy, in the form of heat or electricity, by the now of currents through conductors; in this way all the power in the steam passes through the boiler-plates from the furnace into the boiler. Of course, each one of these means includes an infinite variety of detailed contrivances, more or less dissimilar. But there is good reason for classing them under these three heads, for all the contrivances under each of these heads are subject to the same general limits, whether those of efficiency or distance. There is one thing in common to all these means of transmission, and that is that they all involve a material medium. The quantity of matter required constitutes a primary consideration in all of them. This quantity of matter is fixed by what we may call the properties of matter, one of the most important of which, as regards the first two means, is the possible strength of material. Looking round, we see the effect of the limited strength of material in all nature's works. Of course it may be that we shall be able to work with stronger materials than we have at present. Organic materials, such as the feathers and tissues of animals, are stronger than steel, weight for weight, so that there is a possibility of improvement, but that man will go beyond nature in constructing organic fibre seems improbable, and such possibility of improvement as exists may be discounted. At present we may set down our strongest working material as steel, the art of working in which is so perfect, that we may calculate on nearly the greatest strength for all purposes. I have taken fifteen tons on the square inch as the limit of safe working tension, in making the estimates which I shall now bring before you. First of all, I will ask your attention to the possibilities of transporting power in a stored form. The question of economy in the conveyance of energy in a stored form is simply one of the intensity with which it can be stored. If we want to carry energy about, we must have it stored in some material form and this material has to be carried by ordinary means so that the question of economy is simply the amount of available energy that we can store in a given amount of material. If energy, stored in a particular manner, is more readily available for some special purpose than that stored in another, then it may, on the whole, be more economical to carry it in that form. This is abundantly illustrated in our watch-springs. The greatest amount of energy that can be stored in a given weight of steel is very small, compared with other means. To take a familiar unit, to 45] THE TRANSMISSION OF ENERGY. 119 store the energy necessary to maintain one horse-power for one hour would require no less than fifty tons of steel that is to say, fifty tons of steel in the form of watch-springs, all fresh wound-up, would not supply one horse- power for one hour ; and yet this is the commonest form in which energy is carried about. It is the adaptability of the spring, and the readiness with which energy can be put in and taken out, which recommend the steel spring. India-rubber will store much more energy than the same weight of any other material, say, eight or ten times as much as steel ; but of this, several tons would be required to store the horse-power for one hour. A much more capacious reservoir, according to its weight, is compressed air. There are certain difficulties in getting the energy in and out without loss ; but with air, compressed to four times the pressure of the atmosphere, we should only require about 20 Ibs. of air to yield the amount of one horse-power for one hour. Of course, if we were going to carry this air about, to the weight of the air would have to be added the weight of a case to contain it, and such a case, in the form of steel tubes, would weigh something like 230 Ibs.; so that, in any form in which we can carry compressed air about, we shall have about 300 Ibs. to carry for each horse-power per hour. Another means of storing energy, very largely used, is hot water. This is largely used in a way not always recognised. The boiler serves another purpose besides that of converting the energy of the furnace into the power of the steam. It stores the power, and equalises the stream between the fire and the engine, a function the importance of which has been brought to the front in the recent efforts to apply electricity for communication of power, where the want of a similar reservoir between the generator and the motor has, in many cases, proved fatal to the enterprise, a want which secondary batteries are now being used to meet. Hot water has also been employed as an independent reservoir, and as such it is better in some respects than compressed air. The fundamental limits are of much the same kind. In this case, however, the absolute limit is temperature. The vessel in which the water is carried must be strong enough to withstand the pressure, and all materials lose their strength as they get hot. The considerations are here much the same as in the steam-engine, and 400 Fah. appears to be about the limit. At this temperature, for every 4 Ibs. of water the cases would weigh 1 lb., and there would be no advantage of large over small cases ; except as a matter of construction, the proportionate weight would be the same. The gross power of a pound of water, the steam being used without condensation, is about 20,000 foot-pounds, or we should require 50 Ibs. to store 1,000,000; this is the extreme limit again. The present accomplishment would be about 150 Ibs. per 1,000,000 foot-pounds stored 120 THE TRANSMISSION OF ENERGY. [45 rather less than compressed air. The only other means of packing power, that is at present looked to, is that of the much talked about secondary battery. Here there is a great deal of doubt as to what is actually ac- complished ; take the most reliable statements, from which it seems that in order to get 1,000,000 foot-pounds, something like 100 Ibs. of battery is required, which will make this means of storing energy very much the same as compressed air or hot water. It is important to notice that the initial cost of the energy stored by these means differs considerably. This cost is rather difficult to estimate ; but a practical estimate may be formed in this way : Taking the power, as delivered by the steam-engine, as 1 , how much of this power will be given out after secondary storage ? Here the hot water has an advantage, for it is heated directly by the coal, and is all on its way to the steam-engine. With compressed air, there are three operations, each as costly as the steam-engine, and at least half the initial power is spent during the com- pression, storage, and expansion ; so that the energy is at least double as costly in coal, and six times as costly in machinery. I have put it down as three times as costly as the energy in hot water, but this is considerably below the mark. The electricity has also to go through three operations, and cannot be less costly than compressed air. Now, if we revert for one moment to the consideration of the main transmission of power, we see at what an immense disadvantage any form of packed energy is, compared with coal or corn ; as at present packed it weighs at least 100 times as much. While the limits imposed by the strength of material render it certain, as far as compressed air and hot water are concerned, that the weight can never be reduced by more than half, these limits are sufficient to show that packed energy cannot be transported over long distances, even if it can be obtained directly from such falls as Niagara. But this is no argument against the importance of these means for short distances and special purposes. As I have already pointed out, our watches show that circum- stances may render the very heaviest means the best for particular purposes. And if in any of its forms packed energy were directly available for house- hold purposes, though it cost ten or twenty times as much as power direct from the steam-engine, its use would still be assured. One fact should be noticed, that in all these forms the power is packed, and needs nothing but drawing off, whereas corn or coal do not contain the power. The oxygen is an equally essential ingredient. In this fact lies the 45] THE TRANSMISSION OF ENERGY. 121 great advantage of corn and coal for transportation. They are_really, so to speak, but cheques for power, which can be cashed at any spot where a bank, in the form of a steam-engine or a horse, exists. But, of course, not being energy, they are not generally current in fact they are worthless, except where the bank exists, and even there when they represent such small amounts that the banks refuse them. Now these forms of packed power are, so to speak, generally current; that is to say, they are available under almost all circumstances, and in greater or less degrees of smallness; from the very smallest, which is the watch-spring in our pockets, which supplies a con- tinuous stream of power in less than one ten thousand millionth of a horse- power ; or the Whitehead torpedo, which carries some million foot-pounds of energy under the sea. Perhaps the most pressing purpose for which these forms of packed energy are wanting is that of locomotion. The distance which a locomotive body, be it animal or machine, can travel, loaded or free, is limited by the ratio of the power which it carries to its gross weight. The speed which it can attain is limited by the rate at which it can use its energy compared with its weight. Hence there are two particulars in which we can compare the different forms of stored energy for locomotive purposes. Let us take the horse and the locomotive. A full-sized horse weighs, say, 1,500 Ibs., and, at a rate of 2 miles an hour, will go five hours without food, doing about 10,000,000 foot-pounds of work, including the work neces- sary to move itself; this represents the largest result, or about 150 Ibs. per 1,000,000 foot-pounds. If the horse is put to ten miles an hour, it will not do more than 1'5 million foot-pounds in a single journey, besides moving itself. Probably the greatest rate at which a horse can use its energy is about 4,000,000 foot-pounds per hour, or 750 Ibs. per horse-power. A locomotive with its tender, say, weighing sixty tons, exerts 500 horse- power gross 270 Ibs. per horse-power; so that a first-class locomotive with tender is about one-fifth as heavy for its power as the horse ; but then the horse cannot go more than ten miles an hour. Now, in a general way, passenger locomotives carry coal and water for eighty or one hundred miles, i.e. two hours; or the locomotive already mentioned expends at one run about 2,000,000,000 foot-pounds; which means that the gross weight of the locomotive is about 60 Ibs. or 70 Ibs. per 1,000,000 foot-pounds of power with which the locomotive starts. In thus taking the gross weight of the horse or locomotive, we must remember that this includes the weight of carriage and machinery, and that in whatever form the energy is carried, this weight must be added. In the locomotive the weight of water and coal in the tender for two hours' journey 122 THE TRANSMISSION OF ENERGY. [45 weighs about one-quarter the gross load ; and if we add the weight of the boiler, we may consider the carriage and machinery at one-half to one-third the gross load. Taking the latter, and substituting for the boiler, coal, and water, energy in either of the above forms, the coal, water, and boiler would be about 40 Ibs. per 1,000,000 : so that, if we took compressed air instead, we should have one-fourth the power ; or the engine would run for thirty minutes instead of two hours, a distance of twenty-five miles instead of a hundred. A fireless locomotive might do more than this, say, thirty-five minutes, or thirty miles, at the same speed as the locomotive. Faure's battery, if it could be made to work at all, would carry the locomotive forty- eight minutes, or thirty-five to forty miles. These figures seem to show that the locomotive has little to fear from any of these rivals, that is, under circumstances where the smoke and steam are no harm, and where a full-sized locomotive is required. But there are already some cases where the locomotive is required and where the burning of coal is impossible. Should the Channel Tunnel be made, there will be a great field for some form of packed energy. As regards horses, however, there is nothing to show why the horse should not be rivalled by some one of the forms of packed energy. There have been inventors who have constructed carriages to go by clockwork. This has now become possible, substituting hot water, compressed air, or a battery for the spring, and such means have already rivalled the horse on tramways. The fact that horses are at all used for tramcars is a matter of as much surprise as that steam should be used on underground railways. For locomotives driven by compressed air might certainly be made cheaper and better in every way. At the present time it would probably answer well, from a pecuniary point of view, to supply in compressed air energy at the rate of 2d. or 3d. per million foot-pounds, provided a sufficient quantity could be required ; so that if simple and efficient means of applying such energy to perform the heavier part of manual labour could be found, we might get as much power for 6d. as a man will do in a day at 2s. But it is the means of applying it that is wanting. Even for horse work except where there is a railway or tramway the mechanical means are wanting. We have no mechanical substitute for the horse's foot. So that there are more than a million horses in this country continually engaged in the operations of husbandry, where they work in groups so as to get three or four horse-power at one operation, an amount of power not too small for the direct application of steam power ; and although for twenty-five years steam-engine makers have been doing their very best to adapt the power of the steam-engine to this labour, which exceeds any other actual application of power, the possibility of steam 45] THE TRANSMISSION OF ENERGY. 123 ploughing with economy is still a question. The use of steam on paved or on macadam roads is much the same, so that, until steam has been applied to such purposes, there is little hope for other forms of stored energy. Coming back for a moment to Faure's battery, I would carefully point out that the result which I have put down 100 Ibs. per 1,000,000 foot-pounds of energy refers to what has been already accomplished, and not to any possible limit. The principles involved in the chemical action of these batteries, in fact in all batteries, are well understood ; and so far as these principles are involved, it is easy to define limits ; but there are a number of secondary actions which are not so well understood, and which have hitherto prevented any approach to the theoretical limits. In the Faure's battery, the theoretical limits are about 3 Ibs. per 1,000,000 foot-pounds. That is to say, the oxidisation of 1 Ib. of lead to litharge, and the deoxidisation of 1 Ib. of peroxide, together, yield 360,000 foot-pounds. How far, at present, Faure's battery is within this limit, at once appears something like twenty- four times. Should this be accomplished, power could be packed at the rate of 1,000,000 foot-pounds for 3 Ibs., or say 6 Ibs. weight, to allow for wastes, a result which would most certainly displace steam in the locomotive, but which would still leave coal and corn six times the lightest vehicle of power. It should be noticed, however, that although the means of doing so are still entirely wanting, could other metals, such as iron or zinc, be used instead of lead, the results would be much greater. This is shown by the relative amount of power necessary to oxidise or deoxidise these materials, which we see for iron and zinc are five or six times greater than for lead ; here is an apparent opportunity for art. Should this be realised, then, indeed, coal might be displaced as the cheapest medium for the transmission of power, but that would be a small matter compared with the change that would occur in our ways of applying power. For the dream of Jules Verne, of 20,000 miles under the sea, would become a reality, and, instead of steamboats, we should travel in submarine monsters as yet unnamed, which we may call steam-fish. But if science as yet imposes no limits beyond those I have mentioned, at the same time it holds out no prospect. The chemistry of these batteries has been very deeply considered, and those who have studied the subject most deeply apparently see no direction in which to direct their efforts ; so that any great advance in this art must entail a great discovery in science. There now only remains for me to consider the transmission of power as power, or by electricity, a most important branch of my subject, which I must take in my next lecture. 124 THE TRANSMISSION OF ENERGY. [45 III. (Delivered May 7, 1883.) So far I have spoken only of the conveyance of power by means of coal, corn, or in one or other of the several forms of packed energy. To-night I come to consider the transmission of power by what are more distinctly mechanical methods, and by currents along pipes and conductors. These are the means by which power is almost always distributed, i.e., trans- mitted from the acting agent, be it horse, water-wheel, or steam-engine, to its operation, whatever it may be. In most cases the distance of such transmission is so short as to be the subject of small consideration in de- termining the means to be employed. That is to say, the means are chosen rather by their adaptability to receive and render up the power than by the efficiency with which they transmit it. Thus, if we take an ordinary mill, the shaft which receives the power from the engine is generally driven at that speed which is best adapted to receive the power from the engine, and deliver it to the machinery in the mill, without considering whether a much smaller shaft might be used if it were caused to run at a much higher speed. Thus, in a mill driven by an engine of two or three hundred horse-power, the shaft which receives the power will generally be five or six inches in diameter, whereas it would be possible to use a shaft of two inches diameter if the efficiency of the shaft were the only con- sideration. Or, again, take a screw steamboat. The distance from the engines to the screw may be 250 feet, the power 10,000 horse. This could be transmitted by a shaft twelve inches in diameter, if allowed sufficient speed, but the screw has to make sixty revolutions per minute, and this determines the speed at which the shaft is made to run, and hence the shaft is made thirty inches instead of twelve inches. This is because, owing to the smallness of the distance, the efficiency of the means of transmitting the power is a small consideration. There are, however, many circumstances under which it is impossible to bring the source of power close to its work, and then either mechanical power is not used, or the efficiency of the means becomes a consideration. In other cases it is a question whether it is better to distribute the sources of power, such as steam-engines, so that they may be near their work, or to use one large source, and distribute the power by some me- chanical means. This rivalry exists in almost all engineering work which covers a large area, and, generally, a compromise is come to, engines being distributed about the works, and the power of these distributed to the 45] THE TRANSMISSION OF ENERGY. 125 machines by means of shafting. In many cases separate engines are used for each machine, but not often separate boilers, the power being distributed by steam-pipes. Dockyards have long afforded a field for the competition of the various means of distributing power. Here, generally, the distances between the operating machines, such as cranes and capstans, is considerable, and the work required from each machine very casual. And every means of dis- tribution is or has been in use, from a separate engine and boiler to each machine as at Glasgow, separate engines drawing their steam from central boilers, to a complete system of hydraulic transmission from a central pumping station, as at Grimsby or Birkenhead. But the question between centralisation or distribution of steam-engines is not by any means the only one, or most important one, which depends on mechanical means of distributing power. Every improvement in the means of distributing power from a central engine opens a fresh field for its use. The considerations relating to this subject are numerous. Hitherto, as regards the main transmission of power, the principal consideration has been the percentage of loss according to the distance ; but, as regards the final distribution of power, the form in which it is distributed must be such as admits of its being at once available for its purpose. Thus hydraulic distribution is favoured in dockyards, because it is required for heavy forces and slow motions, but where rapid motion is required, hydraulic distribution gives place to some other. Again, where the quantity of power that has to be distributed is a most important consideration, the distribution by means of water or compressed air will generally be the most efficient, whereas these would be by far the most costly means for small quantities. It thus has to be remembered that, besides the general question of efficiency, each means has particular recommendations for particular purposes. It is not, however, with these particular recommendations that I am concerned. My object is to show the limits within which the use of each means is confined, however fit it may be for its purpose. Taking first the mechanical means, which are shafts and ropes, we find that the possible limits to both these means are absolutely defined by the strength of material. The amount of power any piece of material will transmit by motion against resistance, is simply the mean product of the stress or force acting in the direction of motion on the section multiplied by the velocity, so that, if the stress is uniform over the section, the work is the product of the area and intensity of stress and the velocity. 126 THE TRANSMISSION OF ENERGY. [45 In a revolving shaft, neither the stress nor the velocity is uniform over the section, both varying uniformly from nothing in the middle to their greatest value on the outside ; so that their mean product is exactly half the product of the greatest values. The greatest power per square unit of section a shaft can transmit is half the product of the greatest stress into the velocity at the outside of the shaft. Taking, then, the greatest safe working stress for steel at 15,000 Ibs. on the square inch ; taking what is the greatest practical velocity at the surface, 10 feet per second (the speed of railway journals); the work trans- mitted is 75,000 foot-pounds per second per square inch of section 135 horse-power ; so that we should have to have a shaft of upwards of 7 square inches in section to transmit 1,000 horse-power, that is, a shaft of over 3 inch diameter. The friction between such a shaft and lubricated bearings is well known, '04 ; so that, calculating the weight of the shaft 24 Ibs. per foot, we have power spent in friction about 52,000 foot-pounds per mile, that is one-tenth the total power the shaft will transmit. That is, if we put 1,000 horse-power into a 3-inch shaft, making 500 revolutions per minute, we ought, at the end of a mile, to be able to take 900 horse- power out of it. If we had to go farther, the size of the shaft might be diminished, so that in the next mile we should again lose a tenth, and if we repeat this process seven times, we shall, at the end of seven miles, have left about half the original power put in. It will be thought, perhaps, that a 3-inch shaft is very small to transmit so large a force ; this is because the speed of 500 revolutions per minute is inconveniently high for purposes of employing the power ; but if it were merely a question of transmission, it would be about the best speed. This, then, shows the limit of the capacity of shafts as transmitters of work. Turning now to steel ropes, these have a great advantage over shafts, for the stress on the section will be uniform, the velocity will be uniform, and may be at least ten to fifteen times as great as with shafts say 100 feet per second ; the rope is carried on friction pulleys, which may be at distances of five or six hundred feet, so that the coefficient of friction will not be more than '015, instead of '04. Taking all this into account, and turning to actual results, the work transmitted per inch would be 1,500,000 foot-pounds per second ; or that a |-inch rope is all that is necessary to transmit 1,000 horse-power in one direction, this would make the loss per mile only l-60th. But in practice, rope has to be worked backwards and forwards, and the tension in the backward portion of the rope must be half the tension in the forward portion. This reduces the performance from l-60th to l-20th, which would cause half the work to 45] THE TRANSMISSION OF ENERGY. 127 be lost in ten miles. If we use a bigger rope, and run at lower speed, then the coefficient of friction would be reduced to '01, and the distance extended to fifteen miles. Experience with ropes is large, and they have been found, without question, to have been the most efficient mechanical means of transmitting power to long distances, but their use is subject to drawbacks. The ropes wear somewhat rapidly, as do also the pulleys on which they run, and this circumstance is very much against their use in any permanent work. Nevertheless, they are used for working mines, steep inclines, and steam- ploughs ; while at Schaff hausen they have been used for transmitting power to considerable distances. Turning to the transmission of power along pipes, we find the conditions somewhat modified. The formula for the amount of power transmitted by water is the same, namely, the product of the pressure and area of section into the velocity. But the resistance obeys different laws. In the case of shafts and ropes, we have seen that the distance is subject to an absolute limit. In the case of fluid in pipes this is not so. No matter how long a pipe may be, if there is no leakage, water would flow along the pipe until the level of its surface were the same at both ends. But the rate of flow would diminish with the length and diameter of the pipe. Thus we can transmit power through a perfectly tight pipe, however small, and however long; but when we come to consider the gross power that can be trans- mitted through a given pipe, with a given percentage of loss, the question is different. Given the size and strength of the pipe, the gross amount of power, and the percentage of loss, and the limits are fixed. Thus, taking a 12-inch pipe capable of standing 1,400 Ibs. on the square inch, the loss in transmitting 1,000 horse-power would be about 5 per cent, per mile, at first increasing as the pressure fell to 700 Ibs. to 10 per cent. We should thus have lost half the power in about seven miles. We cannot say that seven miles is the absolute limit, for with a 24 inch pipe, which would cost four times as much per mile, we could transmit the same power thirty times as far with the same loss. The cost of laving a 12-inch pipe for seven miles, however, would probably be as much as even 1,000 horse- power would stand ; while a 24-inch pipe for 200 miles would be out of all proportion. Then there is the consideration of leakage, which, although very small for short lengths, is larger for greater lengths. Seven miles is at present an outside economical limit of hydraulic trans- mission, even for such a large amount of power : but with air the case is different. This flows so much easier than water, that the cost of trans- mitting the same power through the same distance, with the same loss, would be about 12 per cent., or, at the same cost per mile, the air may 128 THE TRANSMISSION OF ENERGY. [45 be transmitted 100 times as far with the same loss. The total cost, however, would thus be 100 times as great, which would exceed the eco- nomical limit ; but not only theory but practice has shown that power may be economically transmitted five times as far by air as by water something like thirty miles. But on comparing these two means, one circumstance must not be lost sight of, and that is, that getting the power into the pipe in the form of compressed air, will cost twice as much as getting it in in the form of water. This is a great advantage for water where the distance is short, but where the distance is long, the greater efficiency of air more than compensates for this initial loss. Like water, air can only be transmitted economically where the quantity is large, the friction being proportionately greater in small pipes than in large, varying as the four-fifths power of the diameter. This is a great drawback, both as regards hydraulic and compressed air transmission. It does not affect ropes and shafts in the same way, but even in these cases considerations of durability prevent these means being used efficiently for the transmission of small quantities of power to con- siderable distances, so that, with the possibility already mentioned, there remains an opening for any means that will enable power to be transmitted efficiently in small quantities, and such a means we have in the flow of electricity along wires or conductors. In considering electricity, we may well start with the questions, (1) Will electricity enable us to transmit power in large quantities more efficiently than the foregoing means ? (2) Will it enable us to transmit small quantities ? These questions may be more definitely answered than they could a few weeks ago. Thanks to the ex- periments of M. Deprez, who appears to have been the only one, out of all those who are advocating the use of electricity, who has had the courage to try and see what can be done, we can now say with certainty that a current of electricity, equivalent to 5 horse-power, may be sent along a telegraph wire l-6th of an inch in diameter, some ten miles long (there and back) with an expenditure of 29 per cent, of the power, because this has already been done. In order to do this, it would seem that M. Deprez has perfected his apparatus so as to have nearly reached the possible limit. Compared with wire rope, this means falls short in actual efficiency, as M. Hirn sends 500 horse-power along a f-inch rope. To carry this amount, as in the experiment of Deprez, one hundred telegraph wires would be required ; these wound into a rope would make it more than T4 inches in diameter, four times the weight of M. Hirn's rope. With the moving rope the loss per mile is only 1*4 per cent., while with the electricity it was nearly 6 ; so that, as regards weight of conductor and efficiency, the electric transmission is far inferior to the flying rope. Nor is this all. With 45] THE TRANSMISSION OF ENERGY. 129 the flying belt, M. Hirn found the loss at the ends, in getting_the power into and out of the rope, 2 per cent. ; whereas, in M. Deprez's experiment, 30 per cent, was lost in the electric machinery alone, which is very small as such machinery goes. But this is not all. No account is here taken of the loss of power in the transmission to and from the electric machinery, a matter which is, I believe, very much under-estimated. The machines made revolutions at 1,000 and 700, much too high for direct connection with either a steam-engine or any mechanical operator : the power, then, had at each end to be transmitted through gearing, or a system of belts. And supposing this alteration of speed to have been five or six at each end, experience tells us that a loss of at least 15 per cent, must ensue. This loss was indeed apparent, for the dynamometer was connected with the machine with a belt, which showed a loss from this one belt alone of 20 per cent. Taking the whole result, it does not appear that more than 15 or 20 per cent, of the work done by the steam-engine could have been applied to any mechanical operation at the other end of the line, as against 90 per cent, which might have been realised with wire rope transmission. To set off against this, electricity has the enormous advantage in the conductor being fixed, and in the fact that it is likely to be, if anything, less costly and more efficient for small quantities of power than for large. These advantages will certainly insure a very large use for electricity in the distribution of power, particularly for high speed machinery. There is yet another means of communicating and distributing energy now coming rapidly into vogue. This is by the transmission of coal-gas along pipes. The distances, often many miles, through which the gas is often transmitted before reaching the engine, are such that, with any other means of distributing power, would considerably enhance the cost of the power. But in the case of gas, it does not appear that these distances are at all a matter of consideration. This may be at once explained. It takes about ten cubic feet of gas to develop 1,000,000 foot-pounds in a gas-engine, whereas of air compressed in the ordinary way it would require something like 140 cubic feet to yield the same power. Hence the comparative cost of transmission is the cost of transmitting ten cubic feet of gas against that of 140 cubic feet of compressed air, and these would be about as one to twenty-five ; so, as a means of distributing energy, gas is twenty-five times more efficient than compressed air. I have now placed before you, as far as circumstances will allow, the various means by which energy, in a form available for power, may be trans- mitted over long distances, together with the circumstances which limit such transmission. By means of the railway and steamboat, corn and coal can o. K. ii. 9 130 THE TRANSMISSION OF ENERGY. [45 be, nay is, transmitted half-way round the earth with an expenditure of power less than half the power represented by the coal carried, but this can only be done where the quantity to be transmitted is very large. At present this efficiency is unrivalled, no means of packed energy or of current energy approaching even 1 per cent. And further, there is apparent room for a large diminution in the present expenditure, small as it is, in the improvement of the steam-engine as a means of directing the energy of coal. For the distribution of power, this means ceases to be efficient, nor can it be employed to transmit energy which has already taken the form of power. For these purposes other means have to be employed. These various means, although they differ greatly in efficiency, all fall so far below the efficiency of coal and corn, that a hundred miles appears to be the outside limit any economical transmission of power in quantity for mechanical purposes, could be at present effected ; and hence any power, be it derived from wind or water, must be used within this radius of its source ; and, except in places far out of the reach of rail or water, this limit may be divided by ten. So far as efficiency of transmission in considerable quantities, neither secondary batteries nor electrical transmission are more efficient than com- pressed air or belts, but when it comes to transmitting small quantities, then electric transmission has a decided advantage. The cost of the electric conductor diminishes with the quantity to be transmitted, and by making the conductor sufficiently large, its efficiency may be increased to any extent. At the present time, electric conductors are continuous half-way round the world, and whenever a message is sent from England to Australia direct energy is transmitted 10,000 miles, but in what quantity ? The energy of the current, as it arrives, is not much more than sufficient to keep a watch going, at any rate not more than I'lOOO millionths of horse-power. The value of such energy, estimated at 17 per minute, would be equivalent to a billion pounds per horse-power per hour, whereas the highest price paid for animal labour in Australia or England is not more than 6d. per horse- power per hour. This shows the difference between the transmission of electricity for telegraphic purposes and its transmission for mechanical purposes. Energy differs in value greatly, but for operations that can be performed by men or horses, the price of energy must be regulated by the highest price of corn. The prosperity of any spot in the past depended on the fertility of the adjacent soil. But the use of coal has altered this, and now the present prosperity of this country is owing to the adjacency of our coal-fields, these 45] THE TRANSMISSION OF ENERGY. 131 having rendered it possible to bring our food across the earth.- The im- proved means of transmitting coal and corn, it would seem, have, or may again, change this, and if, instead of looking on the life of this country as limited by the life of our coal-fields, we look boldly forward, and foster every means, political, social, and mechanical, which may render this a favourite spot to live upon, we need not fear that the necessity of bringing our coal from a distance will make a difference which will counterbalance the ad- vantage we shall derive from the mechanical facilities we shall have here. 92 46. [Read before Section A at the "British Association," 1883.] ON THE EQUATIONS OF MOTION AND THE BOUNDARY CONDITIONS FOR VISCOUS FLUIDS. TAKING the ordinary equations of motion for viscous fluid, and supposing a tube indefinitely broad in the direction z, bounded by solid surfaces y=<> .................................... (i). which tube may be supposed continued in a circle so as to make a circular trough. Suppose it full of water, at rest, and subject to an acceleration X , the equation of motion gives du d?u , r di=*w +x ................................ (2)> or by altering the arrangement, instead of X we may have --- f- Now initially u = 0, .-. $* = dy dy* du or right up to the surface. But by the boundary condition at the solid surface u = always ; which shows that the boundary conditions are at variance with the equation of motion. 46] ON THE EQUATIONS OF MOTION, ETC. 133 The equations simplify to du d*u 1 dp Si =l *dy>-- p dx + X ........................... < 3 >' with the boundary condition that u is always zero at the boundaries. For initial conditions we will take p constant, and u uniformly zero. The equation (2) then becomes * .................................... w X If now we suppose X to have a uniform value the equation of motion gives -j- = X throughout the fluid, i.e. at the boundaries. This is contrary to the boundary conditions, for if u is always zero at the boundaries -^- must (TC also be zero. The functions wanting were rendered evident in the following manner. By differentiating equation (3) with respect to t, remembering that X , du du and p are constants, and that TTT = -?r , ot at d du _ d* /du\ dt~dt~ t *dy*\di)" an equation of which the integrals are well known, f t = Z(A^ + ^) .............................. (6), and which may be determined to suit the initial and boundary conditions. du But it does not follow that the value of -j- in equation (6) is the same CLt as in (4) because the integral of (5) includes an arbitrary function of y, du d-u If we determine f(y) to suit equation (4) then equation (7) will not fit the initial boundary conditions. If however we determine f(y) so that equation (4) shall be satisfied at some small distance r from the boundary, and the boundary condition satisfied we have _ X(l-e '} ........................... (8), where p is a numerical large quantity. 134 ON THE EQUATIONS OF MOTION [46 Such a function satisfies all the boundary conditions, but the general value of the function would be where 2X = X ................................. (9). The addition of such a function to the equation of motion would meet the initial conditions of the case in question, in which X is independent of t, but this is all, for the boundary conditions include that we must have all the differential coefficients of u with respect to t zero at the boundary, to meet which case it would be necessary to have a function Po (y c) Piy^c r t) + &c ............. (10). Instead of adding such functions, however, it seems better to consider in what way the equation of motion can be modified so that these functions result from integration. This would be the case if instead of equation (3) we had the equation du V r . d* (du\ dp v d*u - + *+' l .............. (1 where %A = 1. That is, if we add the term 2 A~, - to the equation, it becomes com- p dif at patible with the boundary conditions, and the term itself is of the same order as others which have been neglected in constructing the equations of motion, and the strong presumption is that such terms have been neglected. The case pursued here is the simplest possible, but by a similar method it may be shown that the general case for a fluid at constant density will be met if the equations of motion be modified as follows : du r 2 _ du _ dp dt p* dt dx dv r 2 dv dp v _., i / 19 \ 777 i v "35 = ~j + Y + p\-v > (>*;. dt p* dt dy dw r 2 dw dp _ dt p* dt ~ dz ^ The equations of motion were not originally the outcome of any com- plete hypothesis of the molecular constitution of fluids. They involved certain assumptions which would enter into such an hypothesis, but by no means completely define it, any more than would the phenomena of approximately steady motion suffice to define the complete phenomena of motion. 46] AND THE BOUNDARY CONDITIONS FOR VISCOUS FLUIDS. 135 The original basis of the equations of motion for viscous fluids were certain experimental phenomena, and it is important to notice that all these phenomena belong to what may be called approximately steady motions. So that neither the experimental verification of these equations, nor the molecular hypothesis on which they were originally based, was in any sense complete or general. And if the original framers of these equations had attempted to carry them to the second order of small quantities, it would only have been done by further molecular assumptions and anything like a complete experimental verification was entirely wanting. This aspect of the case was changed by the foundation of a complete molecular hypothesis of gases, for founding as it did the dynamical theory of gases on complete fundamental assumptions, the equations of motion followed as a consequence of these assumptions and although not attempted, could have been obtained to any degree of small quantities. Maxwell contented himself with showing that the equations of motion resulting from his assumptions agreed with the equations of motion obtained by Stokes to the first order of small quantities*, but it was perfectly possible to have pursued his reasoning to the second order of small quantities. Having then found that certain terms of the second order were wanting in the equations of motion to meet the boundary conditions as shown by experi- i IK nt, the most probable method of defining these terms seemed to be to any the dynamical theory of gases to the second order of small quantities. For this investigation I adopted the same method as that which I have explained in my paper on the dimensional properties of matter in the gaseous state f, merely extending the method to meet the case of varying motion. The result was that I found terms of the form required, but they entered into the equations with the opposite sign to those required to meet the boundary conditions, and would thus only introduce arbitrary constants of a periodic character. Besides which, these terms clearly vanished at the boundaries, i.e. if the boundary were regarded as a plane of total reflection ; while according to the theory, as regards the first order of small quantities, the boundary produced no tangential effects whatever. Having considerable confidence in the method I was using in deducing the equations of motion from the fundamental assumption; it naturally occurred to me to re-examine the fundamental assumptions, to see if these had been introduced into the theory in their fulness. It was then I observed that the theory, both as applied by Maxwell, and myself, neglected any possible dimensions of a molecule, and it became clear that by neglecting * Phil. Trniin., 1867, p. 81. t See Paper 33, Vol. I., pp. 257 ff. 136 ON THE EQUATIONS OF MOTION [46 this we had neglected that which made it possible for the boundary to produce an acceleration on the fluid. By neglecting the dimensions of a molecule, the cause of transference of momentum across a surface reduces itself to the carriage of momentum by the moving molecules, whereas if we take the size of molecules into account, a certain portion of the area of any ideal surface drawn through the gas must be occupied by the solid matter of the molecules, and the stresses in these molecules will be the cause of the transference of momentum across the surface. This cause of the transference of momentum across a plane had been ignored with the dimensions of the molecule in the theory of gases*. It became necessary therefore to take this into account to see what effect it had on the equations of motion. It is clear that this effect would involve the elasticity of the molecules themselves, as the rate at which momentum would traverse them would be that of the propagation of sound in a solid, but considering the relative elasticities of solids and gases, it seemed legitimate to take the elasticity of the molecules as infinite compared with that of the gas, i.e. to assume the molecules as absolutely rigid and the same for groups of molecules in contact, either directly or through other molecules with a solid surface. Now if we imagine a surface plane for the instant to be moving with the mean velocity of the matter which it traverses, and suppose that in molecules are cut by an unit of this plane and if the m molecules, cut by a plane parallel to the first and at a distance r, (the diameter of a molecule) are in contact with those on the first, then if we have a third plane also at a ?71 distance r from the second, of the molecule cut by this will be directly in 772- contact with the second, and indirectly in contact with those on the first, so that of the m molecules on two planes at a distance y from each other -V JW mq r = me r , where p = log e q, will be indirectly in contact with each other. Now since according to the assumptions, we regard this connection as * See Paper 33, Vol. I., pp. 257 ff., "On Certain Dimensional Properties of Matter in the Gaseous State." 46] AND THE BOUNDARY CONDITIONS FOR VISCOUS FLUIDS. 137 rigid, we see that if p is the density of the matter on any plane^this matter is rigidly connected with matter = pe r , at a distance y on either side of the plane, which therefore is an expression for the rigidity of a gas. And it may be noticed that, although the distance to which this rigidity extends is limited by the value of - at a surface such as y = 0, it is absolute, so that at a solid surface all the matter (not molecules) in contact with the surface, has the mean motion of this surface 7j whatever may be the value of -. _py The expression pe r has been obtained on the hypothesis of the distri- bution of molecules in a gas, and even so, without any very great degree of refining. It is impossible without a more definite hypothesis than has been pro- pounded at present as to the constitution of a liquid, to say what form the expression for rigidity might there take, but it is reasonable to suppose that as regards the law of molecular contact, it would be the same as that of a gas, only p instead of being large would be very small, but as regards the rigidity, the same assumption could not be made any more than a whole fishing-rod can be considered rigid in the same degree as a single joint of such a rod. 47. ON THE GENERAL THEORY OF THERMO-DYNAMICS. A LECTURE DELIVERED TO THE INSTITUTION OF CIVIL ENGINEERS. 15 NOVEMBER, 1883. From " The Proceedings of the Institution of Civil Engineers, 1883." IN lecturing on any subject, it seems to be a natural course to begin with a clear explanation of the nature, purpose, and scope of the subject. But in answer to the question What is thermo-dynamics ? I feel tempted to reply It is a very difficult subject, nearly, if not quite, unfit for a lecture. The reasoning involved is such as can only be expressed in mathematical language. But this alone should not preclude the discussion of the leading features in popular language. The physical theories of astronomy, light, and sound involve even more complex reasoning, and yet these have been rendered popular, to the very great improvement of the theories. Had it appeared to me that it was the necessity for mathematical expression which alone stood in the way of a general comprehension of this subject, I should have felt compelled to decline to deliver this lecture, honourable as I acknow- ledge the task to be. What I conceive to be the real difficulty in the apprehension of the leading features of thermo-dynamics is, that it deals with a thing or entity (if I may so call heat) which, although we can recognise and measure its effects, is yet of such a nature that we cannot with any of our senses perceive its mode of operation. Imagine, for a moment, that clocks had been the work of Nature, and that the mechanism had been on such a small scale as to be imperceptible even with the highest microscope. The task of Galileo would then have been reversed ; instead of inventing machinery to perform a certain object, his task would have been from the observed motion of the hands to have 47] ON THE GENERAL THEORY OF THERMO-DYNAMICS. 139 discovered the mechanical principles and actions of which these motions were the result. Such an effort of reason would be strictly parallel to that which was required for the discovery of the mechanical principles and actions of which the phenomena of heat were the result. In the imaginary case of the clock, the discovery might have been made in either of two ways. The scientific method would have been to have observed that the motion of the hands of the clock depended on uniform intermittent motion ; this would have led to the principle of the uniformity of the period of vibrating bodies, and on this principle the whole theory of dynamics might have been founded. Such a theory would have been as obscure, but not more obscure, than the theory of thermo-dynamics. But there was another method in the case of timekeepers, the one by which the theory of dynamics was actually brought to light namely, the invention of an artificial clock, the action of which could be seen, and, so to speak, under- stood. It was from the pendulum that the constancy of the periods of vibrating bodies was discovered, and from this followed the dynamical theories of astronomy, light, and sound. There is no great difficulty in the apprehension of these theories, because they do not call for the creation of a mental picture, but merely for the exaggeration or diminution of what we can actually see in the clock. As regards the mechanical theory of heat, however, no visible mechanical contrivance was discovered or recognised which afforded an example of this action ; apparently, therefore, the only possible method was the scientific method namely, the discovery of the laws of its action from the observation of the phenomena of heat, and accepting these laws, without forming any mental image of the dynamical origin, was the only method open. This is what the present theory of thermo-dynamics purports to be. But although the theory of thermo-dynamics may be said to have been discovered in the form in which it is now put forward, this is not quite true. For one of the discoverers of the second law, and the one who had priority over the others, worked by the aid of a definite mechanical hypothesis as to the actual molecular motions and forces on which the phenomena of heat depend, and many of the most important steps in the theory are solely to be attributed to his labours. But to return to the theory. This may be defined as including all the reasoning based on two perfectly general experimental laws, without any hypothesis as to the mechanical origin of heat. In this form thermo-dynamics is a purely mathematical subject and unfit for a lecture. But as no one who has studied the subject doubts for a moment the mechanical origin of these laws, I shall be following the spirit, if not the letter of my subject, if I introduce a conception of the mechanical actions from which these laws spring. And this I shall do, although I should hardly 140 ON THE GENERAL THEORY OF THERMO-DYNAMICS. [47 have ventured, had it not been that, while considering this lecture, I hit on certain mechanical contrivances which afford sensible examples of the action of heat, in the same way as the pendulum is an example of the same principles as those involved in the phenomena of sound and light. These examples, thanks to the ready aid of Mr Forster in constructing the apparatus, I am in a position to show you, and I am not without hope that these kinetic engines may in a great measure remove the source of obscurity on which I have dwelt. The general action of heat to cause matter to expand, or to tend to expand, is sufficiently obvious and popular. That the expanding matter will do work is also sufficiently obvious, but the exact part which the heat plays in doing this work is very obscure. It is now known that heat performs two, and it may well be said three, distinct parts in doing the work. These are (1) To suppty the energy equivalent to the work done. (2) To give the matter the elasticity which enables it to expand, i.e., to convert the inert matter into an acting machine. (3) To convey itself (i.e., heat) in and out of the matter. This third function is generally taken for granted in the theory of thermo- dynamics. In order to make any use of therrno-dynamics, a knowledge of the experimental phenomena of heat is necessary ; but as time will not permit of my entering largely into these, I have had some of the leading facts suspended as diagrams. One or two it will be well to mention. Heat as a quantity is independent of temperature, the thermal unit taken being the amount of heat necessary to raise 1 Ib. of matter 1 Fahrenheit. Temperature represents the intensity of heat in matter. Matter in most of its forms expands more or less uniformly as we add heat to it ; hence the expansion of matter measures temperature. Gases such as air expand in absolute proportion to the heat added under a constant pressure. Absolute temperature is an idea derived from the observed rate of contraction of gases ; they would vanish to nothing with the temperature 461 below zero Fahrenheit. For the other phenomena I must refer to the diagrams as I proceed. Our knowledge of these facts has been accumulating during the last two hundred years, and it was in a very complete condition forty years ago, 47] ON THE GENERAL THEORY OF THERMO-DYNAMICS. 141 before ther mo-dynamics was born. The birth of this science may be con- sidered as the result of the recognition of work motion against resistance as a true measure of mechanical action, and of accumulated work or energy as the potency of all sources of power. These ideas have now become extremely popular, and all are able to recognise in the raised weight, the bent spring, the moving hammer, the same thing, energy, which is measured by the amount of work which can be derived from any of these sources. Before the recognition of this means of measuring mechanical potency, any definite idea of the true mechanical action of heat was impossible, for we had not recognised the only mechanical action by which it can be measured. In 1843 Joule conclusively proved that, by the expenditure of 772 ft.-lbs. a thermal unit of heat must be produced, provided all the work was spent in producing heat. The simplicity of the ideas here involved, and the com- pleteness of Joule's proof, acted at once to render the first law popular. No language can be too strong in which to express the importance of this discovery ; yet, as was long ago pointed out by Rankine, the very popularity of Joule's law went a long way to obscure the fact that it did not constitute the sole foundation of the theory of thermo-dynamics. Before Joule's dis- covery it was recognised that heat acted a part in causing work to be performed. It was clearly seen that it was heat which caused the water to expand into steam, against the resistance of the engine, and the necessity of heat to cause matter to expand was recognised. To make matter do work it was only necessary to heat it. It would expand, raising a weight ; and since after doing its work the matter was still hot, it was supposed that the only necessity for the heat was to add increased elasticity to matter. It was seen that the heat that had once been used was so degraded in temperature that it could not be all used again. So that, although there was no idea that heat was actually consumed in doing the work, it was seen that for continuous work a continuous supply of heat at a high temperature was necessary. As regards the exact proportion of heat required for the supply of elasticity, to perform a certain quantity of work, fairly clear ideas prevailed. It was seen that this depended on various circumstances. These were formulated by Carnot, who in 1828 gave a formula, which is equivalent to our second law of thermo-dynamics, of which it was the parent. Now this idea that heat merely caused work to be done was not absunl, as is sometimes supposed. Indeed we may say that the present popular idea that the whole heat is convertible into work is more erroneous than the old idea in the ratio of 10 to 1 ; because the old idea that the function of heat 142 ON THE GENERAL THEORY OF THERMO-DYNAMICS. [47 is to supply elasticity was right, as far as it went. Although the present idea that the function of heat is to supply energy from which the work is drawn is also right, yet in any known possible heat-engine ten times more heat is necessary for the purpose of giving elasticity to matter than is converted into work by elasticity. This error, which seems to be very general amongst those who have not made a special study of the subject, may, 1 think, be attributed first, to the popularity of the first law of thermo- dynamics, and secondly to the fact that although the second law of thermo- dynamics is nothing more nor less than a statement of the proportion which the quantity of heat necessary to produce elasticity bears to the quantity which this elasticity will convert into work, yet that it is the invariable custom in stating this law to omit all attempt to explain the purpose which this excess of heat serves ; the reason for this omission being that experiment only shows that this heat is necessary, and hence this is all that we have a right to say. If such an error prevails it is only a popular error, for it certainly did not affect the progress of the science. No sooner did Joule's law become known than it was taken up by Rankine, who, in 1849, published a complete theory of thermo-dynamics, based, as I have said, on a hypothetical constitution of matter. This was almost simultaneously followed by theories based on an improved form of Carnot's reasoning by Thomson and Clausius. Rankine's theory was based on a hypothetical constitution of matter. He invented a system of molecular motions and constraints, which he called molecular vortices, and he then calculated the effects of these motions by the theory of mechanics. The fact that his reasoning was based on a hypothesis was considered by many as a fault in his reasoning. But on the other hand the clear idea thus obtained, as to the reason of everything he was doing, gave him such an advantage over those who were working by experimental laws, of the meaning of which they would venture no opinion, that he was led to make discovery after discovery in advance of his competitors, while some of his discoveries are still beyond the reach of experiment. There was, however, a difficulty Rankine had to face ; some properties of matter were pointed out which his hypothetical matter did not possess. This was not much to be wondered at, for although Rankine had invented machinery which would account for the mechanical action of heat, there was no reason to suppose this to be the only machinery. Rankine, with a view to the difficult calculations he had to make, had chosen machinery as simple as possible. Instead, however, of trying to complicate it, he, yielding to the opinion of his cotemporaries, adopted the general conclusions to which 47] ON THE GENERAL THEORY OF THERMO-DYNAMICS. 143 it had led him as axiomatic laws, and so cut himself adrift from his hypothesis. It comes to be, then, that the student of thermo-dynamics finds as a reason why we must pass a large amount of heat through his engine, besides that which is converted into work, he is to accept an axiomatic law as to the greatest possible amount that can be converted under the circumstances. To tell a child who asks why he cannot have more food, that he can only have 6 oz. a day, would be considered cruel. So to tell a student who wants to know why, out of the ten million foot-lbs. in 1 Ib. of coal, a steam-engine T T can only give one million as work, that he is only allowed -=^ ~j~ , is cruel, J 1 ~T~ yet this is all he can have from the theory of thermo-dynamics based on its experimental laws. Rankine, when compelled to abandon his hypothesis as the foundation of his theory by the objections justly urged against it, pointed out the great disadvantage of a mechanical theory conveying no conception of the mechanical basis of its laws ; and called on all those who taught the subject, to try and find some popular means of illustrating the second law. This call was made twenty years ago ; but, I believe, up to the present time no such illustration has been forthcoming. When undertaking this lecture I had no idea of such an illustration, and I did not intend to say much as to the reason of the second law. But, as I have said, three weeks ago an idea occurred to me. It arose in this way : Heat acts in matter to transform heat into work by molecular mechanism. Having much studied the subject, I have in my mind a picture, right or wrong, of the mechanism, and the part which heat acts. The question occurred Is there no way of making a machine such that, although the parts are in visible motion, and the energy transformed to work is visible energy, yet the energy supplied shall have the characteristics of heat-energy, and the machine shall act simply in virtue of the elasticity caused by the motion of its parts ? The question had no sooner arisen than several ways of carrying out the idea presented themselves. The general idea of the mechanical condition which we call heat is, that the particles of matter are in active motion ; but it is the motion of the individuals in a mob, with no common direction or aim. Rankine assumed the motion to be rotatory, but it now appears more probable that the motion in the particles is oscillatory, undulatory, rotatory, and all kinds of motion, whatsoever; so that the communication of heat to matter means the com- munication of internal agitation mob agitation. If, then, we are to make a machine to act the part of hot matter, we must make a machine to perform 144 ON THE GENERAL THEORY OF THERMO-DYNAMICS. [47 its work in virtue of the communication of internal promiscuous motion amongst its parts. The action of heat-mechanism to do work is simply that of expansion of volume, or the increased effort to expand owing to increased agitation. I first tried to think of some working arrangements of small bodies which should forcibly expand when shaken ; but it appeared that it would be much easier to effect a contraction. This was as good. As long as any definite alteration in shape could be produced against resistances by a definite amount of agitation in its parts, we should have a machine illustrat- ing the action of the heat-engine. Suppose we want to raise a bucket from a well. Our best way is to pull or wind up the rope, but that is because the energy we employ is in a completely directable form. Suppose we had no such directable energy, but could only shake the rope, it having been first made fast at the top (Fig. 1, next page). Then, it being a heavy rope, a chain is better ; suppose we shake the chain laterally, waves will run down the chain, and, if we go on shaking, the chain will assume a continuously changing sinuous form (Figs. 2 and 3); and, as the chain does not stretch, the bucket must be raised to allow for the sinuosities. The chain will have changed its mechanical character, and from being a tight line or tie in a vertical direction, will possess kinetic elasticity, that is, elasticity in virtue of its motion, causing it to contract its vertical length. The bucket will be raised, although not to the top of the well, and work will have been done in raising it, but the work spent in shaking the chain will be not only the equivalent of the work spent in raising the bucket, but also of all the kinetic agitation in the chain necessary to raise the bucket. Having raised the bucket as far as possible with a certain power of agitation, if the supply of agitation be cut off, then that already in the chain will sustain the bucket until it is destroyed by friction, when the bucket will gradually descend. But if we want to do more work, to raise another bucket, we may take that which is raised off at the level at which it is raised ; then, to get the chain down again, we must allow it to cool, i.e., allow the agitation to die out ; then, attaching another bucket, to raise this, we shall again have to supply the same heat, perform the same work, i.e., the work to raise the bucket, and the agitation-energy of the chain. Thus we see that the energy necessary to the working of the machine serves two purposes, it supplies the energy necessary to raise the bucket, and the energy necessary to convert the chain from an inextensible tie into an elastic contracting system, capable of raising the weight, neither of which portions of energy is again serviceable after the bucket has been raised. The one portion is already converted into work, and the other, although still in existence in the chain 47] ON THE GENERAL THEORY OF THERMO-DYNAMICS. 145 as energy, can only sustain the position of the chain. Before it could be used to do more work it must be got out of the chain and back again, which Fig. i. Fig. 2. Fig. 3. is just the thing you cannot do ; we can get some of it out and some of it back, but not all. It must not be supposed that this method of raising a bucket by shaking the rope is recommended as the best means. No one would dream of using it if we could get a direct pull, but that is nothing to the point. We are considering the action of heat, and we have limited ourselves to using energy of the same kind that heat supplies ; that is, energy in the form of promis- cuous agitation, absolutely without direction, so that the question is, how can we raise the bucket by shaking ? I feel that there is a childish simplicity about this illustration, that may at first raise the feeling of "Abana and Pharpar, rivers of Damascus," in the minds of some of rny hearers, but, should this be the case, I have every confidence that calm reflection will have the same effect as on Naaman. The case of the shaken rope, as I have put it, is no mere illustration of the action of heat, but an instance of the same application of the same principles. The sensible energy in the shaking rope only differs from the energy of heat, i.e., a bar of metal is the scale of the motion ; we see that in the chain but not in the bar, not because the molecules of the bar are moving slower, but because the scale of motion is infinitely smaller. The temperature of the bar from absolute zero measures the mean square of the velocity of all its parts, multiplied by some constant depending on the mass of the parts which are moving together ; so the mean square of the velocity of the chain multiplied by the weight per foot of the chain really represents the absolute temperature of the sensible energy in the chain. The apparatus which I have on the table is an obvious adaptation of the rope and the bucket. There are three different illustrations apparently very different in form, but all working by the same principle. O. R. II. 10 146 ON THE GENERAL THEORY OF THERMO-DYNAMICS. [47 Here is the chain (Figs. 1, 2, 3), by the shaking of which (addition of promiscuous energy) a weight of 2 Ibs. is raised 3 feet, or 6 foot-lbs. of work done ; here is another sort of chain, a series of parallel horizontal bars of wood, connected and suspended by two strings (Figs. 4, 5, and 6). By giving a circular oscillation to the upper bar, the whole apparatus is set into a Fig. 4. Fig. 5. Fig. 6. twisting motion (agitation); the strings are continually bent, and the vertical length of the whole system is shortened, and a weight of 10 Ibs. or the bucket of the pump is caused to rise, raising water just as if we boiled water under the piston of a steam-engine. To get the bucket down again for another stroke, we must quiet or cool the chain, just as we must condense the steam, and the energy taken out of the chain in cooling corresponds exactly with the heat that must be taken out of the steam in order to condense it. The waves of the sea constitute a source of energy in the form of sensible agitation ; but this energy cannot be used to work continuously one of these kinetic-machines, for exactly the same reason as the heat in the bodies at the mean temperature of the earth's surface cannot be used to work heat- engines. A chain attached to a ship's mast in a rough sea would become elastic with agitation, but this elasticity could not be used to raise cargo out of the hold, because it would be a constant quantity as long as the roughness of the sea lasted. In practical mechanics we have no source of energy consisting of sensible agitation, besides the waves of the sea ; so that there has been no demand for these kinetic engines to transform sensible mob-energy into work ; had there been, I might have patented my idea, though probably it would have long ago been discovered. But there has been a demand for what we may call sensible kinetic elasticity, to perform for sensible motion the part which 47] ON THE GENERAL THEORY OF THERMO-DYNAMICS. 147 the heat elasticity performs in the thermometer, and for this purpose the principle of the kinetic machine was long ago applied by Watt. The common governor of a steam-engine acts by kinetic elasticity, which elasticity, depending on the speed at which the governor is driven, enables the governor to contract as the speed increases. The motion of the governor is not of the form of promiscuous agitation, but, though systematic, all the motion is at right angles to the direction of operation, so that the principle of its action is the same. The kinetic elasticity of the governor performs the same part as the heat elasticity in the matter of the thermometer ; the first measures by contraction the velocity of the engine, and the other measures by expansion the velocity of the molecules of the matter by which it is surrounded, so that we now see that while measuring the speed of sensible revolution, we are performing on a different scale the same operation as measuring the temperature of bodies which depends on the molecular velocities, and that quite unconsciously we have constructed instruments to perform the two similar operations which act by means of the same mechanical action, namely, kinetic elasticity. These kinetic examples of the action of heat must not be expected to simplify the theory, except in so far as they give the mind something definite to grasp ; what they do is to substitute something we can see for what we can barely conceive. The theory of thermo-dynamics can be deduced from any one of these kinetic examples by the application of the principles of mechanics ; such application involves complex dynamical reasoning, such as can only be executed by the aid of mathematics, and would be altogether unfit to intro- duce into a lecture. I shall therefore pass on to some considerations resulting from the theory of thermo-dynamics. The discovery of the two laws has enabled us to perfect and complete our experimental knowledge of the phenomena of heat. But probably the greatest practical use is that these two laws enable us to calculate with certainty, from the experimental properties of any matter, the extreme potency of any source of power. Thus we find by experiment that a pound of coal burnt in a furnace yields fourteen to sixteen thousand thermal units of heat. The first law, Joule's law, tells us at once that this is equivalent to from 11,000,000 to 13,000,000 foot-lbs. of energy. But this is not, as seems to be generally supposed, the power of coal. The second law of thermo-dynamics tells us that in order that this energy might be realised, it must be capable of being developed at an infinite temperature, whereas we know that this cannot be 102 148 ON THE GENERAL THEORY OF THERMO-DYNAMICS. [47 the case ; and there is a growing idea that the temperature at which coal will burn is not so extremely high, about 3,000 Fahrenheit. Taking this temperature, and assuming the temperature of the atmosphere to be 60, we have for the proportion of the heat of coal, that we could with a perfect engine call power, j, about 80 per cent., or from 9,000,000 to 11,000,000 O^rO J. foot-lbs. Again, we know the heat properties of all known liquids and gases, so that we can, by the second law, tell the greatest possible proportion of the heat received, which can be converted into power by any of these agents. In the steam-engine, for instance, we see that the present limits of art restrict the temperatures absolutely to 400, and practically the limits are much less; while the lowest temperature that can be worked to in a condenser is 100. Then, as the limit to the possibility, we have one-third as the greatest proportion, or three out of the nine million foot-lbs. The greatest actual achievement by Mr Perkins has been about two millions, while the best engines in use only give us a little over one million, or about one-ninth of the possible realizable portion between 3,000 and the mean temperature of the earth's surface. I cannot here enter upon these, but the reasons why higher temperatures cannot be used in the steam-engine are obvious enough. The same reasons do not apply to hot air as an agent. This may be worked at much greater temperatures ; and about thirty years ago, as soon as it appeared from the science of thermo-dynainics that the limit of efficiency depended on the range of temperature, attention was much directed to air as a substitute for steam. The attempts then made failed through what were then called practical, or art difficulties. Just at the present time the possibility of other heat-engines than steam-engines has again come to the front ; and as this is so, it seems desirable to call attention to a circumstance connected with heat-engines which has as yet occupied quite a subordinate place in the theory of heat- engines. This is the law as to the rate at which heat can be made to do work by an agent, such as steam or air. The greatest possible efficiency of the agent, i.e., the proportion which the work done bears to the mechanical equivalent of the heat spent, is a matter of fundamental importance ; but the rapidity with which the heat can be so transformed with a given amount of apparatus, as an engine of a given weight, is a matter of at least as great importance. Which would be the best engine for a steamboat ; one that would develop 47] ON THE GENERAL THEORY OF THERMO-DYNAMICS. 149 20 H.P. for every ton gross weight, consuming 2 Ibs. of coal j>er H.P. per hour, or one that only gave 2 H.P. per ton weight, and only consumed 1 Ib. of coal ? Unquestionably the former ; yet hitherto the question of heat economy has been considered theoretically, to the exclusion of time economy. Yet the latter forms a legitimate part of the subject of thermo-dynamics, and has played a greater part in the selection of stearn as the fittest agent than the consideration of the heat-economy. In the theory of thermo-dynamics it is assumed that the working agent, be it water or any other, can be heated up and cooled down at pleasure, without any consideration as to the time taken for these operations, which are considered to be mere mechanical details. Yet in the science of heat a great amount of labour has been spent ; a great amount of knowledge gained as to the rate at which heat will traverse matter. And more than this ; it is well known that heat cannot be made to enter and leave matter without a certain loss of power, i.e., a certain lowering of the working range of temperature. It is by heat that heat is carried into the substance ; and hence, as I have indicated, there is a third law of thermo-dynamics relative to this transmission. Heat only flows down the gradient of temperature, and in any particular substance the rate at which heat flows is proportional to the gradient of temperature. Hence to get the heat from the source or furnace into the working substance a certain time must be consumed, and this time diminishes as the difference of temperature of the furnace and the working substance increases. The examples of the kinetic engines which I have shown you well illustrate this. If we shake the end of a chain, the wriggle passes along the chain at a given speed. It appears that an interval must elapse between the first shaking of the chain and the establishment of sufficient agitation to move the bucket ; a further interval before the bucket is completely raised ; and further still, another interval must elapse before the chain can be cooled again for another stroke ; so that this kinetic engine will only work at a given rate. I can increase this rate by shaking harder, but then I expend more energy in proportion to the work done. This exactly corresponds with what goes on in the steam-engine, only, owing to the agent water being heated, expanded, and cooled severally in the boiler, cylinder and condenser, the connection is somewhat confused. But it is clear that for every H.P. something like 15 million foot-pounds of power have to pass from the furnace into the boiler. As out of this 15 we cannot use more than 2 million, the remaining 13 are available for forcing the heat from the products of combustion into the water, and out of the steam into the condensing water, and they are usefully employed for this purpose. 150 ON THE GENERAL THEORY OF THERMO-DYNAMICS. [47 The boilers are made small enough to produce sufficient steam, and this size is determined by the difference of the internal temperature of the gases in the furnace and the water in the boiler, and whatever diminishes this difference would necessarily increase the size of the heating surface, i.e., the weight of the engine. The power which this difference of temperature represents cannot be realised in the steam-engine, so that it is most usefully employed in diminishing the necessary size of the boiler. Still it is an important fact to recognise that our present steam-engines require the expenditure of more than five times as much of the power of the heat (not of the heat) in getting the heat into the working substance as in performing the actual operation. This loss of power does not so much occur in the resistance of the metal which separates the furnace from the water as in the resistance of the gases. Gas is a very bad conductor; and though a thin layer adjacent to the plates is always considerably cooled, little further cooling goes on until, by the internal currents, this layer is removed, and a fresh hot layer substituted in its place. Similar resistance would occur inside the boiler between the water and the hot plate, nay does occur, until the water begins to boil, but then the evaporation of the water takes place at the hot surface, and every particle of water boiled absorbs a great deal of heat, which leaves the surface in the form of bubbles, allowing fresh water to come up. If we had air inside the boiler instead of water, we should require from five to ten times the surface to carry off the same heat, which is a sufficient reason why what are called hot-air engines cannot answer, even did not the same argument hold with enormously greater force in the condenser. Steam is as bad a conductor of heat as air as long as it does not condense, but, in condensing, steam will conduct heat to a cold surface at an almost infinite rate, for as the steam comes up to the surface it is virtually anni- hilated, leaving room for fresh steam to follow, which it will do if necessary with the velocity of sound. If, however, there is the least incondensable air in the steam this will be left as a layer against the fresh steam. Some years ago I made some experiments on this subject, which showed that 5 or 10 per cent, of air in the steam would virtually prevent condensation. If a flask be boiled till all the air is out, and nothing but pure steam is left, and if the flask be then closed and a few drops of cold water introduced, the pressure instantly falls to zero, though it immediately recovers from the boiling of the water in the flask. If now a little air be admitted, and allowed to mix with the steam, the few drops of water produce scarcely any effect. The facility with which steam carries heat to a cold surface is both an 47] ON THE GENERAL THEORY OF THERMO-DYNAMICS. 151 enormous advantage and some drawback ; as compared with air it is an enormous advantage in enabling the steam to be cooled in the condenser. But during the working of the steam in the cylinder, when the steam is wanted to keep its heat, the facility with which it condenses is a great draw- back, and necessitates the keeping of the cylinder hotter than the steam by a steam-jacket. For this part of its work the non-conductivity of incon- densable air is a great advantage. In dwelling thus on the conducting powers of air and steam, my purpose has been to prepare the way for a few remarks I wish to make on another form of heat-engine the engine in which the heat is generated in the working substance itself. The combustion-engine, in the form of the cannon, is the oldest form of heat-engine. Here the chemically separate elements in the form of gun- powder are the working substances put into the cylinder ; they take in with them the potential energy of chemical separation, which by means of a spark take the kinetic form of heat. Here there is no conduction, the kinetic elasticity propels the shot, and all the heat over and above that used in imparting energy to the shot is lost. The advantages of this form of engine are two. There is no time necessary for conduction, and as the gas generated is not condensable, there is little loss of heat by conduction to the cold metal. These two advantages are very great, but I should not have mentioned them in reference to guns were it not that there appears to be the dawning of an idea of taming this form of engine so as to substitute it for the steam- engine. To do this it is necessary to introduce coal or coal-gas ; and oxygen in the form of air in place of gunpowder. The thermo-dynamic theory applied to such engines shows that they should possess great advantages over the steam-engine in point of economy. And the considerations I have brought forward as to the loss of the power of heat in the transference of heat from the furnace to the boiler seem to promise such engines an enormous advantage in rate of work, while the substitution of a non-con- densable gas for steam in the cylinder seems to get over the art-difficulty of making cylinders to work under high temperatures. We cannot expect any piston to work in a cylinder of over 800 or 400 temperature, but with non-condensing gases the cylinder may be kept cool with little cooling effect on the gases contained in it, even if the temperature of these is 3,000. This will be the case if the gas in the cylinder is not in a violent state of internal agitation, but it should be remembered that all internal currents much facilitate the conveyance of heat to the walls. There is one drawback shown by the theory of these engines. The simple expansion of the gases resulting from combustion is not sufficient to 152 ON THE GENERAL THEORY OF THERMO-DYNAMICS. [47 cool them to anything like the temperature of 60, and to get the greatest economy some of the remaining heat should be used to heat the fresh charge. To do this, however, would necessitate the extraction of the heat from one mass of gas to communicate it to another, which would introduce all the difficulties of the boiler, increased by having gas instead of water. But even wasting this heat, the theory still shows a large margin of economy for such engines over the present performance of steam-engines, a margin which is said to have been already realised in the gas-engine, which is a form of combustion-engine in a high state of efficiency. Now, by means of Dowson gas, Messrs Crossley seem to have obtained 2,000,000 out of the 10,000,000 ft.-lbs. in 1 Ib. of coal. Further accomplishment in this direction is a question of art ; but while on all other hands science shows impassable barriers not far in advance of the present achievements of art, in this direction thermo-dynamics extended to include the rate of operation shows no known barriers ; while the fact that, as gas-engines, this system of com- bustion heat-engines has already established a footing assures them continual improvement. In conclusion I would say, by way of caution, that the theory of thermo- dynamics does not lead to the conclusion, which seems to be generally held by those who have only realised the first law of the science, that the steam- engine is a semi-barbarous machine, wasting more than it uses, very well for those who know no science, but only waiting until those better educated have time to turn their attention to practical matters, and then to give place to something much better. Thermo-dynamics shows us not the faults but the perfections of the steam-engine, in which there is no waste of power, since all is used either in doing work or in promoting the rate at which the work can be done. Next to the watch, the steam-engine is the highest development of mechanical art, and the science of thermo-dynamics may be said to be the result of the study of the steam-engine. 48. [From the "Proceedings of the Royal Institution of Great Britain," 1884.] (Head March 28, 1884.) IT has long been a matter of very general regret with those who are interested in natural philosophy, that in spite of the most strenuous efforts of the ablest mathematicians, the theory of fluid motion fits very ill with the actual behaviour of fluids ; and this for unexplained reasons. The theory itself appears to be very tolerably complete, and affords the means of calculating the results to be expected in almost every case of fluid motion, but while in many cases the theoretical results agree with those actually obtained, in other cases they are altogether different. If we take a small body such as a raindrop moving through the air, the theory gives us the true law of resistance ; but if we take a large body such as a ship moving through the water, the theoretical law of resistance is altogether out. And what is the most unsatisfactory part of the matter is that the theory affords no clue to the reason why it should apply to the one class more than the other. When, seven years ago, I had the honour of lecturing in this room on the then novel subject of vortex motion, I ventured to insist that the reason why such ill success had attended our theoretical efforts was because, owing to the uniform clearness or opacity of water and air, we can see nothing of the internal motion ; and while exhibiting the phenomena of vortex rings in water, rendered strikingly apparent by partially colouring the water, but otherwise as strikingly invisible, I ventured to predict that the more general application of this method, which I may call the method of colour-bands, 154 ON THE TWO MANNERS OF MOTION OF WATER. [48 would reveal clues to those mysteries of fluid motion which had baffled philosophy. To-night I venture to claim what is at all events a partial verification of that prediction. The fact that we can see as far into fluids as into solids naturally raises the question why the same success should not have been obtained in the case of the theory of fluids as in that of solids ? The answer is plain enough. As a rule, there is no internal motion in solid bodies ; and hence our theory based on the assumption of relative internal rest applies to all cases. It is not, however, impossible that an, at all events seemingly, solid body should have internal motion, and a simple experiment will show that if a class of such bodies existed they would apparently have disobeyed the laws of motion. These two wooden cubes are apparently just alike, each has a string tied to it. Now, if a ball is suspended by a string you all know that it hangs vertically below the point of suspension or swings like a pendulum. You see this one does so. The other you see behaves quite differently, turning up sideways. The effect is very striking so long as you do not know the cause. There is a heavy revolving wheel inside which makes it behave like a top. Now what I wish you to see is, that had such bodies been a work of nature so that we could not see what was going on if, for instance, apples were of this nature while pears were what they are the laws of motion would not have been discovered ; if discovered for pears they would not have applied to apples, and so would hardly have been thought satis- factory. Such is the case with fluids : here are two vessels of water which appear exactly similar even more so than the solids, because you can see right through them and there is nothing unreasonable in supposing that the same laws of motion would apply to both vessels. The application of the method of colour-bands, however, reveals a secret : the water of the one is at rest, while that in the other is in a high state of agitation. I am speaking of the two manners of motion of water not because there are only two motions possible ; looked at by their general appearance the motions of water are infinite in number; but what it is my object to make clear to-night is that all the various phenomena of moving water may be divided into two broadly distinct classes, not according to what with uniform fluids are their apparent motions, but according to the internal motions of the fluids, which are invisible with clear fluids, but which become visible with colour-bands. The phenomena to be shown will, I hope, have some interest in them- 48] ON THE TWO MANNERS OF MOTION OF WATER. 155 selves, but their intrinsic interest is as nothing compared to their philosophical interest. On this, however, I can but slightly touch. I have already pointed out that the problems of fluid-motion may be divided into two classes: those in which the theoretical results agree with the experimental, and those in which they are altogether different. Now what makes the recognition of the two manners of internal motion of fluids so important, is that all those problems to which the theory fits belong to the one class of internal motions. The point before us to-night is simple enough, and may be well expressed by analogy. Most of us have more or less familiarity with the motion of troops, and we can well understand that there exists a science of military tactics which treats of the best manoeuvres and evolutions to meet particular circumstances. Suppose this science proceeds on the assumption that the discipline of the troops is perfect, and hence takes no account of such moral effects as may be produced by the presence of an enemy. Such a theory would stand in the same relation to the movements of troops, as that of hydrodynamics does to the movements of water. For although only the disciplined motion is recognised in military tactics, troops have another manner of motion when anything disturbs their order. And this is precisely how it is with water: it will move in a perfectly direct disciplined manner under some circumstances, while under others it becomes a mass of eddies and cross streams, which may be well likened to the motion of a whirling, struggling mob where each individual particle is obstructing the others. Nor does the analogy end here : the circumstances which determine whether the motion of troops shall be a march or a scramble, are closely analogous to those which determine whether the motion of water shall be direct or sinuous. In both cases there is a certain influence necessary for order : with troops it is discipline ; with water it is viscosity or treacliness. The better the discipline of the troops, or the more treacly the fluid, the less likely is steady motion to be disturbed under any circumstances. On the other hand, speed arid size are in both cases influences conducive to un- steadiness. The larger the army, and the more rapid the evolutions, the greater the chance of disorder ; so with fluid, the larger the channel, and the greater the velocity, the more chance of eddies. With troops some evolutions are much more difficult to effect with steadiness than others, and some evolutions which would be perfectly safe 156 ON THE TWO MANNERS OF MOTION OF WATER. [48 on parade, would be sheer madness in the presence of an enemy. So it is with water. One of my chief objects in introducing this analogy of the troops is to emphasise the fact, that even while executing manoeuvres in a steady manner, there may be a fundamental difference in the condition of the fluid. This is easily realised in the case of troops. Difficult and easy manoeuvres may be executed in equally steady manners if all goes well, but the conditions of the moving troops are essentially different. For while in the one case any slight disarrangement would be easily rectified, in the other it would inevitably lead to a scramble. The source of such a change in the manner of motion under such circumstances, may be ascribed either to the delicacy of the manoeuvre, or to the upsetting disturbance, but as a matter of fact, both of these causes are necessary. In the case of extreme delicacy an indefinitely small disturbance, such as is always to be counted on, will effect the change. Under these circumstances we may well describe the condition of the troops in the simple manoeuvre as stable, while that in the delicate man- oeuvre is unstable, i.e. will break down on the smallest disarrangement. The small disarrangement is the immediate source of the break-down in the same sense as the sound of a voice is sometimes the cause of an avalanche ; but if we regard such disarrangement as certain to occur, then the source of the disturbance is a condition of instability. All this is exactly true for the motion of water. Supposing no disarrange- ment, the water would move in the manner indicated in theory just as, if there is no disturbance, an egg will stand on its end ; but as there is always slight disturbance, it is only when the condition of steady motion is more or less stable that it can exist. In addition then to the theories either of military tactics or of hydrodynamics, it is necessary to know under what circum- stances the manoeuvres of which they treat are stable or unstable. And it is in definitely separating these conditions that the method of colour-bands has done good service which will remove the discredit in which the theory of hydrodynamics has been held. In the first place, it has shown that the property of viscosity or treacliness, possessed more or less by all fluids, is the general influence conclusive to steadiness, while, on the other hand, space and velocity are the counter influence ; and the effect of these influences is subject to one perfectly definite law, which is that a particular evolution becomes unstable for a definite value of the viscosity divided by the product of the velocity and space. This law explains a vast number of phenomena which have hitherto appeared paradoxical. One general conclusion is, that with sufficiently slow motion all manners of motion are stable. 48] ON THE TWO MANNERS OF MOTION OF WATER. 157 The effect of viscosity is well shown by introducing a band of coloured water across a beaker filled with clear water at rest. Now the water is quite still, I turn the beaker round about its axis. The glass turns but not the water, except that which is close to the glass. The coloured water which is close to the glass is drawn out into what looks like a long smear, but it is not a smear, it is simply a colour-band extending from the point in which the colour touched the glass in a spiral manner inwards, showing that the viscosity was slowly communicating the motion of the glass to the water within. To prove this I have only to turn the beaker back, and the colour- band assumes its radial position. Throughout this evolution the motion has been quite steady quite according to the theory. When water flows steadily it flows in streams. Water flowing along a pipe is such a stream bounded by the solid surface of the pipe, but if the water be flowing steadily we can imagine the water to be divided by ideal tubes into a fagot of indefinitely small streams, aoy of which may be coloured without altering its motion, just as one column of infantry may be distin- guished from another by colour. If there is internal motion, it is clear that we cannot consider the whole stream bounded by the pipe as a fagot of elementary streams, as the water is continually crossing the pipe from one side to the other, any more than we can distinguish the streaks of colour in a human stream in the corridor of a theatre. Solid walls are not necessary to form a stream : the jet from a fire hose, the falls of Niagara, are streams bounded by a free surface. A river is a stream half bounded by a solid surface. Streams may be parallel, as in a pipe ; converging, as in a conical mouth- piece ; or when the motion is reversed, diverging. Moreover, the streams may be straight or curved. All these circumstances have their influence on stability in a manner which is indicated in the accompanying table : Circumstances conducive to Direct or Steady Motion. Viscosity or fluid friction which continually destroys disturb- ances. (Treacle is steadier than water.) 2. A free surface. 3. Converging solid boundaries. 4. Curvature with the velocity greatest on the outside. Sinuous or Unsteady Motion. 5. Particular variation of velocity across the stream, as when a stream flows through still water. 6. Solid bounding walls. 7. Diverging solid boundaries. 8. Curvature with the velocity greatest on the inside. 158 ON THE TWO MANNERS OF MOTION OF WATER. [48 It has for a long time been noticed that a stream of fluid through fluid otherwise at rest is in an unstable condition. It is this instability which gives rise to the talking-flame and sensitive-jet with which you have been long familiar in this room. I have here a glass vessel of clear water in front of the lantern, so that any colour-bands will be projected on the screen. You see the ends of two vertical tubes one above the other. Nothing is flowing through these tubes, and the water in the vessel is at rest. I now open two taps, so as to allow a steady stream of coloured water to enter at the lower pipe, water flowing out at the upper. The water enters quite steadily, forms a sort of vortex ring at the end which proceeds across the vessel, and passes out at the lower tube. Now the coloured stream extends straight across the vessel, and fills both pipes. You see no motion ; it looks like a glass rod. The water is, however, flowing slowly along it. The motion is so slow, that the viscosity is paramount, and hence the stream is steady. I increase the speed ; you see a certain wriggling sinuous action in the column ; faster, the column breaks up into beautiful and well-defined eddies, and spreads out into the surrounding water, which, becoming opaque with colour, gradually draws a veil over the experiment. The same is true of all streams bounded by standing water. If the motion is sufficiently slow, according to the size of the stream and the viscosity of the fluid, it is steady and stable. At a certain critical velocity, which is determined by the ratio of the viscosity to the diameter of the stream, the stream becomes unstable. Under any conditions, then, which involve a stream flowing through surrounding water, the motion will be unstable if the velocity is sufficient. Now, one of the most marked facts relating to experimental hydro- dynamics is the difference in the way in which water flows along contract- ing and expanding channels ; these include an enormously large class of the motions of water, but a typical phenomenon is shown by the simple conical tubes. Such a tube is now projected on the screen ; it is surrounded with clear still water. The mouth of the tube at which the water enters is the largest part, and it contracts uniformly for some way down the channel, then the tube expands again gradually until it is nearly as large as at the mouth, and then again contracts to the tube necessary to discharge the water. I draw water through the tube, but you see nothing as to what is going on. I now colour one of the elementary streams outside the mouth ; this colour- band is drawn in with the surrounding water, and will show us what is going on. It enters quite steadily, preserving its clear streak-like character until it has reached the neck where convergence ceases ; now the moment it enters 48] ON THE TWO MANNERS OF MOTION OF WATER. 159 the expanding tube it is altogether broken up into eddies. Thus^the motion is direct in the contracting tube, sinuous in the expanding. The hydrodynamical theory affords no clue to the cause why ; and even by the method of colour-bands the reason for the sinuosity is not at once obvious. If we start the current suddenly, the motion is at first the same in both tubes, its change in the expanding pipe seemed to imply that here the motion was unstable. If so, this ought to appear from the equations ot motion. With this view this case was studied, I am ashamed to say how long, without any light. I then had recourse to the colour-bands again, to try and see how the phenomena came on. It all then became clear : there is an intermediate stage. When the tap is opened, the immediately ensuing motion is nearly the same in both parts ; but while that in the contracting portion maintains its character, that in the expanding portion changes its character. A vortex ring is formed which, moving forward, leaves the motion behind that of a parallel stream through the surrounding water. If the motion be sufficiently slow, as it is now, this stream is stable, as already explained. We thus have steady or direct motion in both the con- tracting and expanding parts of the tube, but the two motions are not similar : the first being one of a fagot of similar elementary contracting streams, the latter being that of one parallel stream through the surround- ing fluid. The first of these is a stable form ; the second an unstable form, and, on increasing the velocity, the first remains, while the second breaks down ; and we have, as before, the expanding part filled with eddies. This experiment is typical of a large class of motions. Wherever fluid flows through a narrow, as it approaches the neck it is steady, after passing, it is sinuous. The same effect is produced by an obstacle in the middle of a stream ; and very nearly the same thing by the motion of a solid object through the water. You see projected on the screen an object not unlike a ship. Here the ship is fixed, and the water flowing past it; but the effect would be the same if we had the ship moving through the water. In the front of the ship the stream is steady, and so till it has passed the middle, then you see the eddies formed behind the ship. It is these eddies which account for the discrepancy between the actual and theoretical resistance of ships. We see, then, that the motion in the expanding channel is sinuous because the only steady motion is that of a stream through water. Numerous cases in which the motion is sinuous may be explained in the same way, but not all. If we have a perfectly parallel channel, neither contracting nor expand- ing, the steady moving stream will be a fagot of perfectly steady parallel 160 ON THE TWO MANNERS OF MOTION OF WATER. [48 elementary streams all in motion, but moving fastest at the centre. Here we have no stream through steady water. Now when this investigation began it was not known, or imperfectly known, whether such a stream was stable or not, but there was a well-known anomaly in the resistance to motion in parallel channels. In rivers, and all pipes of sensible size, experience had shown that the resistance increased as the square of the velocity, whereas in very small pipes, such as represent the smaller veins in animals, Poiseuille had proved the resistance increased as the velocity. Now since the resistance would be as the square of the velocity with sinuous motion, and as the velocity, if direct, it seemed that the discrepancy could be accounted for if the motion could be shown to become unstable for a sufficiently large velocity. This suggested the experiment I am now about to produce before you. You see on the screen a pipe with its end open. It is surrounded by clear water, and by opening a tap I can draw water through it. This makes no difference to the appearance, until I colour one of the elementary streams, when you see a beautiful streak of colour extend all along the pipe. The stream has so far been running steadily, and appears quite stable. I now merely increase the speed ; it is still steady, but the colour-band is drawn down fine. I increase the colour and then again increase the speed. Now you see the colour-band at first vibrates and then mixes so as to fill the tube. This is at a definite velocity ; if the velocity be diminished ever so little the band becomes straight and clear ; increase it again, it breaks up. This critical speed depends on the size of the tube in the exact inverse ratio ; the smaller the tube, the greater the velocity ; also, the more viscous the water the greater the velocity. We have then not only a complete explanation of the difference in the laws of resistance generally experienced, and that found by Poiseuille, but also we have complete evidence of the instability of parallel streams flowing between or over solid surfaces. The cause of the instability is as yet not explained, but this much can be shown, that whereas lateral stiffness in the walls is unimportant, inextensibility or tangential rigidity is essential to the creation of eddies. I cannot show you this, because the only way in which we can produce the necessary conditions without a solid channel, is by a wind blowing over water. When the wind blows over water, it imparts motion to the surface of the water just as a moving solid surface ; moving in this way, however, the water is not susceptible of eddies. It is unstable, but the result of disturbance is waves. This is proved by an experiment long known, but which has recently attracted considerable notice. If oil be put on the surface it spreads out into an indefinitely thin sheet which possesses only one of the characteristics of a solid surface, it offers resistance, very slight, 48] ON THE TWO MANNERS OF MOTION OF WATER. 161 but still resistance to extension and contraction. This, however, is sufficient to entirely alter the character of the motion. It renders the water~unstable internally, and instead of waves, what the wind does is to produce eddies beneath the surface. This has been proved, although I cannot show you the experiments. To those who have observed the phenomena of oil preventing waves, there is probably nothing more striking throughout the region of mechanics. A film of oil so thin that we have no means of illustrating its thickness, and which cannot be perceived except by its effect which possesses no mechanical properties that can be made apparent to our senses is yet able to entirely prevent an action which involves forces the strongest we can con- ceive, which upset our ships and destroy our coasts. This, however, becomes intelligible when we perceive that the action of the oil is not to calm the sea by sheer force, but merely, as by its moral force, to alter the manner of motion produced by the action of the wind, from that of the terrible waves upon the surface, into the harmless eddies below. The wind throws the water into a highly unstable condition, into what morally we should call a condition of great excitement. The oil by an influence we cannot perceive directs this excitement. This influence, though insensibly small, is however now proved of a mechanical kind, and to me it seems that the phenomenon of one of the most powerful mechanical actions of which the forces of nature are capable, being entirely controlled by a mechanical force so slight as to be other- wise quite imperceptible, does away with every argument against the strictly mechanical sources of what we may call mental and moral forces. But to return to the instability in parallel channels. This has been the most complete, as well as the most definite result of the colour-bauds. The circumstances are such as to render definite experiments possible. These have been made, and reveal a definite law of the instability, which law has been tested by reference to all the numerous and important experiments on the resistance in channels by previous observers ; whereupon it is found that waters behave in exactly the same manner whether the channel, as in Poiseuille's experiment, is of the dimensions of a hair, or whether it be the size of a water main or of the Mississippi ; the only difference being that in order that the motions may be compared, the velocity must be inversely as the diameter of the pipe. But this is not the only point explained if we consider other fluids than water. Some fluids, like oil or treacle, apparently flow more slowly and steadily than water. This, however, is only in smaller channels ; the critical velocity increases with the viscosity of the fluid. Thus, while water in comparatively large streams is always above its critical velocity, and the motion always sinuous, the motion of treacle in streams of such size as we see is below its critical velocity, and the motion direct, o. R. ii. 11 162 ON THE TWO MANNERS OF MOTION OF WATER. [48 But if nature had produced rivers of treacle the size of the Thames, for instance, the treacle would have flowed just like water. Thus, in the lava streams from a volcano, although looked at close the lava has the consistence of a pudding, in the large and rapid streams down the mountain sides the lava flows as freely as water. I have now only one circumstance left to which to ask your attention. This is the effect of the curvature of the stream on the stability of the fluid. Here again we see the whole effect altered by very slight causes. If water be flowing in a bent channel in steady streams, the question as to whether it will be stable or not turns on the variation in the velocity from the inside to the outside of the stream. In front of the lantern is a cylinder with glass ends, so that the light passes through in the direction of the axis. The disk of light on the screen being the light which passes through this water, and is bounded by the circular walls of the cylinder. By means of two tubes temporarily attached, a stream of coloured water is introduced right across the cylinder extending from wall to wall; the motion is very slow, and the taps being closed, and the tubes removed, the colour-band is practically stationary. The vessel is now caused to revolve about its axis. At first, only the walls of the cylinder move, but the colour- band shows that the water gradually takes up the motion, the streak being wound off at the ends into a spiral thread, but otherwise remaining still and vertical. When the spirals meet in the middle, the whole water is in motion, but the motion is greatest at the outside, and is therefore stable. The vessel stops, and gradually stops the water, beginning at the outside. If the motion remained steady, the spirals would unwind, and the streak be restored. But the motion being slowest at the outside against the surface, you see eddies form, breaking up the spirals for a certain distance towards the middle, but leaving the middle revolving steadily. Besides indicating the effect of curvature, this experiment really illustrates the action of the surface of the earth on the air.moving over it; the varying temperature having much the same influence as the curvature of the vessel on stability. The air is unstable for a few thousand feet above the surface, and the motion is sinuous, resulting in the mixing of the strata, and pro- ducing the heavy cumulus clouds ; but above this the influence of temperature predominates, and clouds, if there are any, are of the stratus-form, like the inner spirals of colour. But it was not the intention of this lecture to trace the two manners of motion of fluids in the phenomena of Nature and Art, so I thank you for your attention. 49. ON THE THEORY OF THE STEAM-ENGINE INDICATOR*. [From the " Proceedings of the Institution of Civil Engineers, 1885."] "ON THE THEORY OF THE STEAM-ENGINE INDICATOR AND THE ERRORS IN INDICATOR-DIAGRAMS." By OSBORNE REYNOLDS, M.A., LL.D., F.R.S., M. Inst. C.E. SECTION I. INTRODUCTION. IN 1856 Hirn published an experimental comparison of the indicated work, with the work done on the brake, and came to the conclusion that, whatever might be the cause, the indicated work was too small, being only just equal to the brake-work, leaving no margin for the air-pump and the friction of the engine. This conclusion of Hirn's seems to have excited little notice. Rankine mentions it in " The Steam-engine," but expresses doubt whether it accords with subsequent experience, particularly that of marine engines. Since that time many engine-experiments have been made. It does not appear, however, that these have been made with a view to verify the indicator, but rather that the indicator-diagrams have been taken as data from which to determine the efficiency of the engines; nor has, so far as the Author is aware, any definite theory of the disturbances to which the diagram is subjected as yet been published. The importance of studying the disturbances, or, in other words, the errors in the diagrams, becomes evident, when it is considered to what an * Joint paper with A. W. Brightmore, D.Sc. 112 164 ON THE THEORY OF THE INDICATOR. [49 extreme extent the indicator is now trusted to give a true measure of the work on the piston. In ninety-nine cases out of every hundred, there is absolutely no check within 20 or 30 per cent. In some classes of engines (winding and pumping) the work they are performing is of a measurable kind, but rarely or never is the work measurable to within 5 or 10 per cent. The only work which is definitely measurable is that done on the friction- brake as used by the Royal Agricultural Society ; and even then, although the brake may give a measure of the actual work to within 1 per cent, or less, it does not furnish a check on the indicator to within from 5 to 20 per cent., for between the work measured by the indicator and that measured by the brake, is the unknown work done in overcoming the resistance of the engine. This, which varies from 5 to 20 per cent., is an absolutely unknown quantity, except in so far as it is found by subtracting the brake- power from the indicated-power, and hence furnishes no check within its own magnitude on these quantities. There is thus absolutely no check on the indicator, which is now made the sole standard, not only of the performance, but of the value of engines. Considering what this means in mere money, where, as in the case of marine engines, large sums often depend on a margin of power which is a very small percentage of the whole, it becomes evident how important it is that the exact extent to which these instruments can be trusted should be well known. Yet, in spite of Hirn's warning, the results of the indicator appear to be accepted without question, solely on the ground of their general consistency, of the simplicity of its apparent action, and the excellence of its construction. On close examination, it appears in this case, as in others, that the apparent simplicity of action is due to the obscurity of certain facts ; for example, the possible stretching of a piece of string; and that, taking all the circumstances which may affect the diagram into account, its action is by no means a simple matter. It may be that, in some cases, these dis- regarded circumstances only produce an inappreciable effect, but even this cannot be known as long as they are disregarded. The theory of the indicator has now been taught for many years in the engineering classes in Owens College, Manchester, and the calculations to a certain extent have been verified by experiments on the College engine. This engine, though by no means of a high class, has been rendered well adapted for this purpose by the addition of a brake-dynamometer and a speed-indicator. It has long been the Author's intention to publish this theory, but this has been deferred for want of time to make a sufficiently extensive series of experiments. Last year Mr Brightmore, Berkeley Fellow in Owens College, Manchester, undertook the experiments and 49] ON THE THEORY OF THE INDICATOR. 165 carried them out very successfully. The results of his investigation appear to be of considerable importance, and as their interpretation depends on the theory, an account of this is submitted, to be read in conjunction with a Paper by Mr Brightmore. For the diagram to be exact, it is necessary 1. That the pencil of the indicator shall, under every change of pressure, instantly move through a distance in exact proportion to the change of pressure in the cylinder of the engine. 2. That the paper on which the diagram is taken shall change its position in exact accordance with the change of position of the piston of the engine. The first of these is accomplished, so far as it is accomplished, by holding the piston of the indicator by a spring, carefully adjusted, so that the deflection is proportional to the load ; and as there is no great difficulty in making a spring such that this proportion shall be maintained so long as the temperature is constant, and in making the instrument so that the temperature of the spring shall be 212 Fahrenheit, there is no reason to suppose that the indications of the indicator are not within 1 per cent, of the forces at each instant deflecting the spring. But in order that these indications may correspond with the pressures of steam, it is necessary that there should be no other forces acting on the spring. Such forces, however, arise from the inertia of the weights to be moved and the friction, notably that entailed by the necessity of pressing the pencil on the paper. In assuming the indicator as accurate, it is supposed that the forces resulting from inertia and friction are too small to be perceived; whether this is so or not, can only be ascertained by considering these forces. The second of these conditions of exactness is accomplished by connect- ing a revolving drum, by means of mechanism, with the piston of the engine, so that, if there is no yielding in the mechanism, the drum will revolve through distances exactly proportional to the distance moved by the piston of the engine. There is no difficulty in arranging mechanism which will secure the corresponding motion of two bodies, if the forces can be kept constant on the mechanism. This is attempted in the indicator by pulling the drum in one direction by a spring, and connecting it with the piston by means of a cord wound round the drum, so that the spring always keeps the string in tension. Since all strings in fact, all matter is elastic, in order that the position of the drum may always correspond with the position of the engine-piston, it is necessary that the spring shall 166 ON THE THEORY OF THE INDICATOR. [49 exert a constant force in all positions of the drum, and that there shall be no other forces. As a matter of fact, however, the springs used do not exert a constant force, the force increasing as the drum is moved against the spring ; and further, there are forces, namely, the forces arising from the inertia of the drum and the friction of the mechanism, principally of the drum on its supports. The diagram will, therefore, only be accurate in so far as these unequal forces are small ; and the effect of these forces can only be ascertained after careful consideration. It thus appears that there are five principal causes of disturbance ; two of these (1) and (2) affect the motion of the pencil, and three (3) (4) and (5) the motion of the drum. (1) The inertia of the piston of the indicator and its attached weights. (2) The friction of the pencil on the paper and its attached mechanism. (3) Varying action of the spring. (4) Inertia of the drum. (5) Friction of the drum. These will be separately considered. SECTION II. DISTURBANCES ON THE PENCIL. (1) The effect of the inertia of the Pencil and its attached Mechanism. This, although obvious enough in a general way, presents the same problem as the planetary disturbances, which can only be definitely expressed by means of some form of mathematics. As the general solution of the problem is well known to mathematicians, and is unintelligible to those who are not, it will be best here to omit all the steps, and to proceed at once to the results, about which there can be no question. These results may be best expressed in symbols, of which the meaning is as follows ; taking Ibs., feet, and seconds as general units, then put i for the indicated pressure at any instant ; p for the actual pressure corresponding to i ; w for the weight of any particular piece of mechanism attached to the pencil ; r for the ratio which the motion of this weight bears to the motion of the piston of the indicator ; 49] ON THE THEORY OF THE INDICATOR. 167 W for 2 (r*w) where 5 expresses the sum of all the quantities in the brackets ; g for 32'2, the acceleration of gravitation ; e for the number of Ibs. to the inch on the diagram ; a for the area of the piston of the indicator in square inches ; s for the ratio of the motion of the pencil to that of the piston of the indicator. In Richards' indicator a = 0'5 ; F = 0'33; s = 4. For other indicators these may be found by measurement. The relation between i and p, in so far as it is affected by inertia, is expressed by the equation W fJ 2 i - -U n ' nn (~\ } The general solution to this equation is well known, and without going into detail, it will be sufficient to give the solution for the case, which is, N being the number of revolutions of the engine per minute . . . 27rJV, 900xl2a^!^ Sm 3o' + ^ Sm ^O-' &c. (7 sin in \/ I2aesg .t ..(2). t expresses time in seconds ; P! greatest pressure ; p s least back pressure ; A 1} A z are coefficients depending on the shape of the true diagram ; C is a constant depending on the disturbed state of the pencil. From equation (2) it appears that the effect of inertia is to cause two disturbances, corresponding to the two terms on the right-hand side. These may be considered separately. The first term has the factor , . A 1 sm -- t sm . t + &c., which will go through a complete cycle when t changes by 60 JV ' 168 ON THE THEORY OF THE INDICATOR. [49 that is, by the time of revolution of the engine in seconds. This disturbance will be the same during each revolution of the engine, and will be called the cyclic disturbance. Given the shape of the true diagram, it would be possible to determine A lt A 2 so as to find from equation (2) the value of i p. But this would be a very complicated piece of work for such an irregular curve as the diagram, and as the object is not so much to find the magnitude as to find when this is small, it is sufficient to consider a circular or elliptic diagram ; for such a diagram it is found that the mean difference of i and p, written i p, is given by 7T 22 the positive sign to be taken for the forward stroke and the negative for the backward. If this effect were large compared with the mean acting pressure ^ o , then in all probability the area as well as the form of a true diagram would be seriously disturbed ; but if this effect is small, say 1 per cent, in the case of the oval, it will be small for the true diagram. Hence the increase of area is less than 1 per cent, so long as 2WWTT 12 x 900cMW.gr < and from this it is found that the cyclic disturbance may be 1 per cent, for Richards' indicator when N and e have the values in Table I., and as N* this disturbance increases as --, its possible values for all other cases may be found. TABLE I. ENGINE SPEEDS AT WHICH THE ENLARGEMENT OF THE DIAGRAM BY INERTIA BECOMES 1 PER CENT. WITH THE RICHARDS' INDICATOR USED IN THIS INVESTIGATION. Scale of Diagram Number of in Ibs. to an inch. Revolutions. 20 166 30 203 40 237 50 262 60 288 70 312 80 332 90 352 100 371 49] ON THE THEORY OF THE INDICATOR. 169 In the case of the oval or circular diagram the effect of_this cyclic disturbance would be to increase the vertical diameter, as shown by the dotted line in Fig. 1. What it would be on the true diagram is very difficult to express, except to say that it would be to round-off all corners and increase its size much in the same way as in the oval. The second term in equation (2) represents a disturbance which goes through its cycle in an interval of T seconds, where T = W ISaesg .(4). This may be called the vibratory disturbance. The period represented by r is that in which the pencil vibrates when disturbed. Such disturbances are introduced by the departure of the diagram from the true ellipse. Fig. 1. The result of such disturbance is shown by the waving line in Fig. 1. The time occupied in completing each one of these waves as from p l to p t is constant, viz. r equation (4). Hence the number of waves in a complete revolution is given by n = 60 N 27T W I2aesg .(5). 170 ON THE THEORY OF THE INDICATOR. [49 For Richards' indicator In the diagram, owing to the unequal motion of the engine-piston, the lengths of these oscillations increase from the ends to the middle. If, however, a circle be drawn on the atmospheric line AB, having the extreme length of the diagram as diameter, this may be taken to represent the crank-circle on the same scale as AB represents the stroke. Then if the points p l9 p. 2 &c., in which the waving line cuts the mean line, are first projected perpendicularly on to AB in P 1} P 2 &c., and then P l} P 2 projected by means of a radius to represent the connecting-rod on to the crank-circle in the points c l} c 2 &c., it will be found that the arcs CjCa, C 2 c 3 , are all equal, since the crank turns through equal arcs in equal times. But for the effects of friction these oscillations, once set up, would go on for ever ; so that even at low speeds a fair diagram would be impossible. By friction the oscillations are gradually destroyed, so that they are more or less localized to the neighbourhood of the points at which they are produced, i.e., the points where the curvature in the true diagram is sharp, particularly at the point of admission where the rise of pressure being instantaneous acts the part of a live-load, and forces the pencil twice as far as it ought to go. This sets up a series of oscillations. It is seldom that the time of oscillation is exactly commensurable with that of revolution, so that if all the oscillations set up in one revolution are not destroyed by friction before the revolution is complete, the pencil will not describe the same path in two successive revolutions, a fact frequently observed in diagrams taken from locomotives at high speed. The error which these oscillations cause in the area of the diagram depends on their magnitude, but also, and to a greater extent, on the small- ness of n, the number in a revolution. But the evil of these oscillations is not so much an effect on the area, which, even did they exist to the extent shown in Fig. 1, in which n is between six and seven, would still be small. It is the disfigurement and the confusion they produce in the diagram which limits the usefulness of the instrument to cases in which they can be avoided. So long as there are thirty of these oscillations in a cycle the necessary fluid friction of the indicator-piston will so far reduce them as to render a fair diagram possible, but when the number approaches fifteen it becomes necessary to call in the aid of considerable pencil-pressure to prevent their ' destroying the form of the diagram ; and when n is as low as ten it is all the pencil will do to prevent them upsetting the diagram. The Author 49] ON THE THEORY OF THE INDICATOR. 171 has never been able to produce a respectable diagram when th<3 number is as low as ten, but accounts are continually published in which from the speed of the engine and strength of the springs the value of n must be below this. In such cases the pressure of the pencil must have been very great, and it becomes a question how far this cure is a less evil than the disease. (2) The Friction arising from the Pressure of the Pencil. This always acts to oppose the motion of the pencil, and therefore renders it too large during expansion and exhaust, and too small during compression and admission, and thus the general effect is to increase the size of the diagram. In order to understand this effect, it is necessary to notice that this friction consists of two parts : (1) That of the pencil on the paper. (2) That of the mechanism, caused by sustaining the pressure of the pencil. The effect of the actual friction of the pencil is greatly reduced by the motion of the paper. Thus, if while the drum is at rest, the pencil be lifted quietly it will be possible for friction to hold it above or below the atmospheric-line, by a distance depending on the pressure. If, when placed as high or low as it will stand, the drum be moved by the cord, the pencil at once approaches the atmospheric-line, describing a line as shown in Fig. 2 Fig. 2. at first sloping toward the atmospheric-line at 45, but finally becoming parallel. Fig. 2 represents the results with a 20-lb. spring ; the distance at starting was equal to about .4 Ibs., but eventually became about lb., at which it remained constant. The distance at starting represents the extreme friction of pencil and mechanism. The final distance that of the mechanism alone. Fig. 3. These effects on the diagram are different. That of the pencil causes 172 ON THE THEORY OF THE INDICATOR. [49 the pencil to be behind its true position, by a quantity which will bear to the extreme distance, a ratio equal to the sine of the inclination of the curve it is describing at the instant, to the atmospheric-line. The effect of this alone on a rectangular diagram would be to round off the corners as in Fig. 3. With an early cut off, the effect would be as shown in Fig. 4. Fig. 4. The friction of the mechanism causes the pencil to be behind its true position by a nearly constant quantity, and hence during expansion and exhaust the pencil will be too high, and during compression and admission the pencil will be too low. This is shown in Fig. 5. Its effect on the area of the diagram is therefore not very great. Fig. 5. The magnitude of these effects, taken together, on the area of the diagram, depends on the construction of the instrument and on pencil- pressure. From numerous experiments with Richards' and Thomson's indicators, it was found that a comparatively slight alteration of pencil- pressure from that just sufficient to mark the diagram, would cause an excess of 0'5 Ib. during expansion, and an equal fall during compression. While if pencil-pressure were made sufficient to prevent serious oscillations when n=15, the mean acting pressure was affected by as much as 1'5 Ib. Thus it would appear possible to make a difference of as much as 5 per cent, in a locomotive in mid-gear by pencil- friction. The conclusions, then, as regards the motion of the pencil, are, that the general effects of inertia and friction are both to increase the size of the diagram ; that so long as the speeds are such that n is not greater than 15, 49] ON THE THEORY OF THE INDICATOR. 173 the effect of inertia is less than 1 per cent., but that if n is less than 30, oscillations will show themselves unless the pencil-friction He Increased. They may, by this, be kept down till w = 15, but not farther, and then the necessary friction will affect the area of the diagram about 5 per cent. A speed, therefore, which makes w = 15 is about the limiting speed at which diagrams can be taken accurate to 5 per cent., while for the diagrams to be sensibly accurate and free from oscillation the speeds must not be greater than will make n = 30. These speeds for Richards' indicators are given in Table II. TABLE II. N x- e 20 30 40 50 60 70 80 90 100 SECTION III. DISTURBANCES ON THE DRUM. These are the disturbances (3), (4), (5), section (1). They arise from the elasticity of the cord and mechanism connecting the drum with the piston of the engine. In order to express them definitely I is the indicated length of the diagram in inches ; y the yielding of the mechanism in inches per Ib. of the tension ; / the moment of inertia of the drum. (3) The Inertia of the Drum. If the obliquity of the connecting-rod of the engine be disregarded, and x be put for the distance OP (Fig. 1), the force arising from inertia is proportional to N*x and the disturbance arising from this cause will be yIN-x. And as x will be positive or negative ac- cording as P is to the right or left of 0, the diagram will be uniformly elongated. The effect of the obliquity of the connecting-rod would be to increase ?i=30 n=15 69 138 85 170 99 198 105 210 120 240 130 260 139 278 147 294 155 310 174 ON THE THEORY OF THE INDICATOR. [49 this elongation at the back-end and diminish it at the front, increasing the area of the back-end diagram, and diminishing that of the front somewhat, but it is small unless the connecting rod is very short. (4) The effect of the varying Stiffness of the Spring. Let q be the difference of tension of the spring at the extreme ends of the diagram. Then the disturbance of the point P will be I This effect is therefore opposite to that of (3), and the joint effect will be and since IN 2 will be zero at small speeds, and it increases as the square of the speed, when the speed is low the diagram will be qy too short, but as the speed increases this shortening will diminish until at some speed IN Z = j , I and for higher speeds the diagram will be elongated. With the Richards' indicator, the critical speed appears to be 150 = ^. In most diagrams these effects are apparent, but, except when the connecting-rod is short, they do not affect the indicated pressure. (5) The effect of the Friction of the Drum. Let F be the tension on the string necessary to overcome the friction of the drum in either direction. Then during the forward stroke the string will be stretched from this cause yF, and during the backward stroke it will be shortened yF. The effect will be to place the drum always behind its true position by yF. This is shown in Fig. 6. AC&, &c. represent the positions of the crank on its circle, as explained in reference to Fig. 1; but in this case CiC 2 , &c. are chosen so as to correspond with the equidistant positions of the piston. Projecting dc 2 with the con- necting-rod as radius on to the atmospheric-line the points are obtained in which, for a true diagram, the pencil would be when the crank was in the positions C&, &c., but owing to the cause under consideration, as the crank moves from A towards B, the pencil will be (at the points Deflec- tion under lib. .2 '~ '.Q o> a I " H "1 3 || |j S d^ "! 3 to 6 .a a T ~ < qU o IN Reading of Micro- meter. S o rH 3 Weight in Scale Pan. 0-6829 0-4887 0-4888 0-4832 0-4791 0-0246 0-0156 0-0157 0-0099 0-0060 1 0-7075 0-5043 0-5045 0-4931 0-4860 0-0242 0-0156 0-0154 0-0104 0-0061 2 0-7317 0-5199 0-5199 0-5035 0-4921 0-0245 0-0154 0-0154 0-0101 0-0063 3 0-7562 0-5353 0-5353 0-5136 0-4984 0-0243 0-0153 0-0152 0-0101 0-0062 4 0-7805 0-5506 0-5505 0-5237 0-5046 0-0240 0-0156 0-0153 o-oioo 0-0062 5 0-8045 0-5662 0-5658 0-5337 0-5108 0-0240 0-0154 0-0152 o-oioo 0-0062 6 0-8285 0-5816 0-5810 0-5437 0-5170 0-0245 0-0153 0-0153 0-0095 0-0062 7 0-8530 0-5969 0-5963 0-5532 0-5232 0-0250 0-0157 0-0156 0-0099 0-0062 8 0-8780 0-6126 0-6119 0-5631 0-5294 0-0250 0-0157 OO155 0-0098 0-0062 9 0-9030 0-6283 0-6274 0-5729 0-5356 0-0248 0-0157 0-0153 0-0100 0-0061 10 0-9278 0-6440 0-6427 0-5829 0-5417 0-0249 0-0155 0-0157 0-0101 0-0060 11 0-9527 0-6595 0-6584 0-5930 0-5477 0-0247 0-0155 0-0154 0-0099 0-0060 12 0-9774 0-6750 0-6738 0-6029 0-5537 0-0246 0-0158 0-0160 0-0099 0-0059 13 1-0020 0-6908 0-6898 0-6127 0-5596 0-0244 0-0156 0-0156 o-oioo 0-0060 14 1-0264 0-7064 0-7054 0-6227 0-5656 0-0243 0-0158 0-0155 o-oioo 0-0061 15 1-0507 0-7222 0-7209 0-6327 0-5717 0-0246 0-0156 0-0161 o-oioo 0-0059 16 1-0753 0-7378 0-7370 0-6427 0-5776 0-0238 0-0157 0-0159 o-oioo 0-0061 17 1-0991 0-7535 0-7529 0-6527 0-5837 0-0251 0-0156 0-0154 o-oioo 0-0059 18 1-1242 0-7691 0-7683 0-6627 0-5896 0-0244 0-0150 0-0151 o-oioo 0-0062 19 1-1486 0-7841 0-7834 0-6727 0-5958 0-0242 0-0152 0-0154 o-oioo 0-0072 20 1-1728 0-7993 0-7988 0-6827 0-6030 0-0152 0-0156 0-0102 0-0062 21 ... 0-8145 0-8144 0-6929 0-6092 ... 0-0154 0-0155 0-0101 0-0062 22 0-8299 0-8299 0-7030 0-6154 0-0156 0-0153 0-0103 0-0061 23 ... 0-8455 0-8452 0-7133 0-6215 0-0151 0-0155 o-oioo 0-0063 24 ... 0-8606 0-8607 0-7233 0-6278 0-0152 0-0160 0-0101 0-0062 25 ... 0-8768 0-8767 0-7334 0-6340 0-0099 0-0062 26 ... 0-7433 0-6402 ... 0-0102 0-0066 27 ... 0-8535 0-6468 * Different maker. TABLE II. DEFLECTION OF SPRINGS, WHEN HOT. Strength of Spring, 20 32 32* 50 80 Ibs. to the Inch. 60 6 . .> ec 6 . 60 . .0 eco .a 2f2 .2 o t) __ , i ? B y ^ rH *"" pi *"* 8 o"" S 8 b 8 c^ .H b H 8 a'"' Weight in g s "s 0) * .2 QJ |l| ||| 41 S -J M |g| * s 3) '^3 ^ !! S * Scale Pan. * -g 3 * o C 3 a w's 3 rt^ 0-6872 0-4941 0-4937 0-4862 0-5953 01)245 0-0154 0-0162 OO104 0-0063 1 0-7117 0-5095 0-5099 0-4966 0-6016 ()-()-2~)[ 0-0160 0-0157 OO108 0-0065 2 0-7368 0-5255 0-5256 0-5074 0-6081 0-0251 0-0158 0-0157 0-0103 0-0064 3 0-7619 0-5413 0-5413 0-5177 0-6145 0-0249 0-0159 0-0156 0-0103 0-0062 4 0-7868 0-5572 0-5569 0-5280 0-6207 0-0250 0-0157 0-0157 0-0103 0-0063 5 0-8118 0-5729 0-5726 0-5383 0-6270 0-0251 0-0162 0-0156 0-0103 0-0063 6 0-8369 0-5891 0-5882 0-5486 0-6333 0-0253 0-0156 0-0157 0-0098 0-0063 7 0-8622 0-6047 0-6039 0-5584 0-6396 0-0258 0-0162 0-0164 0-0105 0-0064 8 0-8880 0-6209 0-6203 0-5689 0-6460 0-0258 0-0163 0-0159 0-0104 0-0066 9 0-9138 0-6372 0-6362 0-5793 0-6526 0-0256 0-0160 0-0160 0-0099 0-0062 10 0-9394 0-6532 0-6522 0-5892 0-6588 0-0259 0-0163 0-0158 0-0102 0-0062 11 0-9653 0-6695 0-6680 0-5994 0-6650 0-0258 0-0160 0-0162 0-0104 0-0064 12 0-9911 0-6855 0-6842 0-6098 0-6714 0-0250 0-0162 0-0158 0-0098 OO066 13 1-0161 0-7017 0-6700 0-6196 0-6780 0-0250 0-0159 0-0162 0-0106 0-0065 14 1-0411 0-7176 0-7162 0-6302 0-6845 0-0246 0-0163 0-0163 0-0105 0-0062 L5 1-0657 0-7339 0-7325 0-6407 0-6907 OO259 0-0164 0-0161 0-0101 0-0063 16 1-0916 0-7503 0-7486 0-6508 0-6970 0-0258 (t) 0-0161 0-0104 0-0067 17 M174 0-7689 0-7647 0-6612 0-7037 ()-<\-2:r2 0-0160 0-0158 0-0105 0-0064 18 1-1426 0-7849 0-7805 0-6717 0-7101 0-0242 0-0159 0-0157 0-0105 0-0061 19 1-1668 0-8008 0-7962 0-6822 0-7162 . 0-0159 0-0160 0-0106 0-0065 20 ... 0-8167 0-8122 0-6928 0-7227 0-0159 OO157 0-0102 0-0062 21 0-8326 0-8279 0-7030 0-7289 0-0163 0-0170 0-0105 0-0063 22 ... 0-8489 0-8449 0-7135 0-7352 0-0162 0-0156 0-0064 n 0-8(55 1 0-8605 0-7416 ... 0-0159 0-0160 . . 00063 24 ... 0-8810 0-8765 0-7479 . 0-0162 0-0164 ... 0-0063 25 0-8972 0-8929 0-7542 0-0062 26 ... ... 0-7602 Different maker. t Condensed steam let out of cylinder. 188 EXPERIMENTS ON THE STEAM-ENGINE INDICATOR. [49 A EFFECT OF INERTIA OF THE MOVING-PARTS ON THE AREA OF THE DIAGRAM. Having ascertained the errors in the springs, the next question was to find how far the effect of inertia tends to alter the area of the diagram before the oscillations appear. To do this, diagrams were taken at various speeds and with several springs. In Table IV. the efficiences, i.e., the ratios of the brake-pressures to the mean diagram-pressures, are given at the various speeds, instead of the mean pressures as calculated from the diagrams, on account of the difficulty of keeping the load on the brake exactly constant. TABLE III. MEAN DEFLECTIONS OF SPRINGS UNDER 1 Ib. Spring. Experimental Deflection, cold. Experimental Deflection, hot. Deflection from Mark on Spring. Percentage Error. Inch. Inch. Inch. 20 0-0245 0-02525 0-02523 0-08 32 0-0155 0-01600 0-01580 1-25 32* 0-0155 0-01595 0-01580 0-95 50 o-oioo 0-01030 0-01009 2-08 80 0-0062 0-00636 0-00630 0-94 * Different maker. Now if the inertia affects the areas of the diagrams, the areas of the diagrams, and hence the mean diagram- pressures, will vary directly with the velocity, and inversely as the stiffness of the spring (the weight on the brake being constant) ; i.e., the efficiencies will vary directly with the stiffness of the spring and with the inverse of the velocity. However, an examination of the Table shows no appreciable increase of the efficiency with greater stiffness of the spring, and no more decrease, as the velocity increases, than would be accounted for by the greater friction. Table IV. is not filled in for the 20 and 32 springs at the higher speeds, because the oscillations begin to come in. The inference is, " that in a given engine, when the ratio of the speed to the stiffness of the spring, used to indicate it, is not so great as to cause oscillations to appear in the diagram, the area is not appreciably affected by the momentum of the moving parts." This seems natural, for, after the initial disturbance on the admission of the steam to the cylinder, the motion 49 A] EXPERIMENTS ON THE STEAM-ENGINE INDICATOR. 189 of the spring is gradual, and hence its deflection would correspond to the pressure on it. TABLE IV. Speed. Efficiencies. Spring. 20 32 50 80 44 0-94 ... 0-95 ... 0-945 68 0-93 0-94 0-93 ... 0-933 84 ... 0-93 0-93 0-93 0-930 107 0-93 0-94 0-93 0-933 127 ... 0-93 0-92 0-925 OSCILLATIONS. When the ratio of the speed of the engine to the stiffness of the spring, used to indicate it, exceeds a certain value, which is different for different engines, oscillations appear in the diagram. The equation which gives the time of oscillation of the spring, modified by the parallel-motion bars (Fig. 16), devised by Professor Reynolds, is, taking the axis of x vertically upwards : W (1), 190 EXPERIMENTS ON THE STEAM-ENGINE INDICATOR. [49 A k- where W ' = W + (w + w 2 ) + 16^ , CL + (w + w. 2 )- + ^v l ], ct / and e = the stiffness of spring. W = weight of piston + weight of spring. w = weight of rod AD (Fig. 16). w 1 = weight of rod DF. w 2 = weight of rod FH. P = whole pressure of steam on the piston. a = AB. b = AC= GH = distance of centres of gravity of rods AD, FH from A and H respectively. k = radius of gyration of AD, FH, about A and H respectively. W is, in fact, the weight which would have to oscillate at B to be equivalent to the moving-parts, and the expression P Q represents the force which would have to be applied at B, if the parts referred to were removed, to be equivalent to them. Equation (1) is of the well-known form for finding the time of a complete oscillation (T), and then is obtained in the ordinary way g.e Or calling N the number of oscillations per minute 30 Ig7e_ ~ 7T V W" It will be noticed that in equation (1), the rotation of the rod DF, which is very slight, is neglected, as also is the friction of the instrument. In the case of the indicator employed, the values of the above constants were W* = 0-10529 Ib. Tf S2 = 0-10954 Ib. w = 0-00957 Ib. w, = 0-01037 Ib. w, = 0-00866 Ib. a = 0'75 inch. b = 1 inch. A* = 1-83. 49 A] EXPERIMENTS ON THE STEAM-ENGINE INDICATOR. 191 Whence from the above W n = 0'33063 Ib. ^32=0-33488 Ib. ;ui(l from preceding experiments 20 = 475. e : , 2 = 750. NOTE. The suffixes 20 and 32 refer to the springs marked 20 and 32 respectively. Thus N*> = 2050. ^2 = 2560. It will be noticed on substituting for W, that the rod DF has as much influence in causing the oscillations to come in as all the other moving parts together. To verify these results, diagrams were taken with weak springs, in order to bring in oscillations. It must be understood that the diagrams in this Paper are not intended as specimens of good diagrams, but are merely to illustrate the various points considered. The time of oscillation of the indicator-springs may be approximately obtained from such diagrams in the following manner: first, project the crests and hollows of the oscillations vertically down on to the atmospheric- line ; next, with a radius equal to the length of the connecting-rod (reduced to the same scale as the length of the diagram), and centre on the atmo- spheric-line produced, project the points so obtained upon a circle described on the atmospheric-line with the length of diagram as the diameter ; then the arcs of the circle intercepted between alternate intersections represent the angle turned through by the crank during the time of a complete oscillation of the spring. Hence, assuming that the crank-shaft rotates uniformly, these arcs would represent the time of a complete oscillation. There are several reasons why the number of oscillations per minute so obtained should not quite equal the number as obtained above from theory. Firstly, the neglect of the rotation of the bar DF, and of the friction in the equation, would make a slight difference ; but the most important reason is the gradual decrease of pressure in the cylinder of the engine, consequent upon the motion of the piston and initial condensation. This diminution of pressure causes the crests to lie behind, and the hollows to be in advance of their true position (Figs. 17 to 25), by an amount varying with the rate of decrease. Supposing for the moment the lag to be equal in amount for each crest, the projection of it (the lag) upon the crank-circle will include a greater arc towards the ends than in the middle of the diagram ; thus, other things being the same, causing the time of oscillation to appear 192 EXPERIMENTS ON THE STEAM-ENGINE INDICATOR. [49 A too great at one end of the diagram, and too small at the other end of the diagram. However, this tendency is counteracted, at least during the first half of the stroke (and it is during this period chiefly that the time of oscilla- tion is measured), by the retardation of the velocity of oscillation, and consequently the greater effect of the reduction of pressure in causing the crests to lag as the stroke progresses. That the velocity of oscillation decreases with the distance from the point of admission is seen by integrat- ing equation (1), where dx dt qe qe where c = the distance of a hollow from the atmospheric-line. Now 2 f c j is equal to the range of oscillation, as may be seen by again integrating equation (1), and in the case of the diagrams referred to, the range of oscillation, and hence from above, the velocity of oscillation of the spring diminished as the stroke advances, which is almost self-evident, for the time of oscillation is independent of the range, so that if the range be reduced the velocity must be reduced also. From equation (2) it is also seen that, other things being the same, the number of oscillations in a diagram increases with the stiffness of the spring, hence the counteracting effect, just referred to, would be less marked as the stiffness of the spring used is increased, so that for this reason the number of oscillations per minute as obtained from a diagram would be nearer the truth the weaker the spring. Again, the number of oscillations per minute will probably be nearer the truth the greater the speed of the engine ; for the number of oscillations in Fig. 17. Front-end diagram taken with 20 spring at 141 revolutions. a diagram is smaller the greater the speed of the engine, because the time of oscillation of the spring is independent of the speed of the engine, and 49 A] EXPERIMENTS ON THE STEAM-ENGINE INDICATOR. 193 hence the ratio of the velocity of oscillation to the rate of reduction of pressure is less the higher the speed of the engine, hence the counteracting effect referred to is greater. These two points are illustrated in the diagrams, Figs. 17 to 25, and the accompanying Table V. Fig. 18. Front-end diagram taken with 20 spring at 127 revolutions. Fig. 19. Front-end diagram taken with 20 spring at 107 revolutions. Fig. 20. Back-end diagram taken with 20 spring at 144 revolutions, o. it. ii. 13 194 EXPERIMENTS ON THE STEAM-ENGINE INDICATOR. [49 A Fig. 21. Back-end diagram taken with 20 spring at 127 revolutions. Fig. 22. Front-end diagram taken with 32 spring at 144 revolutions. Fig. 23. Front-end diagram taken with 32 spring at 127 revolutions. 49 A] EXPERIMENTS ON THE STEAM-ENGINE INDICATOR. 195 Fig. 24. Back-end diagram taken with 32 spring at 144 revolutions. Fig. 25. Back-end diagram taken with 32 spring at 127 revolutions. TABLE V. Speed. Revolutions per minute. End. Sprin. Number of Oscillations. Difference per cent. From Diagram. From Formula. 144 (Fig. 17) Front 20 1,950 2,050 5-0 127 (Fig. 18) 20 1,920 6'5 107 (Fig. 191 20 1,883 8-5 144 (Fig. 20) Back 20 1,950 5-0 127 (Fig. 21) 20 1,930 6-0 144 (Fig. 22) Front 32 2,370 2,560 7-5 127 (Fig. 23) 32 2,300 10-0 144 (Fig. 24) Back 32 2,300 10-0 127 (Fig. 25) > 32 2,300 lO'O In calculating the oscillations from the diagrams a mean value was taken. The distance to which the oscillations extend depends on the range of 132 196 EXPERIMENTS ON THE STEAM-ENGINE INDICATOR. [49 A the first one, and on the friction of the pencil. The range of the first oscil- lation is great if the period of a semi-oscillation nearly coincides with the time the steam takes to attain its maximum pressure on admission ; this happens when the engine is running fast. It is small when the time of attaining the greatest pressure of steam and the time of a semi-oscillation are not nearly equal. Thus, when the steam is wire-drawn on entering the cylinder of an engine, that engine would give a better diagram at high speeds than if this were not the case. Again, if the steam be throttled on entering the indicator, the time of the steam attaining its maximum pressure in the indicator-cylinder will be lengthened; hence the extent of the first oscillation will be reduced, and therefore the oscillations in the diagram will be reduced ; but the diagram so obtained does not give a correct idea of the work done, but is too small in proportion to the amount of throttling. The effect of the friction of the pencil in lessening the extent of the oscillations varies with the pressure on the pencil. When the oscillations are thus reduced by pressing the pencil on the paper an indefiniteness is introduced into the results, owing to the pencil sticking either too high or too low, and the results cannot be relied on. To illustrate this point diagrams were taken under the same conditions, of which the results are given in Table VI. In the case of the weaker springs, 20 Ibs. and 32 Ibs., the pencil was pressed on the diagram-paper so as to reduce the oscillations. Diagrams were taken with stiffer springs, in which oscillations do not perceptibly enter, to check the results so obtained. TABLE VI. FRONT-END EFFICIENCIES. Speed. 20 Spring (pencil pressed). 32 Spring (pencil pressed). 50 Spring. 80 Spring. 69 0-932 0-927 0-959 0-958 87 0-931 0-942 0-954 0-954 108 0-918 0-907 0-955 0-954 Mean efficiencies 0-927 0-925 0-956 0-955 Table VI. shows that in those experiments in which the pencil was pressed on to the paper the results are too small by more than 3 per cent. No 49 Aj EXPERIMENTS ON THE STEAM-ENGINE INDICATOR. 197 doubt if the engine had cut-off earlier, and been working with a higher pressure of steam, the results would have been still more discordant. Probably the most accurate method of arriving at the mean pressure when the oscillations extend a good way into the diagram, at least when the cut-off occurs late in the stroke as in the present case, is to draw a line midway between the crests and hollows, and to the value for the mean pressure obtained by taking this line add an amount, which in the case of indicators similar to the one employed in these experiments is 0'35 Ib. To see the reason for this, referring back to equation (1), and integrating it twice Iw Substituting in this t = IT A/ (time of half oscillation) 20 x = - c, e i.e., Q is the arithmetical mean of ex and ec. Substituting in this expression the value for Q, and taking the area of the indicator piston as 0'5 square inch, the following value for the intensity of pressure (p) is obtained : p = e (x + c) + 2 ( W+ (w -f w a ) - + 4wj ) . \ & / Hence if a line midway between the crests and hollows be taken as repre- senting the pressure, the mean pressure so obtained will be too small by the amount of the second term on the right, which for the indicator employed = 0'35 Ib. This would be negligible for any considerable pressure. It was found that with the indicator used, a diagram tolerably free from oscillations could be taken from the engine up to a speed of about 90 revolu- tions per minute, with a spring of 20 Ibs. to the inch. Hence, since the time the steam takes to attain its maximum pressure in the cylinder varies with the speed of the engine (in different engines it would also vary with the arrangement of the slide-valve), it might be expected to obtain a diagram tolerably free from oscillations at a speed of from 400 to 500 revo- lutions per minute, with an indicator having a parallel-motion in which the rod corresponding to DF is absent, and in which the other moving-parts are as light again as in the present case. This would be the case with an indicator of smaller diameter, in which a much stronger spring could be used for the same weight. For much higher speeds than this, unless the relative time occupied in attaining the maximum pressure increased with 198 EXPERIMENTS ON THE STEAM-ENGINE INDICATOR. [49 the speed, it would appear that the diagrams would be affected to a great but unknown extent by the oscillations of the spring. VITIATION OF THE DIAGRAM BY THE STRETCHING OF THE INDICATOR-CORD. The effect of the stretching of the cord varies greatly with the shape of the diagram, and with the state of lubrication of the paper drum. Owing to the late cut-off, the engine employed in the experiments was not well suited for showing this effect. However, in some experiments, when the paper-drum wanted oiling, the diagram given with the cord was more than 7 per cent, smaller than that given with the steel wire. The effect is in all cases to reduce the area, though not necessarily to reduce the mean pressure calculated from it. To ascertain if the diagrams from the engine in question would show much difference when taken with cord and with wire, the experiments summarised in Table VII. were made. The lengths and efficiencies given are the mean of the front- and back-end diagrams. TABLE VII. Speed. Wire. String. Length. Efficiency. Length. Efficiency. 68 Inches. 5-11 0-93 Inches. 4-78 0-94 84 5-11 0-93 4-80 0-94 107 5-13 0-94 4-80 0-94 127 5-12 0-93 4-80 0-97 Although the efficiency as calculated from the two sets of diagrams is inconsiderable, yet the difference in their lengths points to a large difference in their areas. The difference in the tension of the indicator-cord at various parts of the stroke may be shown by considering the equation of motion of the indicator-drum. This equation during the outward stroke is dt* = Ta- J\L - 49 A] EXPERIMENTS ON THE STEAM-ENGINE INDICATOR. 199 where / = the moment of inertia of the drum about its axis. T = the tension in the cord. a = radius of drum. M g = moment of resistance of the drum-spring about the axis of drum. Jiff = moment of friction about the same line. Hence r -s(' 3? +* + ri-R ~r , the angular acceleration of the drum about its axis, is a maximum to begin with, and continues to decrease during the stroke, becoming zero near the middle of the stroke. M s is constant during the stroke. Mf is a maximum on starting, then suddenly decreases and then varies directly with some power of the velocity, increasing therefore until about the middle of the stroke, and then diminishing. Thus it is evident that during the outward stroke the tension T is a maximum to begin with, decreases rapidly about the middle of the stroke, and more slowly towards the end. At the end of the stroke the friction suddenly changes sign, thus causing a sudden diminution in the tension at the commencement of the inward stroke ; afterwards the tension increases rapidly about the middle of the stroke, and more slowly towards the end. Hence it might be expected that that part of a diagram taken during the outward stroke would be shortened to commence with, then slightly stretched, and slightly shortened at the end; and that that part taken during the inward stroke, would be first shortened, then lengthened a little, and slightly shortened towards the end, almost as in the case of the outward stroke. To show that this actually takes place, an arrangement was devised by Professor Reynolds, the object of which was to prick holes in the diagram corresponding to eleven equidistant positions of the piston. For this purpose a Grove battery (D Fig. 26) of five cells, in conjunction with a Ruhmkorrf coil, was used. But in order to get the holes pricked in their proper positions, instead of the ordinary arrangement for making and breaking contact, the following plan was adopted, the hammer of the coil being held back. The wire from one pole of the battery was connected with one of the binding-screws (//) of the primary coil as usual, but the wire from the other pole of the battery was connected with the engine. A wire from the other 200 EXPERIMENTS ON THE STEAM-ENGINE INDICATOR. [4-9 A binding-screw (G) of the primary coil was attached to the contact-breaker (B). This consisted of a smooth piece of wood, into which eleven pieces of wire were inserted at equal distances, and filed level with the wood, the Fig. 26. ELECTRICAL APPARATUS FOR SHOWING THE DISTORTION OF A DIAGRAM BY THE INDICATOR-COKD. distance between the first and the last wires being the length of the stroke of the engine. The contact-breaker was fixed on the lower slide bar, so that the central wire should be at the middle of the stroke, and so that a pointer (A), which was secured to the cross-head, should slide on the smooth piece of wood. Hence every time the pointer crossed a wire on the contact-breaker the circuit of the primary current was complete, and a spark of the induced current passed through the diagram-paper. To bring this about one wire of the induced current was connected with the metallic drum (E), and the other to a cup of mercury (F), into which the metallic pencil dipped, thus completing the circuit of the induced current when the pencil touched the paper. In the diagrams, Figs. 27 to 34, which were taken in this manner, the position of the pricked holes, corresponding to the eleven equidistant posi- tions of the piston, are indicated by small circles. The relative positions of these circles show which parts of the diagrams are lengthened, and which are shortened. An examination shows that the effect is not merely to shorten Fig. 27. Front-end pricked diagram taken with wire at 107 revolutions. Fig. 28. Back-end pricked diagram taken with wire at 107 revolutions. 49 A] EXPERIMENTS ON THE STEAM-ENGINE INDICATOR. Fig. 29. Front-end pricked diagram taken with string at 107 revolutions. o o e PO o Fig. 30. Bcick-end pricked diagram taken with string at 107 revolutions. Fig. 31. Front-end pricked diagram taken with wire at 127 revolutions. 201 -e o-

p x I + sin $ ' where tf> is the angle of repose, we cannot assume that tan has any relation to the actual friction between the molecules. The extreme value of is a matter of arrangement ; as in the case of shot, which would pile equally well although without friction. Supposing the grains rigid, the relations between distortion and dilata- tion are independent of friction ; that is to say, the same distortion of any bounding surfaces must mean the same internal distortion whatever the friction may be. The only possible effect of friction would be to render the grains stable under circumstances under which they would not otherwise be stable ; and hence we might, with friction, be able to bring about an alteration of the boundaries other than the alteration possible without friction ; and thus we 50] COMPOSED OF RIGID PARTICLES IN CONTACT. 207 might possibly obtain a dilatation due to friction. How far this is the case can be best ascertained by experiment. In the case of a granular medium, friction may always be relaxed by relieving the mass of stress, and any stability due to this cause would be shown by shaking the mass when in a condition of no stress. But before applying this test, it is necessary to make perfectly sure that during the shaking the boundary spheres do not change position. Another test of the effect of friction is, by comparing the relative dilatation and distortion with different degrees of friction. If the dilatation were in any sense a consequence of friction, it would be greater when the coefficient of friction between the spheres was greater. Where the granular mass is bounded by solid surfaces, the friction of the grains against these surfaces will considerably modify the results. The problem presented by frictionless balls is much simpler than that presented in the case of friction. In the former case the theoretical problem may be attacked with some hope of success. With friction the property is most easily studied by experiment. As a matter of fact, if we take means to measure the volume of a mass of solid grains more or less approximately spheres, the property of dilatancy is evident enough, and its effects are very striking, affording an explanation of many well-known phenomena. If we have in a canvas bag any hard grains or balls, so long as the bag is not nearly full it will change its shape as it is moved about ; but when the sack is approximately full, a small change of shape causes it to become perfectly hard. There is perhaps nothing surprising in this, even apart from familiarity ; because an inextensible sack has a rigid shape when extended to the full, any deformation diminishing its capacity, so that contents which did not fill the sack at its greatest extension fill it when deformed. On careful consideration, however, many curious questions present them- selves. If, instead of a canvas bag, we have an extremely flexible bag of india- rubber, this envelope, when filled with heavy spheres (No. 6 shot), imposes no sensible restraint on their distortion ; standing on the table it takes nearly the form of a heap of shot. This is apparently accounted for by the fact that the capacity of the bag does not diminish as it is deformed. In this condition it really shows us less of the qualities of its granular contents than the canvas bag. But as it is impervious to fluid, it will enable me to measure exactly the volume of its contents. Filling up the interstices between the shot with water, so that the bag is 208 ON THE DILATANCY OF MEDIA [50 quite full of water and shot, no bubble of air in it, and carefully closing the mouth, I now find that the bag has become absolutely rigid in whatever form it happened to be when closed. It is clear that the envelope now imposes no distortional constraint on the shot within it, nor does the water. What, then, converts the heap of loose shot into an absolutely rigid body ? Clearly the limit which is imposed on the volume by the pressure of the atmosphere. So long as the arrangement of the shot is such that there is enough water to fill the interstices, the shot are free, but any arrangement which requires more room, is absolutely prevented by the pressure of the atmosphere. If there is an excess of water in the bag when the shot are in their maximum density, the bag will change its shape quite freely for a limited extent, but then becomes instantly rigid, supporting 56 Ib. without further change. By connecting the bag with a graduated vessel of water, so that the quantity which flows in and out can be measured, the bag again becomes susceptible of any amount of distortion. Getting the bag into a spherical form, and its contents at maximum density, and then squeezing it between two planes, the moment the squeezing begins the water begins to flow in, and flows in at a diminishing rate until it ceases to draw more water. The material in the bag is in a condition of minimum density under the circumstances. This does not mean that all the parts are in a condition of minimum density, because the distortion is not the same in all the parts ; but some parts have passed through the condition of maximum, while others have not reached it, so that on further distortion the dilatations of the latter balance the contractions of the former. If we continue to squeeze, water begins to flow out until about half as much has run out as came in ; then again it begins to flow in. We cannot by squeezing get it back into a con- dition of uniform maximum density, because the strain is not homogeneous. This is just what would occur if the shot were frictionless ; so that it is not surprising to find that, using oil instead of water, or, better (on account of the india-rubber), a strong solution of soap and water, which greatly diminishes the friction, the results are not altered. On measuring the quantities of water, we find that the greatest quantity drawn in is about 10 per cent, of the volume of the bag ; this is about one- third of the difference between the volumes of the shot at minimum and maximum density. = : 1, or 30 per cent, of the latter. v 2 50] COMPOSED OF RIGID PARTICLES IN CONTACT. 209 On easing the bag it might be supposed that the shot would return to their initial condition. But that does not follow : the elasticity of form of the bag is so slight compared with its elasticity of volume, that resti- tution will only take place as long as it is accompanied with contraction of volume. So long as the point of maximum volume has not been reached, approxi- mate restitution follows quite as nearly as could be expected, considering that friction opposes restitution. But when the squeezing has been carried past the point of maximum volume, then restitution requires expansion ; and this the elasticity of shape is not equal to accomplish, so that the bag retains its flattened condition. This experiment has been varied in a great variety of ways. The very finest quartz sand, or glass balls f inch in diameter, all give the same results. Sand is, on the whole, the most convenient material, and its extreme fineness reduces any effect of the squeezing of the india-rubber between the interstices of the balls at the boundaries ; which effect is very apparent with the balloon bags, and shot as large as No. 6. A well-marked phenomenon receives its explanation at once from the existence of dilatancy in sand. When the falling tide leaves the sand firm, as the foot falls on it the sand whitens, or appears momentarily to dry round the foot. When this happens the sand is full of water, the surface of which is kept up to that of the sand by capillary attraction ; the pressure of the foot causing dilatation of the sand, more water is required, which has to be obtained either by depressing the level of the surface against the capillary attraction, or by drawing water through the interstices of the surrounding sand. This latter requires time to accomplish, so that for the moment the capillary forces are overcome ; the surface of the water is lowered below that of the sand, leaving the latter white or dryer until a sufficient supply has been obtained from below, when the surface rises and wets the sand again. On raising the foot it is generally seen that the sand under the foot and around becomes momentarily wet ; this is because, on the distorting forces being removed, the sand again contracts, and the excess of water finds momentary relief at the surface. Leaving out of account the effect of friction between the balls and the envelope, the results obtained with actual balls, as regards the relation between distortion and dilatation, appear to be the same as would follow if the balls were smooth. The friction at the boundaries is not important as long as the strain over the boundaries is homogeneous, and particularly if the balls indent them- selves into the boundaries, as they do in the case of india-rubber. But with o. K. ii. 14 210 ON THE DILATANCY OF MEDIA [50 a plane surface, the balls at the boundaries are in another condition from the balls within. The layer of balls at the surface can only vary its density from 2/V3 to 1. This means that the layer of balls at a surface can slide between that surface and the adjacent layer, causing much less dilatation than would be caused by the sliding of an internal layer within the mass. Hence, where two parts of the mass are connected by such a surface, certain con- ditions of strain of the boundaries may be accommodated by a continuous stream of balls adjacent to the surface. This fact made itself evident in two very different experiments. In order to examine the formation which the shot went through, an ordinary glass funnel was filled with shot and oil, and held vertical while more shot were forced up the spout of the funnel. It was expected that the shot in the funnel would rise as a body, expanding laterally so as to keep the funnel full. This seems to have been the effect at the commencement of the experiment ; but after a small quantity had passed up it appeared, looking at the side of the funnel, that the shot were rising much too fast, for which, on looking into the top of the funnel, the reason became apparent. A sheet of shot adjacent to the funnel was rising steadily all round, leaving the interior shot at the same level with only a slight disturbance. In another experiment one india-rubber ball was filled with sand and water ; at the centre of this ball was another much smaller ball, communi- cating through the sides of the outer envelope by means of a glass pipe with an hydraulic pump. It was expected that, on expanding the interior ball by water, the sand in the outer ball would dilate, expanding the outer ball and drawing more water into the intervening sand. This it did, but not to the extent expected. It was then observed that the outer envelope, instead of expanding, generally bulged in the immediate neighbourhood of the point where the glass tube passed through it ; showing that this tube acted as a conductor for the sand from the immediate neighbourhood of the interior ball to the outer envelope, just as the glass sides of the funnel had acted for the shot. As regards any results which may be expected to follow from the recog- nition of this property of dilatancy, In a practical point of view, it will place the theory of earth-pressures on a true foundation. But inasmuch as the present theory is founded on the angle of repose, which is certainly not altered by the recognition of dilatancy, its effect will be mainly to show the real reason for the angle of repose. The greatest results are likely to follow in philosophy, and it was with a view to these results that the investigation was undertaken. 50] COMPOSED OF RIGID PARTICLES IN CONTACT. 211 The recognition of this property of dilatancy places a hitherto^ unrecog- nized mechanical contrivance at the command of those who would explain the fundamental arrangement of the universe, and one which, so far as I have been able to look into it, seems to promise great things, besides possessing the inherent advantage of extreme simplicity. Hitherto no medium has ever been suggested which would cause a statical force of attraction between two bodies at a distance. Such attraction would be caused by granular media in virtue of this dilatancy and stress. More than this, when two bodies in a granular medium under stress are near together, the effect of dilatancy is to cause forces between the bodies, in very striking accordance with those necessary to explain coherence of matter. Suppose an outer envelope of sufficiently large extent, at first not abso- lutely rigid, filled with granular media, at its maximum density. Suppose one of the grains of the media commences to grow into a larger sphere ; as it grows, the surrounding medium will be pushed outwards radially from the centre of the expanding sphere. Considering spherical envelopes following the grains of the medium, these will expand as the grains move outwards. This fixes the distortion of the medium, which must be contraction along the radii, and expansion along all tangents. The consequent amount of dilatation depends on the relation of distortion and dilatation, and on the arrangement of the grains in the medium. At first the entire medium will undergo dilatation, which will diminish as the distance from the centre increases. As the expansion goes on, the medium immediately adjacent to the sphere will first arrive at a condition of minimum density; and for further expansion this will be returning to a maximum density, while that a little further away will have reached a minimum. The effect of continued growth will therefore be, to institute concentric undula- tions of density from maximum to minimum density, which will move outwards ; so that after considerable growth, the sphere will be surrounded with a series of envelopes of alternately maximum and minimum density, the medium at a great distance being at maximum density. At a definite distance from the centre of the sphere not more than where R is the radius of the sphere, the density will be a minimum, and between this and the sphere there may be a number of alternations, depending on the relative diameters of the grains and the spheres. The distance between these alternations will diminish rapidly as the sphere is approached. The distance of the next maximum is I'ZR, the next minimum is given by T09.R, and the next maximum T06.R. 142 212 ON THE DILATANCY OF MEDIA [50 The general condition of the medium around a sphere which has expanded in the medium, is shown in Fig. 3, which has been arrived at on the sup- position that the sphere is large compared with the grains. Curve, vf DenJ? Fig. 3. From a radius about 1'4R the variation is periodic, with a rapidly diminishing period. In this condition, supposing the medium of unlimited extent and the sphere smooth, the sphere may move without causing further expansion, merely changing the position of the distortion in the medium ; for the grains, slipping over the sphere, would come back to their original positions. It thus appears that smooth bodies would move without resistance, if the relation between the size of the grains and bodies is such, that the energy due to the relative motion of the grains in immediate proximity may 50] COMPOSED OF RIGID PARTICLES IN CONTACT. 213 be neglected. The kinetic energy of the motion of the medium would be proportional to the volume of the ball, multiplied by the density of the medium, and the square of the velocity. But the momentum might be infinite, supposing the medium infinite in extent, in which case a single sphere would be held rigidly fixed. If we suppose two balls to expand instead of one, and suppose the dis- tortion of the medium for one ball to be the same as if the other were not there, the result will be a compound distortion. Since, however, the dilatation does not bear a linear relation to the distortion, the dilatation resulting from the compound distortion will not be the sum of the dilatations for the separate distortions, unless we neglect the squares and products of the distortions as small. Supposing the bodies so far apart that one or other of the separate distortions caused at any point is small, then, retaining squares and products, it appears that the resultant dilatation at any point will be less than the sum of the separate dilatations, by quantities which are proportional to the products of the separate distortions. The integrals of these terms through the space bounded by spheres of radii R and L, are expressed by finite terms, and terms inversely propor- tional to L, which latter vanish if L is infinite. Thus, while the total separate dilatations are infinite, the compound dilatations differ from the sum of the separate by finite terms, and these are functions of the product of the volumes, and the reciprocal of the distance. Assuming stress in the medium, the difference in the value of these finite terms for two relative positions of the bodies, multiplied by the stresses, represents an amount of work which must be done by the bodies on the medium in moving from one position to another. To get rid of the difficulty of infinite extent of medium, if for the moment we assume the envelope sufficiently large and imposing a normal pressure upon the medium, then, since the work done will be proportional to the dilatation, the force between the bodies will be proportional to the rate at which this dilatation varies with the distance between them. The force between the bodies would depend on the character of the elasticity, as well as on the dilatation. It is not necessary to assume the outer envelope elastic ; this may be absolutely rigid, and one or both the balls elastic. In such case the two balls are connected by a definite kinematic relation. As they approach they must expand, doing work which is spent in producing 214 ON THE DILATANCY OF MEDIA [50 energy of motion ; as they recede, the kinetic energy is spent in the work of compressing the balls. As already stated, the momentum of the infinite medium for a single ball in finite motion may be infinite, and proportional to the product of the volume of the ball by the velocity; but with two balls moving in opposite directions, with velocities inversely as the masses, the momentum of the system is zero. Therefore such motion may be the only motion possible in a medium of infinite extent. When the distance between the balls is of the same order as their dimensions, the law of attraction changes with the law of the compound dilatations, and becomes periodic, corresponding to the undulations of density surrounding the balls. Thus, before actual contact was reached, the balls would suffer alternate repulsion and attraction, with positions of equilibrium more or less stable between, as shown in Figs. 4 and 5. Fig. 4. Fig. 5. We have thus a possible explanation of the cohesion and chemical combination of molecules, which I think is far more in accordance with actual experience than anything hitherto suggested. It was the observation of these envelopes of maximum and minimum density, which led me to look more fully into the property of dilatancy. The assumed elasticity of the surrounding envelope, or of the balls, has only been introduced to make the argument clear. The medium itself may be supposed to possess kinetic elasticity arising from internal distortional motion, such as would arise from the transmission of waves, in which the motion of the medium is in the plane of their fronts. The fitness of a dilatant medium to transmit such waves is only less striking than its property of causing attraction, because in the first respect it is not unique. 50] COMPOSED OF RIGID PARTICLES IN CONTACT. 215 But, as far as I can see, such transmission is not possible in a medium composed of uniform grains. If, however, we have comparatively large grains uniformly interspersed, then such transmission becomes possible. If, notwith- standing the large grains, the medium is at maximum density, the large grains will not be free to move without causing further dilatation ; and it seems that the medium would transmit distortional vibrations, in which the distortions of the two sets of grains are opposite. Such waves, although the motion would be essentially in the plane of the wave, would cause dilatation, just as waves in a chain cause contraction in the reach of the chain. They would in fact impart elasticity to the medium, exactly as, in the case of a slack chain having its ends fixed but otherwise not subject to forces, any lateral motion imparted to the chain will cause tension, proportional to the energy of disturbance divided by the slackness or free length of chain. Distortional waves therefore, travelling through dilatant material which does not quite occupy the space in which it is confined when at maximum density, would render the medium uniformly elastic to distortion, but not in the same degree to compression or extension. The tension caused by such waves would depend on the gross energy of motion of the waves, divided by the total dilatation from maximum density consequent on the wave- motion. All such waves, whatever might be their length, would therefore move with the same velocity. If, when rendered elastic by such waves, the medium were thrown into a state of distortion by some external cause, this would diminish the possible dilatation caused by the waves. Thus work would have to be done on the medium in producing the external distortion, which would be spent in in- creasing the energy of the waves. For instance, the separation of two bodies in such a medium, which, as already shown, would increase the statical distortion, would increase the energy of the waves and vice versd. As far as the integrations have been carried for this condition of elasticity, it appears, with a certain arrangement of large and small grains, that the forces between the bodies would be proportional to the product of the volumes divided by the square of the distance ; i.e. that the state of stress of the medium may be the same as Maxwell has shown must exist in the ether to account for gravity. We have thus an instance of a medium, transmitting waves similar to heat-waves, and causing force between bodies similar to the forces of gravitation and cohesion, in such a manner as to constitute a conservative system. More than this, by the separation of the two sets of grains, there would result phenomena similar to those resulting from the separation of the two electricities. The observed conducting power of a continuous surface for the grains of a medium, closely resembles the 216 ON THE DILATANCY OF MEDIA, ETC. [50 conduction of electricity. And such a composite medium would be suscep- tible of a state in which the arrangement of the two sets of grains were thrown into opposite distortions, which state, so far as it has yet been examined, appears to coincide with the state of a medium necessary to explain electrodynamic and magnetic phenomena according to Maxwell's theory. In this short sketch of the results which it appears to me may follow from the recognition of the property of dilatancy, I have not attempted to follow the exact reasoning even so far as I have carried it. In the preliminary acceptance of a theory, the mind must be guided rather by a general view of its adaptability, than by its definite accordance with some out of many observed facts. And as it seems, after a preliminary investigation, that in space filled with discrete particles, endowed with rigidity, smoothness, and inertia, the property of dilatancy would cause amongst other bodies, not only one property, but all the fundamental proper- ties of matter, I have, in pointing out the existence of dilatancy, ventured to call attention to this dilatant or kinematic theory of ether, without waiting for the completion of the definite integrations, which must take long, although it is by these that the fitness of the hypotheses must be eventually tested. 51. EXPERIMENTS SHOWING DILATANCY, A PROPERTY OF GRANULAR MATERIAL, POSSIBLY CONNECTED WITH GRAVITATION. [From the " Proceedings of the Royal Institution of Great Britain."] (Read February 12, 1886.) IN commencing this discourse, the author said, My principal object to-night is to show you certain experiments which I have ventured to think would interest you on account of their novelty, and of their paradoxical character. It is not, however, solely or chiefly on account of their being curious that I venture to call your attention to them. Let them have been never so striking, you would not have been troubled with them, had it not been that they afford evidence of a fact of real importance in mechanical philosophy. This newly recognised property of granular masses, which I have called dilatancy, will, it may be hoped, be rendered intelligible by the experiments, but it was not by these experiments that it was discovered. This discovery, if I may so call it, was the result of an attempt to conceive the mechanical properties a medium must possess, in order that it might fulfil the functions of an all-pervading ether not only in transmitting waves of light, and refusing to transmit waves like those of sound, but in causing the force of gravitation between distant bodies, and actions of cohesion, elasticity, and friction between adjacent molecules, together with the electric and magnetic properties of matter, and at the same time allowing the free motion of bodies. It will be well known to those who attend the lectures in this room, that although a vast increase has been achieved in knowledge of the actions called 218 EXPERIMENTS SHOWING DILATANCY, A PROPERTY OF [51 the physical properties of matter, we have as yet no satisfactory explanation as to the prima causa of these actions themselves ; that to explain the trans- mission of light and heat, it has been found necessary to assume space filled with material possessing the properties of an elastic jelly, the existence of which, though it accounts for the transmission of light, has hitherto seemed inconsistent with the free motion of matter, and failed to afford the slightest reason for the gravitation, cohesion, and other physical properties of matter. To explain these, other forms of ether have been invented, as in the corpuscular theory and the celebrated hypothesis of La Sage, the im- possibilities of which hypotheses have been finally proved by the late Professor Maxwell, to whom we owe so much of our definite knowledge of the fundamental physics. Maxwell insisted on the fact, that even if each of the physical properties could be explained by a special ether, it would not advance philosophy, as each of these ethers would require another ether to explain its existence, ad infinitum. Maxwell clearly contemplated the existence of one medium, but it was a medium which would cause not one but all the physical properties of matter. His writings are full of definite investigations as to what the mechanical properties of this ether must be, to account for the laws of gravitation, electricity, magnetism, and the trans- mission of light, and he has proved very clear and definite properties, although, as he distinctly states, he was unable to conceive a mechanism which should possess these properties. As the result of a long-continued effort to conceive a mechanical system possessing the properties assigned by Maxwell, and further, which would account for the cohesion of the molecules of matter, it became apparent that the simplest conceivable medium a mass of rigid granules in contact with each other would answer not one but all the known requirements, provided the shape and mutual fit of the grains were such, that while the grains rigidly preserved their shape, the medium should possess the apparently paradoxical, or anti-sponge property, of swelling in bulk as its shape was altered. I may here remark, that if ether is atomic or granular, that it should be a mass of grains holding each other in position by contact, like the grains in the sack of corn, is one of only two possible conceptions ; the other being that of La Sage, or the corpuscular theory that the grains are free like bullets, moving in space in all directions. Nor, in spite of its paradoxical sound, is there any great difficulty of con- ceiving the swelling in bulk. When the grains are in contact, it appears at once that the mechanical properties of the medium must be to some extent affected by the shape and fit of the grains. And having arrived at the con- clusion, that in order to act the part of ether, this shape and fit must be such 51] GRANULAR MATERIAL, POSSIBLY CONNECTED WITH GRAVITATION. 219 that the mass could not change its shape, without changing its volume or space occupied, the next thing was to see what possible shape could be given to the grains, so that while these rigidly preserved their shape, the medium might possess this property of dilatancy. It was obvious that the grains must so interlock, that when any change of shape of the mass occurred, the interstices between the grains should increase. This would be possessed by grains shaped to fit into each other's interstices in one particular arrangement. In an ordinary mass of brickwork or masonry well bonded without mortar, the blocks fit so as to have no interstices ; but if the pile be in any way distorted, interstices appear, which shows that the space occupied by the entire mass has increased. (Shown by a model.) At first it appeared that there must be something special and systematic, as in the brick wall, in the fit of the grain of ether, but subsequent con- sideration revealed the striking fact, that a medium composed of grains, of any possible shape, possessed this property of dilatancy, so long as one im- portant condition was satisfied. This condition is, that the medium should be continuous, infinite in extent, or that the grains at the boundary should be so held as to prevent a rearrangement commencing. All that is wanted is a mass of hard smooth grains, each grain being held by the adjacent grains, and the grains on the outside prevented from rearranging. Smooth hard spheres arranged as an ordinary pile of shot are in their closest order, the interstices occupying a space about one-third that occupied by the spheres themselves. By forcing the outside shot so as to give the pile a different shape, the inside spheres are forced by those on the outside, and the interstices increase. Thus by shaping the outside of the pile, the interstices may be increased to any extent, until they occupy about nine- tenths of the volume of the spheres : this is the most open formation. A further change of shape in the same direction causes a contraction of the interstices, until a minimum volume is reached, and then again an expansion, and so on. The point to be realised is, that in any of these arrangements, if the whole of the spheres on the outside of the group are fixed, those inside will be fixed also. (Shown by a model.) An interior portion of a mass of smooth hard spheres therefore cannot have its shape changed by the surrounding spheres, without altering the room it occupies, and the same is true for any granular mass, whatever be the shape of the grains. Considering the generality of this conclusion, the non-discovery of this property as existing in tangible matter, requires a word of explanation. 220 EXPERIMENTS SHOWING DILATANCY, A PROPERTY OF [51 The physical properties of elasticity, adhesion, and friction, so far render the molecules of ordinary matter incapable of behaving as a system of parts with the sole property of keeping their shape, and so prevent evidence of dilatancy in solids and fluids. This is quite consistent with dilatancy in the ether, for the properties of elasticity, cohesion, and friction, in tangible matter, are due to the presence of the ether, so that it would be illogical for the elementary atoms of the ether to possess these properties. This, although a sufficient reason why dilatancy has not been recognised as a property of solid and fluid matter, does not explain its non-existence in masses of solid, hard, free grains, as of corn, shot, and sand. To understand why it has not been observed in these, it must be remembered that, to ordinary observation, these present only an outside appearance, and that the condition essential for dilatancy, that the outside grains should not be free to rearrange, is seldom fulfilled. Also these granular forms of matter, though commonplace, have not been the subjects of physical research, and hence such evidence as they do afford has escaped detection. Once, however, having recognised dilatancy as a universal property of granular masses, it was obvious that if evidence of it was to be sought from tangible matter, it must be sought in what have hitherto been the most commonplace and least interesting arrangements. That an important geometrical and mechanical property of a material system should have been hidden for thousands of years, even in sand and corn, is such a striking thought, that it required no little faith in mechanical principles to undertake the search for it, and although finding nothing but what was strictly in accordance with the conclusions previously arrived at, the evidence obtained of this long-hidden property was as much a matter of visual surprise to the lecturer, as it can be to any of the audience. To render the dilatancy of a granular mass evident, it was necessary to accomplish two things: (1) the outside grains must be controlled so that they could not rearrange, and this without preventing change of shape and bulk of the mass; (2) the changes of bulk or volume of the mass, or of the interstices between the grains, must be rendered evident by some method of measurement which did not depend on the shape of the mass. A very simple means a thin india-rubber envelope or boundary answered both these purposes to perfection. The thin india-rubber closed over the outside grains sufficiently to prevent their change of position, and the impervious character of the bag allowed of a continuous measure of the volume of the contents, by measuring the quantity of air or water necessary to fill the interstices. Taking an india-rubber bag which will hold six pints of water, without stretching, and having only a small tubular aperture, getting it quite dry, 51] GRANULAR MATERIAL, POSSIBLY CONNECTED WITH GRAVITATION. 221 and putting into it six pints of dry sea sand, such as will run_in an hour- glass, sharp river sand, dry corn, shot or glass marbles, it presents no very striking appearance, but all the same when filled with any of these materials, it cannot have its form changed, as by squeezing between two boards, without changing its volume. These changes of volume are not sufficient to be noticeable while the squeezing is going on, but they may be rendered apparent. It is sufficient to do this with the bag full of clean dry Calais sand, such as is used in an hour-glass. The tube from the bag is connected with a mercurial pressure-gauge, so that the bag is closed by the mercury. The actual volume occupied by the quartz grains is four and a half pints. The remaining space, one and a half pints, is occupied by the interstices between the grains in their closest order ; these interstices are full of air, so that three-quarters of the bag are occupied by quartz, and one-quarter by air. Since the bag is closed, and no more air can get in, if interstices are increased from one pint and a half to two pints, the air must expand, and its pressure will fall from that of the atmosphere to three-quarters of an atmosphere. As soon as squeezing begins, the mercury rises on the side connected with the bag, and steadily rises as the bag flattens, until it has risen seven inches, showing that the bag has increased in capacity by half a pint, or one-twelfth of its initial capacity. That by squeezing a porous mass like sand we should diminish the pressure of the air in the pores is paradoxical, and shows the anti-sponginess of the granular material ; had there been a sponge in the bag, the pressure of the air would have increased with the squeezing. This experiment has been mainly introduced to prevent a possible im- pression that the fluid filling the interstices has anything to do with the dilatation besides measuring it. Water affords a more definite measure of volume than air. Taking a small india-rubber bottle with a glass neck full of shot and water, so that the water stands well into the neck. If instead of shot the bag were full of water, or had anything of the nature of a sponge in it, when the bag was squeezed the water would be forced up the neck. With the shot the opposite result is obtained ; as I squeeze the bag, the water decidedly shrinks in the neck. This experiment, which you see is on a very small scale, was not designed to show to an audience ; it was the original experiment which was made for my own satisfaction, when the idea of dilatancy first presented itself. The result, but for the knowledge of dilatancy, would appear paradoxical, not to say magical. When we squeeze a sponge between two planes, water 222 EXPERIMENTS SHOWING DILATANCY, A PROPERTY OF [51 is squeezed out ; when we squeeze sand, shot, or granular material, water is drawn in. Taking a larger apparatus, a bag which holds six pints of sand, the interstices of which are full of water without any air the glass neck being graduated so as to measure the water drawn in. On squeezing the bag with a large pair of pincers, a pint of water is drawn from the neck into the bag. This is the maximum dilatation ; the grains of sand are now in the most open order into which they can be brought by this squeezing ; further squeezing causes them to take closer order, the interstices diminish, and the water runs out into the vessel, arid for still further squeezing is drawn back again, showing that as the change of form continues, the medium passes through maximum and minimum dilatations. This experiment may be repeated with granules of any size or shape, provided they are hard, and shows the universality of dilatancy. Although not more definite, perhaps more striking evidence of dilatancy is afforded by the means which the non-expansibility of water affords of limiting the volume of the bag. An impervious bag full of sand and water without air cannot have its contents enlarged without creating a vacuum inside it the interstices of the sand are therefore strictly limited to the volume of the water inside it, unless forces are brought to bear sufficient to overcome the pressure of the atmosphere and create a vacuum. Since then, owing to this property of dilatancy, the shape of a granular mass at its greatest density cannot change without enlarging the interstices, if we prevent this enlargement by closing the bag we prevent change of shape. Taking the same bag, the sand being at its closest order and closing the neck so that it cannot draw more water. A severe pinch is put on the bag, but it does not change its shape at all ; the shape cannot alter without enlarging the interstices, which cannot enlarge without drawing more water, and this is prevented. To show that there is an effort to enlarge going on, it is only necessary to open a communication with a pressure- gauge, as in the experiment with air. The mercury rises on the side of the bag, showing when the pinch is hardest (about 200 Ibs. on the planes) that the pressure in the bag is less by 27 inches of mercury than the pressure of the atmosphere ; a little more squeezing and there is a vacuum in the bag. Without a knowledge of the property of dilatancy such a method of producing a vacuum would sound somewhat paradoxical. Opening the neck to allow the entrance of water, the bag at once yields to a slight pressure, changing shape, but this change at once stops when the supply is cut off, preventing further dilatation. In these experiments neither the thickness of the bag, nor the character of the fluid, has anything to do with the dilatation of the contents, 51] GRANULAR MATERIAL, POSSIBLY CONNECTED WITH GRAVITATION. 223 considered as forming an interior group of a continuous medium, the bag merely controlling the outside members as they would be controlled by surrounding grains, and the fluid merely measuring or limiting the volume of the interstices. It has, however, been absence of such control of the outside grains, and such means of measuring the volume of the interstices, that has prevented the dilatancy revealing itself as a general mechanical property of granular material ; as a mechanical property, because dilatancy has long been known to those who buy and sell corn. It is seldom left for the philosopher to discover anything which has a direct influence on pecuniary interests ; and when corn was bought and sold by measure, it was in the interest of the vendor to make the interstices as large as possible, and of the vendee to make them as small ; of the vendor to make the corn lie as lightly as possible, and of the vendee to get it as dense as possible. These interests are obvious ; but the methods of getting corn dense and light are paradoxical when compared with the methods for other material. If we want to get any elastic material light we shake it up, as a pillow or a feather bed, or a basket of dried fruit ; to get these dense we squeeze them into the measure. With corn it is the reverse ; it is no good squeezing it to get it dense ; if we try to press it into the measure we make it light to get it dense we must shake it which, owing to the surface of the measure being free, causes a rearrangement in which the grains take the closest order. At the present day the measure for corn has been replaced by the scales, but years ago corn was bought and sold by measure only, and measuring was then an art which is still preserved. It is understood that the corn is to be measured light, and the method employed is now seen to have made use of the property of dilatancy. The measure is filled over full and the top struck with a round pin called the strake or strickle. The universal art is to put the strake end on into the measure before commencing to fill it. Then when heaped full, to pull the strake gently out and strike the top; if now the measure be shaken it will be seen that it is only nine-tenths full. Sand presents many striking phenomena well known but not hitherto explained, which are now seen to be simply evidence of dilatancy. Every one who walks on the strand must have been painfully struck with the difference in the firmness and softness of the sand at different times ; letting alone when it is quite dry and loose. At one time it will be so firm and hard that you may walk with high heels without leaving a footprint ; while at others, although the sand is not dry, one sinks in so as to make walking painful. Had you noticed you would have found that the sand is firm as the tide falls, and becomes soft again after it has been left dry for some hours. The reason for this difference is exactly the same as 224 EXPERIMENTS SHOWING DILATANCY, A PROPERTY OF [51 that of the closed bags with water and air in the interstices of the sand. The tide leaves the sand, though apparently dry on the surface, with all its interstices perfectly full of water, which is kept up to the surface of the sand by capillary attraction ; at the same time the water is percolating through the sand from the sands above, where the capillary action is not sufficient to hold the water. When the foot falls on this water-saturated sand, it tends to change its shape, but it cannot do this without enlarging the interstices without drawing in more water. This is a work of time, so that the foot is gone again before the sand has yielded. If you stand still, you will find that your feet sink more or less, and that when you move, the sand becomes wet all round the space you stood on, which is the excess of water you have drawn in, set free by the sand regaining its densest form. One phenomenon attending walking on firm sand is very striking ; as the foot falls, the sand all round appears to shoot white or dry momentarily, soon becoming dark again. This is the suction into the enlarging interstices below the foot, which for the moment depresses the capillary surface of the water below that of the sand. After the tide has left the sand for a sufficient time, the greater part of the water has run out of the interstices, leaving them full of air, w r hich by expanding allows the interstices to enlarge, and the foot to sink in far enough to make walking unpleasant. If we walk on sand under water, it is always more or less soft, for the interstices can enlarge, drawing in water from above. The firmness of the sand is thus seen to be due to the interstices being full of water, and to the capillary action or surface tension of the water at the surface of the sand. This capillary action will hold the water up in the sand for some inches or feet, according to the fineness of the sand. This is shown by a somewhat striking experiment. If sand running in a stream from a small hole in the bottom of a vessel, as in an hour-glass, fall into a vessel containing a slight depth of water, the sand at first forms an island, which rises above the water. The sand which then falls on the top of this island is dry as it falls, but capillary action draws up the water which fills the interstices and gives the sand coherence. The island grows vertically, very fast, and assumes the form of a column, sometimes with branches like a tree or a fern, some inches or even a foot high. The strength of these consists in the surface tension of the water preventing air from being drawn in to enlarge the interstices, which therefore cannot change shape ; it is therefore another evidence of dilatancy. By substituting an impervious envelope for the surface of water, firmness of sand saturated with water may be rendered very striking. 51] GRANULAR MATERIAL, POSSIBLY CONNECTED WITH GRAVITATION. 225 Thin india-rubber balloons, which may be easily expanded with the mouth, afford an almost transparent envelope. Taking one containing about six pints of sand and water, closed without air, there being more water than will fill the interstices at the densest, but not enough to allow them to enlarge to the full extent. When standing on the table, the elasticity of the envelope gives it a rounded shape. The sand has settled down to the bottom, and the excess of water appears above the sand, the surface of which is free. The bag may be squeezed and its shape altered, apparently as though it had no firmness, but this is only so long as the surface is free. But taking it between two vertical plates and squeezing, at first it submits, apparently without resistance, when all at once it comes to a dead stop. Turning it on to its side, a 56-lb. weight produces no further alteration of shape ; but on removing the weight, the bag at once returns to its almost rounded shape. Putting the bag now between two vertical plates, and slightly shaking while squeezing, so as to keep the sand at its densest, while it still has a free surface, it can be pressed out until it is a broad flat plate. It is still soft as long as it is squeezed, but the moment the pressure is removed, the elasticity of the bag tends to draw it back to its rounded form, changing its shape, enlarging the interstices, and absorbing the excess of water; this is soon gone, and the bag remains a flat cake with peculiar properties. To pressures on its sides it at once yields, such pressures having nothing to overcome but the elasticity of the bag, for change of shape in that direction causes the sand to contract. To radial pressures on its rim, however, it is perfectly rigid, as such pressures tend further to dilate the sand ; when placed on its edge, it bears one cwt. without flinching. If, however, while supporting the weight it is pressed sufficiently on the sides, all strength vanishes, and it is again a rounded bag of loose sand and water. By shaking the bag into a mould, it can be made to take any shape ; then, by drawing off the excess of water and closing the bag, the sand becomes perfectly rigid, and will not change its shape without the envelope be torn ; no amount of shaking will effect a change. In this way bricks can be made of sand or fine shot full of water and the thinnest india-rubber envelope, which will stand as much pressure as ordinary bricks without change of shape ; also permanent casts of figures may be taken. I have now shown, as fully as time will allow, the experiments which afford evidence of the existence of the property of dilatancy, and how it explains natural phenomena hitherto but little noticed. Beyond affording evidence of the existence of the property dilatancy, o. R. ii. 15 226 EXPERIMENTS SHOWING DILATANCY, A PROPERTY OF [51 these experiments have no direct connection with gravitation or the physical properties of matter. These properties cannot be deduced by direct experiment on granular material, for the simple reason that the grains of the medium which con- stitutes the ether must be free from friction, while the grains with which we work are subject to friction. These properties can only be deduced by mathematical reasoning, into which I will not drag you to-night. I will merely point out two or three facts, which may serve to convey an idea of how dilatancy should have such a bearing on the foundation of the universe. If you look at this diagram, you see it represents a ball surrounded by a continuous mass of grain, the density of the grains being indicated by the depth of colour. If that ball were to grow in volume, it would have to push out the medium on all sides, and in that way it would distort the groups of grains, or change their form, causing the interstices to increase ; those nearer the ball would be distorted more than those further away. Then the inter- stices of these would grow the most rapidly, and those adjacent to the ball would first come to their openest order for further growth ; these would contract somewhat, those a little further away would reach the openest order, and if the process of growth steadily continued, we should have a series of undulations of density, commencing at the ball and moving outwards ; the first of these waves of open order would not, however, get beyond half the diameter of the ball away. The diagram represents the interstices that would result, if a single grain of the material had grown to the size of the ball, pushing the medium out before it. It is not necessary that the ball should have grown, to produce this result; however the ball were originally placed, if it were moved away from its original place, it would assume this arrange- ment, and with this arrangement it would be free to move. Now, although I cannot attempt to enter upon the relation between the density of the medium, and the force of attraction between two bodies in it, I may call your attention to this fact, that the dilatation as calculated, varies exactly as the force of gravitation, inversely as the square of the distance from an infinite distance till close to the ball, and then goes through several undulations, corresponding exactly to the variations in the attraction of bodies necessary to explain the elasticity and cohesion of molecules. As is shown in the other diagrams, these undulations in density, which may be experimentally produced, not only appear to afford a clear explanation of cohesion, but are the only suggestion of an explanation ever made. And further, similar undulations have been found necessary to explain one of the phenomena of light. My reason for calling your attention to them was partly an experiment, which, although not the most striking, is. the most advanced experiment in the direction of dilatancy. 51] GRANULAR MATERIAL, POSSIBLY CONNECTED WITH GRAVITATION. 227 The apparatus is that represented in the diagram; the medium is con- tained in the large elastic bag ; in the middle of this bag is a small hollow elastic ball, which can be expanded by water forced in through a tube passing through the medium and outside ball ; the quantity of water which passes in is measured by a mercury gauge, the water being forced in by the pressure of the mercury. The medium between the two balls is sand and water, and is connected with a gauge, the water drawn from which measures the dilatation. The full pressure of 30 inches is on the interior ball, but produces no expansion, because the medium outside cannot dilate, as the supply of water is now cut off; opening the tap to admit water to the outer ball, it at once draws water. It has now drawn 3 oz. ; in the meantime the mercury has fallen, showing that an ounce and a half was admitted to the interior ball, the expansion of which drew the water into the outer envelope. This experiment is not striking, but it is definite, and enables us to measure the dilatation consequent on a given distortion. It is impossible for me to go further into this explanation, so I will merely state that the ability of the grains of a medium to slide over a smooth surface has been experimentally shown to produce phenomena closely resembling the conduction of electricity, to complete which it is only necessary to construct the medium of two different sorts of grains, different in size or different in shape, the separation of which would afford the two electricities, and be a simple way out of the difficulty hitherto found in explaining the non-exhaus- tibility of the electricity in a body. Hitherto the two electric fluids have been supposed to reside together in the matter of the machine, which, how- ever much has been withdrawn, has never shown signs of exhaustion. In the dilatant hypothesis, these electricities are the two constituents of the ether which the machine separates, and it is worth noticing that the ordinary electrical machine resembles in all essential particulars the machines used by seedsmen for separating two kinds of seed, trefoil and rye-grass, which grow together : as long as there is a supply of the mixture, the machine is never exhausted. This dilatant hypothesis of ether is very promising, although it cannot be put forward as proved until it has been worked out in detail, which will take long. In the meantime it is put forward mainly to excite interest in the property of dilatancy, to the discovery of which it has led. This property, now that it has once been recognised, is quite independent of any hypothesis, and offers a new field for philosophical and mathematical research quite independent of the ether. 152 52. ON THE THEORY OF LUBRICATION AND ITS APPLICATION TO MR BEAUCHAMP TOWER'S EXPERIMENTS, INCLUD- ING AN EXPERIMENTAL DETERMINATION OF THE VIS- COSITY OF OLIVE OIL. [From the "Philosophical Transactions of the Royal Society," Part I., 1886.] SECTION I. INTRODUCTORY. 1. LUBRICATION, or the action of oils and other viscous fluids to diminish friction and wear between solid surfaces, does not appear to have hitherto formed a subject for theoretical treatment. Such treatment may have been prevented by the obscurity of the physical actions involved, which belong to a class as yet but little known, namely, the boundary or surface actions of fluids ; but the absence of such treatment has also been owing to the want of any general laws discovered by experiment. The subject is of such fundamental importance in practical mechanics, and the opportunities for observation are so frequent, that it may well be a matter of surprise that any general laws should have for so long escaped detection. Besides the general experience obtained, the friction of lubricated surfaces has been the subject of much experimental investigation by able and careful experimenters. But, although in many cases empirical laws have been propounded, these fail for the most part to agree with each other and with the more general experience. 2. The most recent investigation is that of Mr Beauchamp Tower, under- taken at the instance of the Institution of Mechanical Engineers. Mr Tower's first report was published November, 1883, and his second report in 1884 (Proc. Inst. Mechanical Engineers}. 52] ON THE THEORY OF LUBRICATION, ETC. 229 In these reports Mr Tower, making no attempt to formulate, states the results of experiments apparently conducted with extreme care and under very various and well-chosen circumstances. Those results which were obtained under the ordinary conditions of lubrication so far agree with the results of previous investigators as to show a want of any regularity. But one of the causes of this want of regularity, irregularity in the supply of the lubricant, appears to have occurred to Mr Tower early in his investiga- tion, and led him to include amongst his experiments the unusual circum- stances of surfaces completely immersed in oil. This was very fortunate, for not only do the results so obtained show a great degree of regularity, but while making these experiments he was accidentally led to observe a phenomenon which, taken with the results of his experiments, amounts to a crucial proof that in these experiments with the oil bath the surfaces were completely and continuously separated by a film of oil ; this film being maintained by the motion of the journal, although the pressure in the oil at the crown of the bearing was shown by actual measurement to be as much as 625 Ibs. per sq. inch above the pressure in the oil bath. These results obtained with the oil bath are very important, notwith- standing that the condition is not common in practice. They show that with perfect lubrication a definite law of variation of the friction with the pressure and velocity holds for a particular journal and brass. This strongly implies that the irregularity previously found was due to imperfect lubrication. Mr Tower has brought this out: Substituting for the bath an oily pad, pressed against the free part of the journal, and making it so slightly greasy that it was barely perceptible to the touch, he again found considerable regularity in the results ; these, however, were very different from those with the bath. Then with intermediate lubrication he obtained intermediate results, of which he says : " Indeed, the results, generally speaking, were so uncertain and irregular that they may be summed up in a few words. The friction depends on the quantity and uniform distribution of the oil, and may be anything between the oil bath results and seizing, according to the perfection or imperfection of the lubrication." 3. On reading Mr Tower's report it occurred to the author as possible that, in the case of the oil bath, the film of oil might be sufficiently thick for the unknown boundary actions to disappear, in which case the results would be deducible from the equations of hydrodynamics. Mr Tower appears to have considered this, for he remarks that according to the theory of fluid friction the resistance would be as the square of the velocity, whereas in his results it does not increase according to this law. Considering how very general the law of resistance as the square of the velocity is with fluids, there is nothing remarkable in the assumption of its holding in such a case. 230 ON THE THEORY OF LUBRICATION [52 But the study of the behaviour of fluid in very small channels, and par- ticularly the recent determination by the author of the critical velocity at which this law changes from that of the square of the velocity to that of the simple ratio, shows that with such highly viscous fluids as oils, such small spaces as those existing between the journal and its bearing, and such limited velocity as that of the surface of the journal, the resistance would vary, cceteris paribus, as the velocity. Further, the thickness of the oil film would not be uniform and might be affected by the velocity, and as the resistance would vary, cceteris paribus, inversely as the thickness of the film, the velocity might exert in this way a secondary effect on the resistance ; and, further still, the resistance would depend on the viscosity of the oil, and this depends on the temperature. But as Mr Tower had been careful to make all his experiments in the same series with the journal at a temperature of 90 Fahr., it did not at first appear that there could be any considerable temperature effect in his results. 4. The application of the hydrodynamical equations to circumstances similar in so far as they were known to those of Mr Tower's experiments, at once led to an equation between the variation of pressure over the surface and the velocity, which equation appeared to explain the existence of the film of oil at high pressure. This equation was mentioned in a paper read before Section A. of the British Association at Montreal, 1884. It also appears from a paragraph in the President's Address (Brit. Assoc. Rep., 1884, p. 14) that Professor Stokes and Lord Rayleigh had simultaneously arrived at a similar result. At that time the author had no idea of attempting its integration. On subsequent consideration, however, it appeared that the equation might be transformed so as to be approximately integrated, and the theoretical results thus definitely compared with the experimental. 5. The result of this comparison was to show that with a particular journal and brass, the mean thickness of the film of oil would be sensibly constant, and hence, if the viscosity was constant, the resistance would increase directly as the speed. As this was not in accordance with Mr Tower's experiments, in which the resistance increased at a much slower rate, it appeared that either the boundary actions became sensible, or that there must have been a rise in the temperature of the oil which had escaped the thermometers used to measure the temperature of the journal. That there would be some excess of temperature in the oil film, on which all the work of overcoming the friction is spent, is certain ; and after carefully considering the means of escape of this heat, it seems probable that there would be a difference of several degrees between the oil bath and the film of oil. 52] AND ITS APPLICATION TO MR B. TOWER'S EXPERIMENTS. 231 This increase of temperature would be attended by a diminution of viscosity, so that, as the resistance and temperature increased with the velocity, the viscosity would diminish and cause a departure from the simple ratio. 6. In order to obtain a quantitative estimate of these secondary effects, it was necessary to know exactly the relation between the viscosity and temperature of the lubricant used. For this purpose an experimental deter- mination was made of the viscosity of olive oil at different temperatures as compared with the known viscosity of water. From the results of these experiments an empirical formula has been deduced, by means of which definite expressions have been obtained for the approximate variation of the viscosity with the speed and load. Taking these variations of viscosity into account, the results obtained from the hydrodynamical theory are brought into complete accordance with these experiments of Mr Tower. Thus we have not only an explanation of the very novel phenomena brought to light by these experiments, and what appears to be an important verification of the assumptions on which the theory of hydrodynamics is founded, but we also find, what is not shown in the experiments, how the various circum- stances under which the experiments have been made affect the results. 7. Two circumstances particularly are brought out in the theory as principal circumstances which seem to have hitherto entirely escaped notice, even that of Mr Tower. One of these is the difference in the radii of the journal and of the brass or bearing. It is well known that the fitting between the journal and its bearing produces a great effect on the carrying power of the journal, but this fitting is rather supposed to be a matter of smoothness of surface than a degree of correspondence in radii. The radius of the bearing must always be as much larger than that of the journal as is necessary to secure an easy fit ; but more than this, I think, has never been suggested. Now it appears from the theory that if viscosity were constant the friction would be inversely proportional to the difference in radii of the journal and the bearing, and this although the arc of contact is less than the semicircum- ference. Taking the temperature into account, it appears, from the com- parison of the theoretical results with the experimental, that at a temperature of 70'5 Fahr. the radius of one of the brasses used was '00077 inch greater than that of the journal, while at a temperature of 70 Fahr. that of the other was '00084 inch, or 9 per cent, larger than the first. These two brasses were probably both bedded to the journal in the same way, and had neither of them been subjected to any great amount of wear, so 232 ON THE THEORY OF LUBRICATION [52 that there is nothing surprising in their being so nearly the same fit. It would be extremely interesting to find whether prolonged wear of the brass tends to preserve or destroy the fit. This does not appear from Mr Tower's experiments. It does appear, however, that with an increase of temperature the brass expands more than the journal, and that its radius increases as the load increases in a very definite manner. Another circumstance brought out by the theory, and remarked on both by Lord Rayleigh and the author at Montreal, but not before expected, is that the point of nearest approach of the journal to the brass is not by any means in the line of the load, and, what is still more contrary to common supposition, is on the off* side of the line of load. This circumstance, the reason for which is rendered perfectly clear by the conditions of equilibrium, at once accounts for a singular phenomenon mentioned by Mr Tower, viz., that the journal having been run in one direction until the initial tendency to heat had entirely disappeared, on being reversed it immediately began to heat again ; but this effect stopped when the process had been often repeated. The fact being that running in one direction the brass had been worn to the journal only on the off side for that direction, so that when the motion was reversed the new off side was like a new brass. 7 A. The circumstances which determine the greatest load which a bearing will carry with complete lubrication, i.e., with the film of oil extend- ing between brass and journal throughout the entire arc, are definitely shown in the theory. The effect of increasing the load beyond a certain small value, being to cause the brass to approach nearer to the journal at a point H, which moves Fig. 1. from A towards as the load increases, and when the load is such that the least separating distance is about half the difference of radii, the angular * " On" and "off" sides of the line of load are used by Mr Tower to express respectively the sides of approach and succession, as B and A in the figure, the arrow indicating the direction of rotation. 52] AND ITS APPLICATION TO MR B. TOWER'S EXPERIMENTS. 233 position of H is 40 to the off side of 0, the middle of the brass. At this point the pressure in the oil film is everywhere greater than at A and B, the extremities of the brass, but when the load further increases the pressure towards A on the off side becomes smaller or negative. This, when sufficient, will cause rupture in the oil film, which will then only extend between the brass and journal over a portion of the whole arc, and a smaller portion as the load increases. Thus, since the amount of negative pressure which the oil will bear, depends on circumstances which are uncertain, the limit of the safe load for complete lubrication is that which causes the least separating distance to be half the distance of radii of the brass and journal. The rupture of the oil film does not take place at the point of nearest approach, and hence the brass may still be entirely separate from the journal, and could the integrations be effected it would be possible to deal as definitely with this condition as with that of complete lubrication ; but these difficulties have limited the actual application of the theory to complete lubrication. This however by no means requires an oil bath, but merely sufficient oil on the journal. What happens when the supply of oil is limited, i.e., insufficient for com- plete lubrication, cannot be definitely expressed without further integrations; but sufficient may be seen to show that the brass will still be completely separated from the journal, although the separating film will not touch the brass, except over a limited area; but in this case it is easy to show by general reasoning that in the one extreme, where the supply of oil is limited, the friction increases directly as the load and is independent of the velocity, while in the other, where the oil is abundant, the circumstances are those of the oil bath. The effect of the limited length of the journal is also apparent in the equations, as is also the effect of necking the shaft to form the journal, so that the ends of the brass are against flanges on the shaft. The theory is perfectly applicable to cases in which the direction of the load on the bearing varies, as with the crank pin and with the bearings of the crank shaft of the steam-engine ; but these cases have not been considered, as there are no definite experiments to compare. 8. Although in the main the present investigation has been directed to the circumstances of Mr Tower's experiments, viz., a cylindrical journal revolving in a cylindrical brass, it has, on the one hand, been found necessary to proceed from the general equations of equilibrium of viscous fluids, and, on the other hand, to consider somewhat generally the physical property of viscosity and its dependence on temperature. 234 ON THE THEORY OF LUBRICATION [52 The property of viscosity has been discussed at length in Section II. ; which section also contains the account of an experimental investigation as to the viscosity of olive oil. The general theory deduced from the hydrodynamical equations for viscous fluids, with the methods of application, is given in Sections IV., V., VI, VII., and VIII. As there are some considerations which cannot be taken into account in the more general method, which method also tends to render obscure the more immediate purpose of the investigation, a preliminary discussion of the problem, illustrated by aid of the graphic method, has been introduced as Section III. Finally, the definite application of the theory to Mr Tower's experiments is given in Section IX. SECTION II. THE PROPERTIES OF LUBRICANTS. 9. The Definition of Viscosity. In distinguishing between solid and fluid matter, it is customary to define fluid as a state of matter incapable of sustaining tangential or shearing stress. This definition, however, as is well known, is only true as applied to actual fluids when at rest. The resistance encountered by water and all known fluids flowing steadily along parallel channels, affords definite proof that in certain states of motion all actual fluids will sustain shearing stress. These actual fluids are, therefore, called in the language of mathematics imperfect or viscous fluids. In order to obtain the equations of motion of such fluids, it has been necessary to define clearly the property of viscosity. This definition has been obtained from the consideration that to cause shearing stress in a body it is necessary to submit it to forces tending to change its shape. Forces tending to cause a general motion, whether linear, revolving, uniform expan- sion, or uniform contraction, call forth no shearing stress. Using the term distortion to express change of shape, apart from change of position, uniform expansion, or contraction, the viscosity of a fluid is defined as the shearing stress caused in the fluid while undergoing distor- tion, and the shearing stress divided by the rate of distortion is called the coefficient of viscosity, or, commonly, the viscosity of the fluid. This is best expressed by considering a mass of fluid bounded by two parallel planes at a distance a, and supposing the fluid between these planes 52] AND ITS APPLICATION TO MR B. TOWER'S EXPERIMENTS. 235 to be in motion in a direction parallel to these surfaces with^ a_ velocity which varies uniformly from at one of these surfaces to u at the other. Then the rate of distortion is u a and the shearing stress on a plane parallel to the motion is expressed by /* being the coefficient of viscosity or the modulus of the resistance to distor- tional motion. 10. The Character of Viscosity. In dealing with ideal fluids, it is of course allowable to consider p as being zero or having any conceivable value ; but practically, as regards natural philosophy, the value of any such considerations depends on whether the calculated behaviour of the ideal fluid is found to agree with the behaviour of the actual fluids whether taking a particular fluid, a value of fj, can be found such that the values of f calculated by equation (1) agree with the values off determined by experiments for all values of a and u. In the mathematical theory of viscous fluid, //, is assumed to be constant for a particular fluid. This supposition is sometimes justified by reference to some assumed dynamical constitution of fluids ; but apart from such hypotheses there is no more ground for supposing a constant value for /* than there is for supposing a particular law of gravitation, in other words, there is no ground at all. If a particular value of //. is found to bring the calculated results into agreement with all experimental results, then this value of /j, defines a property of actual fluids, and of course it has been with this object that the mathematical theory of ^ has been studied. The chief question as regards //, is a simple one within a particular fluid is fji constant ? In other words, is viscosity a property of a fluid like inertia which is independent of its motion ? If it is, our equations may be useful ; if it is not then the introduction of p, into the equations renders them so complex that it is almost hopeless to expect anything from them. Another question of scarcely less practical importance relates to the character of //, near the bounding surfaces of the fluid. If /u, is constant in the fluid, does it change its value near the boundary of the fluid ? Is there anything like slipping between the fluid and a solid boundary with which it is in contact ? As regards the answers to these questions the present position is some- what as follows : 236 ON THE THEORY OF LUBRICATION [52 11. The Two Viscosities. The general experience that the resistance varies as the square of the velocity is an absolute proof that /JL is not constant unless a restricted meaning be given to the definition of viscosity, excluding such part of the resistance as may be due, in the way explained by Prof. Stokes*, to internal eddies or cross streams, however insensible these may be, so long as they are not simply molecular motions. On the other hand in the definite experiments made by Colomb, and particularly by Poiseuille, it was found that the resistance was proportional to the velocity, and therefore that //, was absolutely constant i.e., independent of the velocity f. To meet this discordance it has been supposed that //. varied with the rate of distortion i.e., is a function of u/a, but is sensibly constant when u/a is small | . To assume this, however, is to neglect Poiseuille's experiments, in which he found for water the resistance absolutely proportional to the velocity in a tube '6 mm. diameter up to a velocity of 6 metres per second, which corresponds to a value of u/a = 20,000. On the other hand it is found by Darcy and others in large tubes that ni the resistance varies as the square of the velocity for values of - , as low as 1. Thus in a tube of *6 mm. we have p. constant for all rates of distor- tion below 20,000, while in a tube of 500 mm. diameter p is a function of the distortion for all values greater than 1. It is, therefore, clear that if /i is a function of the distortion it must also be a function of the dimensions of the channels, and in that case /j, cannot be considered as a property of the fluid only. The change in the law of resistance from the simple ratio has, however, been shown by the author to be due to a change in the character of the motion of the fluid from that of direct parallel motion to that of sinuous or eddying motion ||. * Stokes's Reprint, vol. i., p. 99. t Paris Mem. Savans Etrang., torn. 9 (1846), p. 434. I Lamb's Motion of fluids, 1879, Art. 180. Recherches Kxpl. Paris, 1852. || "An Experimental Investigation of the circumstances which determine whether the Motion of Water shall be Direct or Sinuous." Phil. Trans., vol. 174 (1883), p. 935. 52] AND ITS APPLICATION TO MR B. TOWER'S EXPERIMENTS. 237 In the latter case, although the mean motion at any point taken over a sufficient time is parallel to the pipe, it is made up of a succession of motions crossing the pipe in different directions. The question as to whether, in the case of sinuous motion, p is to be considered as a function of the velocity or not, depends on whether we regard f as expressing the instantaneous shearing stress at a point, or the mean over a sufficient time. Whether we regard the symbols in the equations of motion as expressing the instantaneous motion or the mean taken over a sufficient time. If the latter, then /* must be held to include, in addition to the mean stress, the momentum per second parallel to u carried by the cross streams in the negative direction across the surface over which / is measured. If, however, we regard the motion at each instant, then we must restrict our definition of viscosity by making f the instantaneous value of the intensity of resistance at a point. This is a quantity which we have and can have no means of measuring except under circumstances which secure that f is constant for all points over a given surface, and for all instants over a given time. It thus appears that there are two essentially distinct viscosities in fluids. The one a mechanical viscosity arising from the molar motion of the fluid, the other a physical property of the fluid. It is worth while to point out that, although the conditions under which the first of these the mechanical viscosity can exist, depend primarily on the physical viscosity, the actual magnitudes of these viscosities are independent, or are only connected in a secondary manner. This is shown by a very striking but little noticed fact. When the motion of the fluid is such that the resistance is as the square of the velocity, the magnitude of this resistance is sensibly quite independent of the character of the fluid in all respects except that of density. Thus, when in a particular pipe the velocity of oil or treacle is sufficient for the resistance to vary as the square of the velocity, the resistance is practically the same as it would be with water at the same velocity, while the physical viscosity of water is more than one hundred times less. The answer, then, to the question as to the constancy of /z, may be clearly given IJL measures a physical property of the fluid which is independent of its motion. But in this sense //, is the coefficient of instantaneous resistance to distortion at a point moving with the fluid. This restriction is equivalent to restricting the applications of the equations of motion for a viscous fluid to the cases in which there are no eddies or sinuosities. 238 ON THE THEORY OF LUBRICATION [52 This, as shown by the author, is the case in parallel channels so long as the product of the velocity, the width of the channel, and the density of the fluid divided by /* is less than a certain constant value. In a round tube this constant is 1400, or At a temperature of 50, we have, with a foot as unit of length, for water ^ = 0-00001428, P Dv < '02, so that if D, the diameter of the channel, be '001 inch, v would have to be at least 240 feet per second for the resistance to vary other than as the velocity. As regards the slipping at the boundaries, Poiseuille's experiments, as well as those of the author, failed to show a trace of this, although f reached the value of 0702 Ib. per square inch, so that within this limit it may be taken as proved that there is no slipping between any solid surface and water. With other fluids, such as mercury in glass tubes, it is possible that the case may be different ; but, as regards oils, the probability seems to be that the limit within which there is no slipping will be much higher than with water. 12. Experimental Determination of the Value of p for Olive Oil. Since the value of p for water is known for all moderate temperatures, in order to obtain the value for oil it is only necessary to ascertain the relative times taken by the same volumes of oil and water to flow through the same channel, care being taken to make the channel such that there are no eddies and that the energy of motion is small compared with the loss of head. These times are proportional to t P' where p is the fall of pressure ; therefore the times multiplied by the respective falls of pressure are proportional to the viscosities. The arrangement of apparatus used is shown in Fig. 2. The test tube (A) containing the fluid to be tested was fixed in a beaker of water, which was heated from below and maintained at any required temperature. 52] AND ITS APPLICATION TO MR B. TOWER'S EXPERIMENTS. 239 A syphon (B), made of glass tube T % inch internal diameter, with the extremity of its short limb drawn down to capillary size for aT length of about 6 inches, this six inches being bent up and down so as only to occupy Fig. 2. some 2 inches at the bottom of the test tube. The long limb of the syphon extended to about 2 feet below the mean level of the fluid in the test tube. Two marks on the test tube at different levels served to show when a definite volume had been withdrawn. The syphon used was the same for each set of experiments on oil and water, so that the pressure urging the fluid through the tube was propor- tioned to the density of the fluids that is, it was 1'915 as great for oil as water, disregarding the effect of the variation of temperature on volume, which in no case amounted to 1 per cent. Experiments were first made with water at different temperatures, the times taken for the water to fall from the first mark to the second being carefully noted. The syphon was then dried and replaced and oil substituted for water. Two sets of similar apparatus were used on different occasions, different samples of oil being used. In the first set the experiments on oil were made at temperatures from 95 to 200 Fahr.; in the second set, from 61 to 120 Fahr. In so far as the temperatures overlapped, the 240 ON THE THEORY OF LUBRICATION [52 viscosities for the two oils agreed to within 4 per cent., but as the law of variation of the viscosity seemed to change rapidly at about 140 Fahr., only the second set have been recorded. These are shown in Table I. TABLE I. VISCOSITY OF OIL COMPARED WITH WATER: 11 April, 1884. Temperature. M Number. Third. Time seconds. 107 experi- mental. experi- mental. log /j. calculated. 107 calculated. Fahrenheit. Centigrade. 1 Water . 60 15-5 25 1-640 2 M M 3 j> M 4 Olive oil 61 16 2040 123-00 5-08990 5-090133 123-06 5 81 1350 81-00 6-90848 6-89807 79-08 6 M 94 1000 60-00 6-77815 6-78290 59-34 7 120 555 33-40 6-52375 6-52375 33-40 From Poiseuille's experiments it is found that, measuring viscosity in pounds on the square inch, for water at a temperature of 61 Fahr., /i=10- 7 x 1-61. Adopting this value of yu, for the experiments on water at 61 Fahr., the other experimental values of fi for water at different temperatures, obtained as being in the ratios of the times, were found to be in very close agreement with those calculated from Poiseuille's law for the respective temperatures. This tested the efficiency of the apparatus. It has not been thought necessary to record any experiment on water except at the temperature of 61 Fahr. The experimental value of p, for oil are in the ratios of the times multi- plied by '915, the specific gravity of oil ; these are given as the experimental values of //- in the table. Another column contains the values of /z for oil, calculated from an empirical formula fitted to the experimental values. This formula was found by comparing the logarithms of the experimental values of \i. It appeared that the differences in these logarithms were nearly proportional to the differences in the corresponding temperatures, or that T being temperature in degrees Fahr., log ,*!- log ,*, = -0096 (Z'.-r,), in degrees Centigrade log K - log ^ = -00535 (T, - r l\) ; whence since '0096 = '0021 Iog 10 e, 00535 = -0123 Iog 10 e, 52] AND ITS APPLICATION TO MR B. TOWER'S EXPERIMENTS. 241 for degrees Fahrenheit for degrees Centigrade 1 _ 0221(2', - 1 g--oi23(r,-r 2 ) .(2). This ratio holds well within the experimental accuracy from temperatures ranging from 61 to 120 Fahr. This is shown in the table, and again in Fig. 3, in which the ordinates are proportional to log/i, the abscissae being proportional to the corresponding temperatures. 5.1 5.0 6.9 6.8 6.G 6.5 70 80 90 100 Temperature Fahrenheit Fig. 3. 110 120 13. The Comparative Values of fi for Different Fluids and Different Systems of Units. The values of p, given by different writers for air and water, are ex- pressed in various units of force and length, so that it is a matter of some trouble to compare them. To facilitate this for the future comparative values are here given. Those for water have been deduced from Poiseuille's formula, for air from Maxwell's formulae, and for olive oil from the experi- ments recorded in the previous article. The units of length, mass, and time, being respectively the centimetre, gramme, and second, in which case the unit of force is the weight of one 980'oth (g) part of a gramme, expressing temperature in degrees Centigrade by T and putting o. R. ii. 0-03367932' +0-00022099367 72 (3), 16 242 ON THE THEORY OF LUBRICATION [52 for water . . . /a = (M)177931P ] air. . . . /* = 0-0001879 (1 + 0'00366T)[ (4). olive oil . . /* = 3'2653e-' 01232 '* j With the same unit of length, but g grammes as unit of mass and 1 gramme as unit of force, the values of yu. are for water .../* = O'OOOOISIP \ air. . . . fi = 0-00000019153 (1 + -003662 7 ) L (5). olive oil . . p, = 0-0033303e- 123r j The units of length and mass being the foot and pound and the temperature in degrees Fahr. for water . . . /* = 0'0011971P air. . . . fi = 0-000011788 (1 + -0020274T) (6). olive oil . . ^ = 0-21943e- 022ir With the same unit of length the unit of mass being g (32'1695) Ibs. and the unit of force 1 Ib. for water .../*- 0'000037166P \ air.-. . . /* = 0-00000036645(1 + '002074^)1 (7). olive oil . . fi = 0-00682 13e-' 022ir J Taking the unit of length 1 inch and the unit of force 1 Ib. for water . . . p = 0'000000258105P \ air. . . . p = 0-0000000025447 (1 + '0020274T) I (8). olive oil . . ^ = 0-00004737e- >0221JI j SECTION III. GENERAL VIEW OF THE ACTION OF LUBRICATION. 14. Tlie case of two nearly Parallel Surfaces separated by a Viscous Fluid. Let AB and CD (Fig. 4) be perpendicular sections of the surfaces, CD being of limited but of great extent compared with the distance h * For olive oil the values of /* have only been tested between the limits of temperature 16 and 49 C. or 61 and 120 Fahr. 52] AND ITS APPLICATION TO MR B. TOWER'S EXPERIMENTS. 243 between the surfaces, both surfaces being of unlimited length in_a direction perpendicular to the paper. Fig. 4. Case 1. Parallel Surfaces in Relative Tangential Motion. In Fig. 5 the surface CD is supposed fixed, while AB moves to the left with a velo- city U. Then by the definition of viscosity (Art. 9) there will be a tangential resistance W U F==fJ 'J' and the tangential motion of the fluid will vary uniformly from U at AB to zero at CD. Thus if FG (Fig. 5) be taken to represent U, then PN will represent the velocity in the fluid at P. Fig. 5. The slope of the line EG therefore may be taken to represent the force F, and the direction of the tangential force on either surface is the same as if EG were in tension. The sloping lines therefore represent the condition of motion and stress throughout the film (Fig. 5). Case 2. Parallel Surfaces approaching with no Tangential Motion. The fluid has to be squeezed out between the surfaces, and since there is no motion at the surface, the horizontal velocity outward will be greatest half-way between the surfaces, nothing at the middle of CD, and greatest at the ends. If in a certain state of the motion (shown by dotted line, Fig. 6) the space between AB and CD be divided into 10 equal parts by vertical lines (Fig. 6, dotted figure), and these lines be supposed to move with the fluid, 162 244 ON THE THEORY OF LUBRICATION [52 they will shortly after assume the positions of the curved lines (Fig. 6), in which the areas included between each pair of curved lines is the same as Fig. 6. Ill the dotted figure. In this case, as in Case 1, the distance QP will represent the motion at any point P, and the slope of the lines will represent the tangential forces in the fluid as if the lines were stretched elastic strings. It is at once seen from this that the fluid will be pulled towards the middle of CD by the viscosity as though by the stretched elastic lines, and hence that the pressure will be greatest at and fall off towards the ends C and D, and would be approximately represented by the curve at the top of the figure. Case 3. Parallel Surfaces approaching with Tangential Motion. The lines representing the motions in Cases 1 and 2 may be superimposed by adding the distances PQ in Fig. 6 to the distances PN in Fig. 5. The result will be as shown in Fig. 7, in which the lines represent in the same way as before the motions and stresses in the fluid where the surfaces are approaching with tangential motion. In this case the distribution of pressure over GD is nearly the same as in Case 2, and the mean tangential force will be the same as in Case 1. The distribution of the friction over CD will, however, be different. This is shown by the inclination of the curves at the points where they meet the surface. Thus on CD the slope is greater on the left and less on the AND ITS APPLICATION TO MR B. TOWER'S EXPERIMENTS. 245 right, which shows that the friction will be greater on the left ancL less on the right than in Case 1. On AB the slope is greater on the right and less on the left, as is also the friction. Case 4. Surfaces inclined with Tangential Movement only. AB is in motion as in Case 1, and CD is inclined as in Fig. 8. The effect in this case will be nearly the same as in the compound movement (Case 3). For if corresponding to the uniform movement U of AB, the velocity of the fluid varied uniformly from the surface AB to CD, then the quantity carried across any section PQ would be U PQx 2' and consequently would be proportional to PQ; but the quantities carried across all sections must be the same, as the surfaces do not change their relative distances ; therefore there must be a general outflow from any vertical sections PQ, P'Q' given by f (PQ-P'Q'X This outflow will take place to the right and left of the section of greatest pressure. Let this be PjQ^ then the flow past any other section PQ is f (PQ-PM to the right or left according as PQ is to the right or left of PjQj. Hence at this section the motion will be one of uniform variation, and to the right and left the lines showing the motion and friction will be nearly as in Fig. 7. This is shown in Fig. 9. This is the explanation of continuous lubrication. The pressure of the intervening film of fluid would cause a force tending to separate the surfaces. 246 ON THE THEORY OF LUBRICATION [52 The mean line or resultant of this force would act through some point 0. This point does not necessarily coincide with P 1? the point of maximum pressure. Fig. 9. For equilibrium of the surface AB, will be in the line of the resultant external force urging the surfaces together, otherwise the surface ACD would change its inclination. The resultant pressure must also be equal to the resultant external force perpendicular to AB (neglecting the obliquity of CD). If the surfaces were free to approach the pressure would adjust itself to the load, for the nearer the surfaces the greater would be the friction and consequent pressure for the same velocity, so that the surfaces would approach until the pressure balanced the load. As the distance between the surfaces diminished would change its position, and therefore, to prevent an alteration of inclination, the surface CD must be constrained so that it could not turn round. It is to be noticed that continuous lubrication between plane surfaces can only take place with continuous motion in one direction, which is the direction of continuous inclination of the surfaces. With reciprocating motion, in order that there may be continuous lubri- cation, the surfaces must be other than plane. 15. Revolving Cylindrical Surface. When the moving surface AB is cylindrical and revolving about its axis, the general motion of the film will differ somewhat from what it is with flat surfaces. Case 5. Revolving Motion, CD flat and symmetrically placed. The surface velocity of AB may be expressed by U as before. The curves of motion found by the same method as in the previous cases are shown in Fig. 10. 52] AND ITS APPLICATION TO MR B. TOWER'S EXPERIMENTS. 247 The curves to the right of GH, the shortest distance between the surfaces, will have the same character as those in Fig. 9 to the right of G, at which is also the shortest distance between the surfaces. Fig. 10. On the left of GH the curves will be exactly similar to those on the right, only drawn the other way about, so that they are concave towards a section at P 2 in a similar position on the left to that occupied by Pj on the right. This is because a uniformly varying motion would carry a quantity of fluid proportional to the thickness of the stratum from right to left, and thus while it would carry more fluid through the sections towards the right than it would carry across GH, necessitating an outward flow from the position P l in both directions, the same motion would carry more fluid away from sections towards G than it would supply past GH, thus necessitating an inward flow towards the position P 2 . Since G is in the middle of CD these two actions, though opposite, will be otherwise symmetrical, and From the convexity of the curves to the section at P 2 it appears that this section would be one of minimum pressure, just as P t is of maximum. Of course this is supposing the lubricant under sufficient pressure at C and D to allow of the pressure falling. The curve of pressure would be similar to that at the top of Fig. 10, in which C and D are points of equal pressure, P 1 HP. 2 the singular points in the curve. Under such conditions the fluid pressure acts to separate the surfaces on the right, but as the pressure is negative on the left the surfaces will be 248 ON THE THEORY OF LUBRICATION [52 drawn together. So that the total effect will be to produce a turning moment on the surface AB. Case 6. The same as Case 5, except that is not in the middle of CD. In this case the curves of motion will be symmetrical on each side of H at equal distances, as shown in Fig. 11. Fig. 11. If C lies between H and P 2 the pressure will be altogether positive, as shown by the curve above Fig. 11 that is, will tend to separate the surfaces. 16. The Effect of a Limiting Supply of Lubricating Material. In the cases already considered C and D have been the actual limits of the upper surface. If the supply of lubricant is limited C and D may be the extreme points to which the separating film reaches on the upper surface, which may be unlimited, as in Fig. 12. Case 7. Supply of Lubricant Limited. If the surface AB be supposed to have been covered with a film of oil, the oil adhering to the surface and moving with it, then the surface CD to have been brought up to a less distance than that occupied by the film of oil, the oil will accumulate as it is brought up by the motion of AB, forming a pad between the surfaces, particularly on the side D. The thickness of the film as it leaves the side C being reduced until the whole surface AB is covered with a film of such thinness that as much leaves at C as is brought up to D, then the condition will be steady. 52] AND ITS APPLICATION TO MR B. TOWER'S EXPERIMENTS. 249 Putting b for the thickness of the film of oil outside the pad, the quantity of oil brought up to D by the motion of this film will be per second bU, Fig. 12. and the quantity which passes the section P 1 Q 1 , across which the velocity varies uniformly, will be 2 Therefore since there is no further accumulation also, since GP 2 =GP, (Fig. 10, Case 5) P 2 Q 2 = 26. And since the quantity which passes P 2 Q 2 will not be sufficient to occupy the larger sections on the left, the fluid will not touch the upper surface to the left of P 2 . The limit will therefore be at P 2 , the fluid passing away with AB in a film of thickness 6. This is the ordinary case of partial lubrication : AB, the surface of the journal, is covered with a film of oil ; CD, the surface of the brass or bearing, is separated from AB by a pad of oil near H, the point of nearest approach. This pad is under pressure, which is a maximum at P lt and slopes away to nothing at D and P 2 , the extremities of the pad, as is shown by the curve above, Fig. 1 2. 250 ON THE THEORY OF LUBRICATION [52 17. The Relation between Resistance, Load, and Speed for Limited Lubrication. In Case 7 a definite quantity of oil must be in the film round the journal, or in the pad between the surfaces. As the surfaces approach, the pad will increase and the film diminish, and vice versd. The resistance increases with the length of the pad, and with the diminution of the distance between the surfaces. The mean intensity of pressure increases with the length of the pad, and inversely with the thickness of the film, but not in either case in the simple ratio. The total pressure, which is equal to the load, increases with the intensity of pressure and the length of the pad. The definite expressions of these relations depend on certain integrations, which have not yet been effected. From the general relations pointed out, it follows that an increase of load will diminish HG and PiQ lt and con- sequently the thickness of the film round the journal, and will increase the length of the pad. It will therefore increase the friction. Thus with a limited supply of oil the friction will increase with the load in some ratio not precisely determined. Further, both the friction and the pressure increase in the direct ratio of the speed, provided the distance between the surfaces and the length of the pad remains constant ; then, if the load remains constant, the thickness of the film must increase, and the length of the pad diminish with the speed ; and both these effects will diminish friction in exactly the same ratio as the reduction of load diminishes friction. Thus if with a speed U a load W and friction F a certain thickness of oil is maintained, the same will be maintained with a speed MU, a load MW, and the friction will be MF. How far this increase of friction is to be attributed to the increased velocity, and how far to the increased load, is not yet shown in the theory for this case ; but, as has been pointed out, if the load be altered from M W to W, the velocity remaining the same, the friction will be altered from MF in the direction of F. Therefore, with the load constant, it does appear from the theory that the friction will not increase as the first power of the velocity. There is nothing therefore in this theory contrary to the experience that, with very limited lubrication, the friction is proportional to the load and independent of the velocity, while the theoretical conclusion that the friction, with any particular load arid speed, will depend on the supply of oil in the pad, is in strict accordance with Mr Tower's conclusion, and with the general disagreement of the coefficients of friction in different experiments. 52] AND ITS APPLICATION TO MR B. TOWER'S EXPERIMENTS. 251 17 A. The Conditions of Equilibrium with Cylindrical Surfaces. So far CB has been considered as a flat surface, in which case the equili- brium of CB requires that it should be so far constrained by external forces that it cannot either change its direction or move horizontally. When AB is a portion of a cylindrical surface, having its axis parallel to that of AB, the only condition of constraint necessary for equilibrium is that CB shall not turn about its axis. This will appear on consideration of the following cases : Case 8. Surfaces Cylindrical and the Supply of Oil Limited. Fig. 13 shows the surfaces AB and CD. Fig. 13. J is the axis of the journal A B. I is the axis of the brass CD. JL is the line in which the load acts. is the point in which JL meets AB. R = JP. R + a = IQ. h = PQ. h = HG. The condition for the equilibrium of 7 the centre of the brass is that the resultant of the oil pressure on DC together with friction shall be in the direction OL, and the magnitude of this resultant shall be equal to the load. 252 ON THE THEORY OF LUBRICATION [52 As regards the magnitude of this resultant, it increases as HG diminishes to a certain limit, i.e., as the surfaces approach, so that in this respect equili- brium is obviously secured, and it is only the direction of the resultant pressure and friction that need be considered. Since the fluid film is in equilibrium under the forces exerted by the two opposite surfaces, these forces must be equal and opposite, so that it is only necessary to consider the forces exerted by AB on the fluid. From what has been already seen in Cases 6 and 7 it appears that the resultant line of pressure JM always lies on the right or on side of GH. The resultant friction clearly acts to the left, so that if JM be taken to represent the resultant pressure and MN the resultant friction, N is to the left of M and JN the resultant of pressure and friction is to the left of JM. Taking LJ to represent the load, then LN will represent the resultant moving force on GD that is on /. Since H will move in the opposite direction to /, and since the direction of the resultant pressure moves in the same direction as H, the effect of a moving force LN on / will be to move N towards L until they coincide. Thus, as long as JM is within the arc covered by the brass, a position of equilibrium is possible and the equilibrium will be stable. So far the condition of equilibrium shows that H will be on the left or off side of the line of load, and this holds whether the supply of oil is abundant or limited ; but while with a very limited supply of oil, i.e., a very short oil pad, H must always be in the immediate neighbourhood of 0, this is by no means the case as the length of the oil pad increases. Case 9. Cylindrical, Surfaces in Oil Bath. If the supply of oil is sufficient, the oil film or pad between the surfaces will extend continuously from the extremities of the brass, unless such extension would cause negative pressure which might lead to discontinuity. In this case the conditions of equilibrium determine the position of H. The conditions of equilibrium are as before 1. That the horizontal component of the oil pressure on the brass shall balance the horizontal component of the friction ; 2. That the vertical components of the pressure and friction shall balance the load. Taking the surface of the brass, as is usual, to embrace nearly half the circumference of the journal and, to commence with, supposing the brass to be unloaded, the movement of H may be traced as the load increases. When there is no load, the conditions of equilibrium are satisfied if the 52] AND ITS APPLICATION TO MR B. TOWERS EXPERIMENTS. 253 position of H is such, that the vertical components of pressure and friction are each zero, and the horizontal components are equal and opposite. This will be when H is at (Fig. 13); for then, as has been shown, Case 5, the pressure on the left of H will be negative, and will be exactly equal to the pressure at corresponding points on the right, so that the vertical com- ponents left and right balance each other. On the other hand the horizontal component of the pressure to the left and right will both act on the brass to the right, and as these will increase as the surfaces approach, the distance JI must be exactly such that these components balance the resultant friction, which by symmetry will be horizontal and acting to the left. It thus appears that when the brass is unloaded its point of nearest approach will be its middle point. This position, together with the curves of pressure, are shown in Fig. 14. Fig. 14. As the load increases, the positive vertical component on the right of GH must overbalance the negative component on the left. This requires that H should be to the left of 0. It is also necessary that the horizontal components of pressure and friction should balance. 254 ON THE THEORY OF LUBRICATION [52 These two conditions determine the position of H and the value of JI. As the load increases it appears from the exact equations (to be discussed in a subsequent article) that OH reaches a maximum value, which places H nearly, but not quite, at the left extremity of the brass, but leaves JI still small as compared with GH. For a further increase of the load H moves back again towards 0. In this condition the load has become so great that the friction, which remains nearly constant, is so small by comparison that it may be neglected, and the condition of equilibrium is that the horizontal component of the pressure is zero, and the vertical component equal to the load. H continues to recede as the load increases. But when H C becomes greater than HP. 2 , the pressure between P 2 arid C would become negative if the condition did not break down by discontinuity in the oil, which is sure to occur when the pressure falls below that of zero, and then the condition becomes the same as that with a limited supply of oil. This is important, as it shows that with extreme loads the oil bath comes to be practically the same as that of a limited supply of oil, and hence that the extreme load which the brass would carry would be the same in both cases as Mr Tower has shown it to be. In all Mr Tower's experiments with the oil bath it appears that the conditions were such that as the load increased H was in retreat from C towards 0, and that, except in the extreme cases, P 2 had not come up to C. Figs. 2, 3, 4, show the exact curves of pressure as calculated by the exact method to be given, for circumstances corresponding very closely with one of Mr Tower's experiments, in which he actually measured the pressure of oil at three points in the film. These measured pressures are shown by the crosses. The result of the calculations for this experiment is to show, what could not indeed be measured, that in Mr Tower's experiment the difference in the radii of the brass and journal at 70, and a load of 100 Ibs. per square inch was : a = -00077 #=000375 (The angle) OJH = 48. 18. The Wear and Heating of Bearings. Before the journal starts the effect of the load will have brought the brass into contact with the journal at 0. At starting the surfaces will be in 52] AND ITS APPLICATION TO MR B. TOWER'S EXPERIMENTS. 255 contact, and the initial friction will be between solid surfaces, causing some abrasion. After motion commences the surfaces gradually separate as the velocity increases, more particularly in the case of the oil bath, in which case at starting the friction will be much the same as with a limited supply of oil. As the speed increases according to the load, GH approaches, according to the supply of oil, to a, and varies but slightly with any further increase of speed ; so that the resistance becomes more nearly proportional to the speed and less affected by the load. When the condition of steady lubrication has been attained, if the surfaces are completely separated by oil, there should be no wear. But if there is wear, as it appears from one cause or another there generally is, it would take place most rapidly where the surfaces are nearest : that is, at GH on the off side of 0. Thus while the motion is in one direction the tendency to wear the surfaces to a fit would be confined to the offside of 0. This appears to offer a very simple and well-founded explanation of the important and common circumstance that new surfaces do not behave so well as old ones ; and of the circumstance, observed by Mr Tower, that in the case of the oil bath, running the journal in one direction does not prepare the brass for carrying a load when the journal is run in the opposite direction. This explanation, however, depends on the effect of misfit in the journal and brass which has yet to be considered. Case 10. Approximately cylindrical surfaces of limited length in the direction of the axis of rotation. Nothing has so far been said of any possible motion of the fluid perpendicular to the direction of motion and parallel to the axis of the journal. It having been assumed that the surfaces were truly cylindrical and of unlimited length in direction of their axes, and in such case there would be no such flow. But in practice brasses are necessarily of limited length, so that the oil can escape from the ends of the brass. Such escape will obviously prevent the pressure of the film of oil from reaching its full height for some distance from the ends of the brass and cause it to fall to nothing at the extreme ends. This was shown by Mr Tower, who measured the pressure at several points along the brass in the line through 0, and found it to follow a curve similar to that, shown in Fig. 15, which corresponds to what might be expected from escape at the free ends. 256 ON THE THEORY OF LUBRICATION [52 If the surfaces are not strictly parallel in the directions TU and VW, the pressure would be greatest in the narrowest parts, causing axial flow from those into the broader spaces. Hence, if the surfaces were considerably Journal Fig. 15. irregular, the lubricant would, by escaping into broader spaces, allow the brass to approach and eventually to touch the journal at the narrowest spaces, and this would be particularly the case near the ends. As a matter" of fact, the general fit of two new surfaces can only be approximate ; and how nearf the approximation is, is a matter of the time and skill spent on preparing, or, as it is called, bedding them. Such bedding as brasses are subject to would not bring them to a condition in which the hills and hollows differed by less than a YIJOOO^^ P art f an i ncn > so that two such surfaces touching each other on the hills would have spaces as great as of an inch between them. This seems a small matter, but not a when compared with the mean width of the interval between the brass and the journal which, as will be subsequently shown, was less than -nfoffth f an inch. It may be assumed, therefore, that such inequalities generally exist in the surfaces of new brasses and journals. And as the surfaces according to their material and manner of support yield to pressure the brass will close on the journal at its ends, where, owing to the escape of oil, there is no pressure to keep them separate. rn'] AND ITS APPLICATION TO MR B. TOWER'S EXPERIMENTS. 257 The section of a new brass and journal taken at GH will therefore be, if sufficiently magnified, as shown in Fig. 16, the thickness of the film, Journal Longitudinal Section Fig. 16. instead of being, say, of y^f^ths of an inch, varies from to y^^ ths, and is less at the ends than at the middle. In this condition the wear will be at the points of contact, which will be in the neighbourhood of GH on the off side of (Fig. 13), so that, if the journal runs in one direction only, the surfaces in the neighbourhood of GH (on the off side) will be gradually worn to a fit, during which wear the friction will be great and attended with heating, more or less, according to the rate of wear and the obstruction to the escape of heat. So long, however, as the journal runs in one direction only GH will be on one side (the off side) of 0, and the wear will be altogether or mainly on this side, according to the distance of H from 0. In the meantime the brass on the on side is not similarly worn, so that if the motion of the journal is reversed, and the point H transferred to the late on side, the wear will have to be gone through again. That this is the true explanation is confirmed if, as seems from Mr Tower's report, the heating effect on first reversing the journal was much more evident in the case of the oil bath. For when the supply of oil is short, HG will be very small, and H will be close to 0. So that the wearing area will probably extend to both sides of 0, and thus the brass be partially, if not altogether, prepared for running in the opposite direction. When the supply of oil is complete, however, as has been shown, H is .10 or 60 from 0, unless the load is in excess, so that the wear in the neighbourhood of H on the one side of would not extend to a point 100 or 120 over to the other side. Even in the case of a perfectly smooth brass, the running of the journal under a sufficient load in one direction should, supposing some wear, ac- cording to the theory render the brass less well able to carry the load when running in the opposite direction. For, as has already appeared, the pressure between the journal and brass depends on the radius of curvature of the o. R. ii. 17 258 ON THE THEORY OF LUBRICATION [52 brass on the on side being greater than that of the journal. If then the effect of wear is to diminish the radius of the brass on the off side, so that when the motion is reversed the radius of the new on side is equal to or less than that of the journal, while the radius of the new off side is greater, the oil pressure would not rise. And this is the effect of wear ; for as will be definitely shown, the effect of the oil pressure is to increase the radius of curvature of the brass, and as the centre of wear is well on the off side, the effect of sufficient wear will be to bring the radius on this side, while the pressure is on, more nearly to that of the journal, so that on the pressure being removed, the brass on this side may resume a radius even less than that of the journal. SECTION IV. THE EQUATIONS OF HYDRODYNAMICS AS APPLIED TO LUBRICATION. 19. According to the usual method of expressing the stress in a viscous fluid (which is the same as in an elastic solid)* : Pzz=~P Pxy = P,,x = /* Pyz=pzy = /* (Y\ fr\ PZX PXZ dv dw dtK dy dz du dv dw du dv du\ ] dx dy 1 dw dv \ + - K. jly dz ) du dw\ dz ~dx ) , (10). In which the left-hand members are the stresses on planes perpendicular to the first suffix in directions parallel to the second, the first three being the normal stresses, the last six the tangential stresses. * Stokes, " On the Theories of the Internal Friction of Fluids in Motion, and of the Equi- librium and Motion of Elastic Solids." Trans. Cambridge Phil. Soc., vol. vni., p. 287. Also reprint, vol. i., p. 84. Also Lamb's Motion of /'/<*/ _ , , { d_\dp_( _/\,ll d_\TT^~y jjy\ Integrating from y = Q to ij = h, and substituting from conditions (15) d I dp\ d / , 3 dj. From equations (16) and (14) (l). Putting f x fz for the shearing stresses at the solid on the surfaces in the directions x arid z respectively, then taking the positive sign when y = h, and the negative when y ...(19). Equations (17) and (19) are the general equations of equilibrium for the lubricant between continuous surfaces at a distance h, where h is any continuous function of & and z, and p is constant. 262 ON THE THEORY OF LUBRICATION [o2 22. For the further integration of these equations it is necessary to know the exact manner in which x and z enter into h, as well as the function which determines the limit of lubricated surfaces. These integrations have been effected either completely or approximately for certain cases, which include the chief case of practical lubrication. Complete integration has been obtained for the case of two parallel circular or elliptical surfaces approaching without tangential motion. This case is interesting from the experiment, treated approximately by Stefan*, of one surface-plate floating on another in virtue of the separating film of air. It is introduced here, however, as being the most complete as well as the simplest case in which to consider the important effect of normal motion in the action of lubricants. It corresponds with Case 2, Section III. Complete integration is also obtained for two plane surfaces 7 7 it *'\ h = /ij 1 + m - \ a/ between the limits at which p = II (the pressure of the atmosphere) x = 0, x = a, the surfaces being unlimited in the direction of z. This corresponds with Case 4, Section III. For the most important case, that of cylindrical surfaces, approximate integration has been effected for the case of complete lubrication with the surfaces unlimited in the direction of z. Case 9, Section III. SECTION V. CASES IN WHICH THE EQUATIONS ARE COMPLETELY INTEGRATED. 23. Two Parallel Plane Surfaces approaching each other, the Surfaces having Elliptical Boundaries. Here h is constant over the surfaces, and when /*i2 ^2 -, + 3-1, p = tt (20), or c~ U , U l are zero. * Wien. Sitz. Ber., vol. 69 (1874), p. 713. ~>2] AND ITS APPLICATION TO MR B. TOWER'S EXPERIMENTS. 263 Equation (17) becomes d d \ 12ft dh The solution of which is Therefore 2 (t) ( + and E l = Q, &c. . . n ) p_n=^^ -- I- + - - ll ^ . ...(24). h 3 a? + c 2 (a 2 c 2 j dt From equations (19) _ _ 24ft a-c 2 dh\ "***[ (25 ), /. _ 24/z a 2 dh * z ~ ~hT a 2 +c 2 '^ rf7/ supposing surfaces horizontal and the upper surface supported solely by the pressure of the fluid. The conditions of equilibrium in this case are obvious by symmetry. The centre of gravity of the load must be vertically over the centre of the ellipse. Since by symmetry n pxdxdz = j ^' __J* " f (26). nf x dxdz=Q I no V (l j) f z dxdz = B 264 ON THE THEORY OF LUBRICATION [52 And ra>/l-- W=l I 'p-Udxdy (27) J 3/i7r a 3 c 3 eM H 3 " a' 2 + c 2 ' dt .(28). Therefore integrating *-/^sr(fi-fi) ( 29 )> (a 2 + c 2 ) If V/*. 2 2 i / being the time occupied in falling from h-^ to 7< 2 . 21. Plane Surfaces of unlimited Length and parallel in t/te Direction of z. The lower surface unlimited in the direction x and moving with a velocity U. The upper surface fixed and extending from x = to x = a. This case corresponds with Case 4, Section III. The boundary conditions are a > (30) y = k U l = V 1 = , / O!\ ft = h 1 + in - \ aj p is a function of x only. And from equation (17), Section IV., by integration being the value of h when x= x l where the pressure is a maximum. Integrating with respect to x, and putting p = II at the boundaries -2- cm t * 1 ~~ o i m {<**) ^ :: : ,.., _*j. (88> I 1 + W - I 1 + VM a \ ( '"' f "i+il 52] AND ITS APPLICATION TO MR B. TOWER'S EXPERIMENTS. 265 or putting W for the load per unit of breadth, TV is a maximum when m = l'2 approximately and again, by equation (19) f, = Pjf (36); therefore I f x dx = , log e (1 + in) (37) ; and if m 1'2 ^=6572^ (38). In order to render the application of equations (35) and (38) clear, a particular case may be assumed. Let fi = 10~ 5 , which is the value for olive oil at a temperature of 70 Fahr., the unit of length being the inch, and that of force the Ib. Let U = 60 (inches per sec.) A, = -0003. Then from (35), the load in Ibs. per square inch of lubricated surface is given by W = 1070a 2 , a and from (38), the frictional resistance in Ibs. per square inch is F_ a This seems to be about the extreme case of perfect lubrication between plane metal surfaces having what appears to be about the minimum value of h,. 266 ON THE THEORY OF LUBRICATION [52 SECTION VI. THE INTEGRATION OF THE EQUATIONS FOR THE CASE OF CYLINDRICAL SURFACES. 25. General Adaptation of the Equations. Fig. 17 represents a section of two circular cylindrical surfaces at right angles to the axes ; as in Art. 17. J is the axis of the journal AB ; / is the axis of the brass CD ; Fig. 17. JO is the line of action of the load cutting the brass symmetrically, and R = JP h = PQ h = HG, the smallest section JI=ca ^JQ = 0i. PI being the point of maximum pressure. ...(39). ">:>] AXl) ITS APPLICATION TO MH B. TOWER'S EXPERIMENTS. 207 Then taking x for distances measured in the direction OA from on the surface AB, and putting r for the distance of any point from /, x = R6 ] J .............................. (40) y = r - .RJ (41). ccT 2 Neglecting quantities of the order ,, Ki h=a{l + c sin (0 - )} ........................ (42). For if / be moved up to /, Q moves through a distance ca in the direction JH. The boundary conditions are such that (1) all quantities are independent of z\ (2) f/o is constant, U and V l = ; (3) putting # = OJA, 0! = OJB, whence by symmetry - l , Putting L for the effect of the external load and M for the external moment per unit of length in the direction z, and assuming that there are no external horizontal forces, the conditions of equilibrium for the brass are fi (psin0-/cos0) dB = (44) J* /, T. {pcos0+fsiu0}d0 = ~^ (45) J e -ti [ e M //"-I <> Substituting from equations (40), (41), (42) in equations (17) and (19), Section IV., putting a/ a K, = ^ (47) and remembering the boundary conditions, these equations become on integration dp = 6RpU c{sm(0- )-siii (, -j,, 8 - 3 208 ON THE THEORY OF LUBRICATION 3/A U Q c (sin (0 - ) + ^ 2;i+1 sin {(2* + 1) (0 - sin (0 - ) + J5 3 cos 2 (6> -) + &c. ll cos 2*' (0 - <) + 5 2M+1 sin \(2x + 1) (0 - ...(50) Putting = sn < 2)(r+l)r(r- 1)... .(52) r=2 + 2 + -"271+1 ~~ ("" */ (2n + 2) (27 22JI+1 22W+1 r=2n+3 r - 2n - 1 2 + .(53). 52] AND ITS APPLICATION TO MR B. TOWERS EXPERIMENTS. 269 , ( r + 2 r=2*> L ( ? + )J 7 ( ) 2 r=2 2 r 2 c r c r+1 x - 3 (2rc r=2< [4-3 (2vi + 2)] c Ml+1 - ]r .(r L) ... ^ 2r-l 2) c 271 - 1 - 2 x 2 r=2o+3 r CJf*- .(54). The coefficients ,4 , .4i, &c., B , B l , &c., are thus expanded in a series of ascending powers of c with numerical coefficients which do not converge. It seems, however, that if c is not greater than - 6 the series are themselves convergent, and it is only necessary to go to the tenth or twelfth term, to which extent they have been calculated, and are as follows : A, = - V5c - 3'75c 3 - 6-565C 5 - 9"85c 7 - 13'51c 9 - 1 7'Gc 11 - {1 + 3c s + 5-625c 4 + 8-7 5c G + 12'225c 8 4- 16'2c 10 } x A, = 1 + 4-oc 2 + 9'375c 4 + 15'23c 6 + 21-92c" + 29"8c 10 + 38'6c 12 + (3c + 7-oc 3 + 1313c 5 + 19-7c 7 + 27'Olc 9 + 35'2c n } % A 2 = I'oc + 5c ! + 9'85c 5 + 15'75c 7 + 22'56c 9 + 20'24c n + |3c s + 7'5c 4 + 13-13c 6 + 19-7C 8 -(- 27'02c 10 + 35'2c 12 } ; A 3 -- I'oc 2 - 4-7c 4 - 9'2c 6 - 14-7c 8 - 21'45c 10 - {2-5c 3 + 6-5Gc 5 + 1 l'78c 7 + 18'03c 9 + 25'4c u } % A, = - 1-25C 3 - 3-94c 5 - 7'875c 7 - 12-88C 9 - 18'8c n - {1-875C 4 + 5'2oc 6 + 9'85c 8 + 15'48c 10 + 22'45c 12 } x \ r /V At = -939C 4 + 3'07c 6 + 6'33c 8 + 10'68c 10 + {2-63c 5 + 3-94c 7 + 776C 9 + 12'Gc 11 ] % (55). 270 ON THE THEORY OF LUBRICATION = I - {2-5c 2 + 4125c 4 + 5-.312.")C 6 + 6"54c 8 } - (3c -f 4-5c 3 + 5-625c 5 + 6'562c 7 + 7'63c 9 } , = 2c+ 6c 3 + 9-75c + ir3125c 7 + {6c 2 + 9c 4 + 11-25C 6 + 12-5c 8 } % 2 = 2-5c 2 + 5-5c 4 + 7-97c" + lO'OGc 8 + (4-5c 8 + 7-5c 5 + 9'85c 7 + 11 -8c 9 } % S = - 2c 3 - 4-37oc 5 - 5625c 7 - 8'61c 9 - {3c 4 + 5'625c 6 + 7-873c 8 + 9'84c 10 } x , = - l-375c 4 - 3-2c 6 - 5'03c 8 - (l-875c 8 + 3'94c 7 + 5'09c 9 } x [52 27. The Integration of the Equations. Integrating equation (50) between the limits and P-PQ _ A ff) fi x K lC ' A^ {cos (0 (j) n ) cos (# <)} + -JT {sin 2 (0 < ) sin 2 (# < rt )} &c. &c. _ 2M+1 {cos f(2n + 1) (0 - <&)! - cos [(2w V^i I I * whence putting 6 = ^ by condition (43) j = ^0^1 - A l sin ^i sin (f> + ~ sin 2^ cos 2< - &c. 2 sin [(2n + 1) 0,] sin (2n + 1) ^ Putting j 1 . / . E = A l cos ^i cos <^>o + ~^~ cos 2^i sin 2< + &c. + ^^T cos K 2w + !) 0il cos ^ 2w + !> 0o .(56). (57), .(58). cos 2w^, sin 2n<6 (59), 52] AND ITS APPLICATION TO MR B. TOWER'S EXPERIMENTS. 271 whence from equations (57) and (58) ~ = E + A - A, cos (0 - ) + 2 sin 2(0- ) (60). Multiplying equation (60) by sin and integrating between and l remembering that n = fl l re, p _ p -j^-- sin 0d0 = 2A (sin 9 l l cos 0J J e a -K-C /sin 20i sin , - g - - 1 sin A 9 f . a sin 30 cos 20\ -y (sm0cos20 -- &c. &c. sin (2n + 2) ^ sin (2n + 1) ^> sin < 2n0 l sin (2?? + 1) ^ 2/1 + 2 2rc AM (sin (2n + 1) O l cos 2w sin (2n + 1)0 cos 2n0| '2n { 2n - 1 ~~2wTT~ ' j Multiplying equation (60) by cos 0, and integrating from to 0^ atn A 2 /sin'30 1 . i A - O j\ e + -^ I - sin 2< f sin sin 20 ) &c. SB \ / A. 2n+l (2// + 1 gin + 2 ) In + 2//' (2w + 1 Multiplying equation (51) by cos# and integrating [ e /cos - y - 1 > = J e -K-2 sn j sn . sin (j - - + l sin /sin 3^ cos 20 . + B. 2 ( g r + sin 9 l cos 20o - &c. D (sin (2?i 4- 1 ) #1 cos 2w0 sin ( 2n 1) 6 l cos 2n + l 2n-l J (sin(2n + 2)g 1 sin(2ra + l)0 sin 2n^ sin (2?t + 1 -" i+1 | 2n+2 2n ..................... (63) 272 ON THE THEORY OF LUBRICATION [52 Multiplying equation (51) by sin 9 and integrating 2<9 1 cos(f> - -sing, sin R (sin (2w + 1) B{ sin 2n(j) sin (2?? 1) O l sin 2re^) i 2?i + 1 "2n I } (sin (2 + 2) ^ cos (2n + 1) ^> sin ^n9 l cos (2?i 1) < ] l ~"~ " "5T ...... (64). Integrating equation (51) - T' fy. = 2B ^ - 25j sin ^ sin <^> -f 25a sin 2^ cos ./ ^ 2 ^ sn nj cos 2?i sin (2/i + 1) 0jsin(2?i + 1) <^> (65). Substituting from equations (61) (65) in the equations of equilibrium (44), (45), (46), there results From (44) = 2 (KjAo + KzB ) sin 0, - ^K^A^ cos (9, K,B,) sin < n - (A^c^ a + /T.,5,) ft, sin + &c. c^an ^ sin (2n - 1) 0, cos ~ " - if r> 1 cos 2n + 1 sin 2n0j sin (2n - -g^- K 1 cA m+1 ^ sin (2n + 1) ^ sin (2n f 1 c m+1 R ^ sn n + sn n + (> V 2w + 1 ~ Aa ^ +1 J ' 2rc + 2 "J ...... ( 52] AND ITS APPLICATION TO MR B. TOWER'S EXPERIMENTS. 273 From (45) - , cos sn>i- cos (2n + 1) ^ (67) From (46) If I 97? ?A - 25, sin ^ sin (/> + - 2 sin 2(9, cos 2 4 sin 2n0 l cos 2w< 2/Jan^i /,- ) ! sin (2w+l) (70). o. B. ii. 18 274 ON THE THEORY OF LUBRICATION [52 Equation (66) gives = (2^0% - 2^T 2 ) sin 0, - 2^0%^ cos ^ - ^c bm ' sin fa sin ............... (71). Equation (67) gives L -, /sin 20j ^ , osfa ..................... (72) and equation (68) gives (73). Also equation (57) P Po = Kc (cos #1 cos fa j(0 cos (6 )} ............ (74). Eliminating ^ between equations (70) and (71) K 2 2 sin #1 The equations (74) and (75) suffice to determine a, c, fa and under the conditions ./ _ an d c small so long as is not small, in which case the terms retained in the equations become so small that some that have been neglected rise into relative importance. To commence with let L = 0. (Cases 5 and 9, Section III.) Then by (72) cos $ = -, i /r,^ sin 1 and by (/O) % = __J > putting for ^ its value sin (fa fa) sinfl, cosfa = j ' .............................. (76). "i Equation (76) gives two equal values of opposite signs for fa. These correspond to the positions of P l and P 2) the points of maximum and minimum pressure. For the extreme cases (77). 52] AND ITS APPLICATION TO MR B. TOWER'S EXPERIMENTS. 275 From equations (73) and (47) tym a =^M ................................. ( 78 > , f a sin B l and f,,m(7o) .- ................. (79). (0i -- - +ism2^j When L increases From (72) and (75) R /sin 20! \ 2 sin fl tan < = JT, ( ? - 4 - - _ - - ...... (80). Id T ( "i - 7j Hence as L increases tan diminishes until the approximation fails. This, however, does not happen so long as c is small. As the load increases from zero, equation (80) shows that G moves away from towards A. It also appears from equation (70) that % and < t diminish as diminishes, and that fa is positive as long as the equations hold. To proceed further it is necessary to retain all terms of the first order of small quantities. Retaining the first power of c only, equations (55) become A = - 1-oc - x B = 1 - 3c % .(81). I Ao = l'5c From (58) sin 20, v(0, + 3csin } sin ) = sin 0, sin r5c0, + I'.V - cos 2<6 n ...(821 /v \ J O r u V / From (66) = } ^K\c (1'oc + %) + 2^ (1 ~ ^ c %)} g i n 0i l'5c + Y) 0i cos0, + [K lC (1 + 3c X ) - 2K 2 c] ~~ 1 sin ^ - [K lC (1 + 3cx) + 2JSTjc} ^, sin s20 ..................... (83). 182 276 ON THE THEORY OF LUBRICATION [52 From (67) L ( sin 2#A = - [Ki c (1 + 3c%) ~ 2/Tgc] ( 0! ~ cos (J sin 30j sin 2< - sin D sin 2< ) (84). T From (68) M - 4c sin 0, sin } (85). In the equations (82) to (85) terms have been retained as far as the second power of c, but these terms have very unequal values. As ^ and sin diminish c increases, and the products of c% or c sin < may be regarded as never becoming important and be omitted when multiplied by K^c or 7T 2 . Making such omissions and eliminating ^ between (82) and (83) (, / . - sin30A ..sin 20,) , sin 0! + c -$0, sin 0! - -^ - - 3 (sin X - X cos 0j) = (86). sn < = 03 f X - sm - 2 (sin 0! - 0! cos 00 sin 1 Equation (86) is a quadratic for c in terms of sin (/> , from which it is clear that as c increases from zero < goes through a minimum value when (< i /f ! , sin 30A , x sin 20 1 ' f 0j sin 0j -- - - 3 (sm 0, - 0, cos ?,) =-- \ o / ^ As the load increases from zero the value of c increases from that of equation (79) to the positive root of (87). As the load continues to increase c further increases, but again increases, so that, as shown by equation (86), for values of < greater than the minimum there are two loads, two values of c, and two values of %. If 0j is nearly - , c will be of the order \ ^ when < is small, and sin & V zzt will be of the order 4c ; so that, so long as \ / ^ is sufficiently small, no error has been introduced by the neglect of products and squares of these quantities. For example 0! - 1-37045 (78 31' 30" as in Tower's experiments) ...... (88). 52] AND ITS APPLICATION TO Mil 13. TOWER'S EXPERIMENTS. 277 By equation (86) sin = 3'934c + T9847 -^ (89). And by (87) at the minimum value of fa a 2R j / a \ sin fa = o'ol * / -y^ Putting x = sin fa sin fa equation (82) becomes sin fa = - -16776c + "5656 ^ . . .(91), Me or, when fa is a minimum, / sin 1 = -682A/^ (92). Therefore x = 4'928 A/^ (93). Equations (84) and (85) give ^ = -11753^0.. (95), J.IT whence equations (47) c = '388a4-. ...(96), Iwl p_ M < U7) - So long then as a -^ is not greater than 0'2, these approximate solutions are sufficiently applicable to any case. For greater values of -^ the solution becomes more difficult, as long how- ever as c is not greater than '5 the solution can be obtained for any particular value of c. 278 ON THE THEORY OF LUBRICATION 29. Further Approximation to the Solution of the Equations for particular Values of c. The process here adopted is to assume a value for c. From equations (53) and (54) to find A Q = A ' + A " X B = B ' + B " X (98), &c. &c. where J./, A", J5/, B" are numerical. These coefficients are then introduced into equations (58) and (66) which on eliminating % give one equation for . The complex manner in which fa enters into the equation renders solution difficult except by trial, in which way values of fa corresponding to different values of c have been found. The value of fa substituted in equations (58) or (66) gives % and fa. The corresponding values of c,, fa and % being thus obtained, a complete table might be calculated. This, however, has not been done, as there does not exist sufficient experimental data to render such a table necessary. What has been done is to obtain fa and fa for c = '5, 6^ having the value 1'3704 as in equation (88) and in Tower's experiments. The value c = '5 was chosen by a process of trial in order to correspond with the experiments in which Mr Tower measured the pressure at different parts of the journal as described in his second report, and as being the greatest value of c for which complete lubrication is certain. Putting c = '5, equations (43) and (44) give A = - 1-5351 -2-3018% # = - '012 -2-304% A^= 3-0723 + 3-0721% B,= 2-143 + 2-286% A 2 = 1-8647 + 1-5360% B.,= 1139+ '896% V (99). A 3 = - -8911- -6571% # 3 =- -455- -316% A t = - -3753- -2582% J5 4 = - -146- -097%1 A s = -1396+ -1343% Taking 1 = 1-3704 or 78 31' 30" j it was found by trial that when [ (10U), = 48 I 52] AND ITS APPLICATION TO MR B. TOWER'S EXPERIMENTS. ^79 and K. 2 was neglected (under the circumstances K 2 was about '0003^), equation (58) gave X = - -82295 or - sin 55 22' 40"\ and equation (66) gave (101). X = - -82274 - sin 55 21' 40" The difference "00021 being in the same direction and about the magnitude of the effect of neglecting K z . This solution was therefore sufficiently accurate, and adopting the value of <,-< '5 renders approxi- mation very laborious, but chiefly because it appears almost obvious that the value of c, which renders the pressure negative at the off extremity of the brass is the largest value of c under which lubrication can be considered certain. The journal may run with considerably higher values of c, the continuity of the film being maintained by the pressure of the atmosphere, which would be most likely to be the case with high speeds. But although the load which makes c = '5 is not necessarily the limit of carrying power of the journal, it would seem to be the limit of the safe working load, a conclusion which, as will appear on considering Mr Tower's experiments, seems to be in accordance with experience. This concludes the hydro-mechanical theory of lubrication so far as it has 282 ON THE THEORY OF LUBRICATION [52 been carried in this investigation. There remain, however, physical consider- ations as to the effect of variations of the speed and load on a and p, which have to be taken into account before applying the theory. SECTION VIII. THE EFFECTS OF HEAT AND ELASTICITY. 30. p> and a are only to be inferred from the experiments. The equations of the last section give directly the friction, the intensity of pressure, and the distance between the cylindrical surfaces, when the velocity, the radii of curvature of the journal and the brass, the length of the brass, and the manner of loading are known (i.e. when U, R, a, 1} L, and JM are known); and, further, if M the moment of friction is known, the equa- tions afford the means of determining a when fj, is known, or //, when a is known. The quantities U, R, 1} and L are of a nature to be easily determined in any experiment or actual case, and M is easily measured in special experi- ments, but with a and fi it is different. By no known means can the difference of radii (a) of the journal and its brass be determined to one ten-thousandth part of an inch, and this would be necessary in order to obtain a precise value of a. As a matter of fact even a rough measurement of a is impossible. To determine a, therefore, it is necessary to know the moment of friction or the distribution of pressure ; then if the value of yu- be known by experiments such as those described for olive oil (Section II.), a can be deduced from the equations for any particular value of p. But although the values of /*, may have been determined for all temperatures for the particular oil used, and that value chosen which corre- sponds with the temperature of the oil bath in the experiment, the question still arises whether the oil bath (or wherever the temperature is measured) is at the same temperature as the oil film. Considering the thinness of the film and the rapid conduction of heat by metal surfaces, it seemed at first sight reasonable to assume that there would be no great difference, but when on applying the equations to determine the value of a for one of the journals and brasses used in Mr Tower's experiments, it was found that the different experiments did not give the same values for a, and that the calculated values of a increased much faster with the velocity when the load was constant than with the load when the velocity was constant, it seemed probable that the temperature of the oil film must have varied in a manner unperceived, increasing with the velocity and diminishing the viscosity, which would account for an apparent increase of a. 52] AND ITS APPLICATION TO MR B. TOWER'S EXPERIMENTS. 283 That a should increase with the load was to be expected, considering that the materials of both journal and brass are elastic, and that the loads range up to as much as 600 Ibs. per square inch, but there does not appear any reason why a should increase with the velocity unless there is an increase of temperature in the metal. If this occurs, the apparent increase of a would be partly real and partly due to the unappreciated diminution of p owing to the rise of temperature. Until some law of this variation of temperature and of the variation of a with the load is found, the results obtained from the equations, with values of fj, corresponding to some measured temperature, such as that of the oil bath or a point in the brass, can only be considered as approximately applicable to actual results. Even so, however, the degree of approximation is not very wide as long as the conditions are such that the journal " runs cool." But, treated so, the equations fail to show in a satisfactory way what is one of the most important matters connected with lubrication the circum- stances which limit the load which a journal will carry. For, although it may be assumed that the limit is reached when ca, the shortest distance between the surfaces, becomes zero or less than a certain value, yet, accord- ing to the equations, assuming a and p to be constant, the value of c ^increases directly as U if the load be constant; so that the limiting load should increase with U. But this is not the case, for it seems from experi- ments that at a certain value of U the limiting load is a maximum if it does not diminish for a further increase of U. Although, therefore, the close agreement of the calculated distribution of the pressure over the bearing with that observed and the approximate agreement of the calculated values of the friction for different speeds and loads, such as result when /* and a are considered constant, seem to afford sufficient verification of the theory, and hence a sufficient insight into the general action of lubrication, without entering into the difficult and some- what conjectural subject of the effects of heat and elasticity, yet the possibility of obtaining definite evidence as to the circumstances which determine the limits to lubrication, which, not having been experimentally discovered, are a great desideratum in practice, seemed to render it worth while making an attempt to find the laws connecting the velocity and load with a and p. As neither the temperature of the oil film nor the interval between the surfaces can be measured, the only plan is to infer the law of the variations of these quantities for such complete series of experiments as Mr Tower's. In attempting this, a probable formula with arbitrary constants is first assumed or deduced from theoretical considerations, and then these constants 284 ON THE THEORY OF LUBRICATION [52 are determined from the experiment and the general agreement tested. In order to determine the actual circumstances on which the constants depend, it is important to obtain the formula from theoretical considerations. This has therefore been done, although these considerations would not be sufficient to establish the formulae without a close agreement with the experiments. 31. The Effect of the Load and Velocity to alter the Value of the Difference of Radii of the Brass and Journal, i.e., of a. The effect of the load is owing to the elasticity of the materials, hence it is probable that the effect will be proportional to the load L. To express this put a=a -\-rnL (109). The effect of the temperature on a is owing to the different coefficients of expansion of brass and iron. Thus : a T -a =(B'-S')(T-T )R, where B' and S' are the respective coefficients of expansion of the bearing and journal. These for brass and iron are : B' = -00001 11 S' = -0000061, therefore putting T T = Tm (the mean rise of temperature due to friction) a T =a (l + -000005 - Tm] ...(110), jy putting E for "000005 - . d If, as seems general, a is about '0005 inch, then with a 4-inch journal #=02 about (Ill), which is sufficiently large to be important. 32. The Effect of Speed on the Temperature. Putting T the temperature of the surroundings and bath, T! the mean temperature of the oil as it is carried out of the film, T m the mean rise in temperature of the film due to friction, Q the volume of the oil carried through per second, 52] AND ITS APPLICATION TO MR B. TOWER'S EXPERIMENTS. 285 D the density, S the specific heat, H the heat generated, H l the heat lost by conduction, C' a coefficient of conduction, taking the inch, lb., and degree Fahr., as units, and 12 J as mechanical equivalent of heat. m _ rn _ -" -"o ~DSQ"' Putting H-H^ = qT m q .............................. (112) where q is a constant depending on the relative values of TT and T m , also on how far the metal of the journal assists the oil in carrying out heat. On substituting the values of H lt H 2 , Q, it appears R6 = 12JC'. There does not appear to be any reason to assume any of the quantities in the denominator to be functions of the temperature except h^. By equation (42) h l =a{l +c sin (fa - <)}. Equation (11 2 A) may thus have the form T m = - L - ........................ (113), ~ where A = 3JDSqa (l +ml) [l+c sin(^ - )} ............... (114). This shows that A is a function of the load, increasing as the first power and diminishing as the second power, but the experiments show the effects of these terms are small, and A is constant except for extreme loads. 286 ON THE THEORY OF LUBRICATION [52 33. The Formulas for Temperature and Friction, and the Interpretation of the Constants. From equation (8), Section II., p^^-cw-T.) ^| [ (115). = -0221 (for olive oil)J From equations (109) and (110) a = (a + ml){l + E(T-T )} (116), or approximately since E (T l T ) is small ^ (117). Whence substituting in the equation which results from (51) c being small f=^U (118). Putting T x for any particular temperature, p, x , a x corresponding values j ~ * {J C/ x ..*.....* (J.-LtJ). From (113) f=A{T m + ET* m }+^T m (120). These equations (119) and (120) are independent, and therefore furnish a check upon each other when the constants are known. Thus substituting the experimental values of U and/ in (120) a series of values of T m are obtained, which when substituted in (119) should give the same value off. In these equations the meaning of the constants is as follows : C is the rate at which viscosity increases with temperature. E is the rate at which a increases with temperature, owing to the different expansion of brass and iron. A U expresses generally the mechanical equivalent of the heat which is carried out of the oil film by the motion of the oil and journal for each degree rise of temperature in the film. B expresses the mechanical equivalent of the heat conducted away through the brass and journal. The respective importance of these two coefficients is easily apparent. When the velocities are low, but little heat will be carried out, and hence 52] AND ITS APPLICATION TO MR B. TOWER'S EXPERIMENTS. 287 the temperature of the journal depends solely on the value of B. But when the velocities are high, B becomes insignificant compared with AU, and it is A alone which keeps the journal cool. The value of A may to some extent be inferred from the quantities which enter into it as in equation (114). Thus in the case of Mr Tower's experiment, since E = 2 {(9 = 1-37 a = -00075 J = -772 D = 0-033 S= 0-31 (for olive oil) A = -00630 (121). It is very difficult to form an estimate of q, but it would seem probable that it has a value not far from 2 ; and, as will be subsequently shown, in the case of Mr Tower's experiments, q is about 3*5 or A =0-0223 (122). As B expresses the rate at which the heat generated in the oil film is carried away, by conduction through the oil and the surrounding metal, any estimate of its value is very difficult. If we could measure the temperature at the surfaces of the metal, B might be made to depend only on the thick- ness and conductivity of the oil film. But before heat can escape from the journal or bearing, it must pass along intricate metal channels formed by the journal or shaft and its supports ; and, on consideration, it appears that in ordinary cases the resistance of such channels would be much greater than the resistance of the oil film itself. For example, in the case of a railway axle, the heat generated must escape either along the journal to the nearest wheel, or through the brass and the cast-iron axle box to the outside surface, so that either way it must traverse at least three or four inches of iron. This is about the best arranged class of journals for cooling. In most other cases heat has much further to go before it can escape. However, in every case B will depend on the surrounding conditions, and can only be determined by experiment. From the experiments, to be considered in the next section, it appears that B=l (about) (123). But it is to be noticed that Mr Tower has introduced a somewhat abnormal condition by heating the oil bath above the surrounding tem- perature. For in this way, letting alone the heat generated by friction, there must have been a continual flow of heat from the bath along the journal to the machinery; and, considering the comparatively limited surface of the journal in contact with the hot oil and the large area of section of the journal, it appears unlikely that the temperature of the journal was raised by the bath to anything like the full temperature of the latter, 288 ON THE THEORY OF LUBRICATION [52 a conclusion which is borne out by Mr Tower's experiments with different temperatures in the bath (Table XII., page 291), which shows that the tem- perature of the bath produced a much smaller effect on the friction than would have followed from the known viscosity of oil had the temperature of the oil film corresponded with the temperature of the bath. Thus the temperature of the film independent of friction is not the temperature of the bath or surrounding objects, and as it is unknown until determined from the experiments, it will be designated as T - L X} and the suffix x used to designate the particular value for T= T x of all those quantities which depend on the temperature as If T be the mean temperature of the film, T-T x =T m .............................. (123 A) where T m is the rise of temperature due to the film. 33 A. The Maximum Load the Journal will carry at any Speed. It has already been pointed out that the carrying power of the journal is at its greatest when c is between '5 and '6. If, therefore, taking the load constant, c passes through a minimum value as the velocity increases with a constant load, then the load which brings c to a constant value will be a maximum for some particular velocity, and if the particular value of c be that at which the carrying power is greatest, the carrying power will be greatest at that particular speed. The question whether, according to the theory, journals have a maximum carrying power at any particular speed turns on whether -jjj (c being constant) is zero for any value of U. This admits of an answer if the values for JJL, a, T x , and equations (119) L f and (120) hold, for when c is constant , T , ,~. Tr , and A are constant, KtU K 2 U whence differentiating and substituting it appears that when c is constant f> ~ 52] AND ITS APPLICATION TO MR E. TOWER'S EXPERIMENTS. 289 where U is to be taken positive, and T increases as U increases, -This shows that ,, for a constant value of c, changes sign for some value of U if T continues to increase with U. Hence, according to the theory, the values of L, which make c constant as U increases, approach a maximum value as U increases, and since this value, when c is about '5, represents the carrying power of the journal, this approaches a maximum as U increases. SECTION IX. APPLICATION OF THE EQUATIONS TO MR TOWER'S EXPERIMENTS. 34. References to Mr Tower's Reports. From the experiments described in Mr Tower's Reports I. and II., in the Minutes of the Institution of Mechanical Engineers, 1884, the journal had a diameter of 4 inches, and the chord of the arc covered by the brass was 3'92 inches, the length of the brass being 6 inches. The loads on the brass in Ibs., divided by 24, are called the nominal load per square inch. The moments of friction in inches and Ibs., divided by 24 x R, are called the nominal friction per square inch. These nominal loads, arid nominal frictions, with the number of revolu- tions from 100 to 450, at which they were taken, are arranged in tables for each kind of oil used, and also for the same oil at different temperatures of the bath. All the tables relative to the oil bath in the first report refer to the same brass and journal. And with this brass, to be here called No. 1, no definite measurements of the actual pressure were made. The second report contains the account of the pressure measurements, but it is to be noticed that these were made with a new brass, here called No. 2, and that the only friction measurements recorded with this brass are three made at velocities five times less than the smallest velocity used in the case of brass No. 1. It thus happens that while by the application of the foregoing theory to the friction experiments on brass No. 1 a value is obtained for a, the difference in radii of brass No. 1 and the journal ; and from the pressure experiments on brass No. 2 a value is obtained for a in the case of brass No. 2; since these are different brasses there is no means of checking the one estimate against the other. o. R. ii. 19 290 ON THE THEORY OF LUBRICATION [52 The following tables extracted from Mr Tower's reports are those to which reference has chiefly to be made. The first of these extracts is a portion of Table I. in Mr Tower's first report ; this related to olive oil, but corresponds very closely with the results for lard oil also, although not quite so close for mineral oil. From TABLE I. (Mr Tower's 1st Report, Brass No. 1.) Bath of Olive Oil. Temperature, 90 deg. Fahr., 4-in. journal, 6-in. long. Chord of arc of contact = 3-92 in. O ^ ~Q rd Nominal friction per square inch of bearing. ll .** 100 rev. 150 rev. 200 rev. 250 rev. 300 rev. 350 rev. 400 rev. 450 rev. j a o >> 105 ft. 157 ft. 209 ft. 262 ft. 314 ft. 360 ft. 419 ft. 471 ft. per mm. per mm. per mm. per mm. per mm. per mm. per mm. per mm. 520 416 520 624 675 728 779 883 468 ... 514 607 654 701 794 841 935 415 498 580 622 705 787 870 995 363 ... 472 580 616 689 725 798 907 310 464 526 588 650 680 742 835 258 361 438 515 592 644 669 747 798 205 368 43 512 572 613 675 736 818 153 351 458 535 611 672 718 764 871 100 36 45 555 63 69 77 82 89 The second extract is Table IX. in the first report ; this shows the effect of temperature on the friction of the journal with lard oil. From Mr Tower's 1st Report, Brass No. 1. TABLE IX. Bath of Lard Oil. Variation of Friction with Temperature. Nominal Load 100 Ibs. per square inch. Nominal friction per square inch of bearing. Temperature. Fahr. 100 rev. 150 rev. 200 rev. 250 rev. 300 rev. ! 350 rev. 400 rev. 450 rev. per mm. per mm. per mm. per mm. per mm. per mm. per nun. per mm. 120 24 29 35 40 44 47 51 54 110 26 32 39 44 50 55 59 64 100 29 37 45 51 58 65 71 77 90 34 43 52 60 69 77 85 93 80 4 52 63 73 83 93 1-02 1-12 70 48 -65 8 92 1-03 1-15 1-24 1-33 60 59 84 1-03 1-19 T30 1-4 1-48 1-56 52] AND ITS APPLICATION TO MR B. TOWER'S EXPERIMENTS. 291 The third extract is from the second report, being Table XIL^-represent- ing the oil pressure at different parts of the bearing as measured with brass No. 2. From Mr Tower's Second Report, Brass No. 2. Heavy Mineral Oil Nominal Load 333 Ibs. per square inch. Number of revolutions 150 per minute. Temperature, 90. TABLE XII. Oil pressure at different points of a bearing. Longitudinal planes on centre off Pressure per square inch Ib. Ib. Ib. Transverse plane, middle 370 625 500 Transverse plane, No. 1 355 615 485 Transverse plane, No. 2 310 565 430 The points at which the pressure was measured were at the intersections of six planes, three parallel vertical longitudinal planes parallel to the axis of the journal, one through the axis, and the other two, one on the on and the other on the off side, both of them at a distance of '975 inch from that through the axis, three transverse planes, one in the middle of the journal, the other two respectively at a distance of one and two inches on the same side of that through the middle. In referring to these experimental results in the subsequent articles, The nominal load per square inch is expressed by L' ; The number of revolutions per minute by The nominal friction by /' ; The effect of the fall of pressure ajflhe ends of the journal on the mean pressure is expressed by - , thus s n is a coefficient depending on the way in which the journal fits the shaft. '. ~ A \ (124). _ 7x60 '1-rrR s= 1-21 192 292 ON THE THEORY OF LUBRICATION [52 35. The Effect of Necking the Journal. The expression (124) for f assumes that the journal was not necked into the shaft. From Mr Tower's reports it does not appear whether or not the brass was fitted into a neck on the shaft ; but since there is no mention of such necking, the theory is applied on the supposition that there was not. If there were, the friction at the ends of the brass would increase the moment of friction. Put b for the depth of the neck and a' for the thickness of the oil film at the ends, then the moment of resistance of these ends would be- Hence if M be the moment of friction of the cylindrical portion of the journal only And from equation (97) TT ft .(126). f ,_ , 0.0 -!> ' -360 a 1 TlT For example, a 5-inch shaft necked down to a 4-inch journal would give b '5 inch. Whence, assuming | and l the relative friction of the ends to that of the journal would be 11 '36 to 31-00, or 28 per cent, of the friction; and the values of a, calculated on the assumption of no necking, would have to be increased in the ratio n T33. Even if there is no necking the value of a will probably not be the same all along the journal, in which case the values of a 2 and a in K l and K 2 will be means, and then the square of the mean will be less than the mean of the squares, so that n will probably have a value greater than unity, although there may be no necking of the shaft. 36. A first Approximation to the Difference in the Radii of the Journal and Brass No. 1. The recorded temperature in Mr Tower's Table I. is 90 Fahr. Accepting this, and taking the value of /A, equation (8), Section II., Hn^IQ-' x 6-81 (128). 52] AND ITS APPLICATION TO MR B. TOWER'S EXPERIMENTS. 293 By equation (97) nu. nM a = 2757W (129) ' Since R = 2 and U ' = oO HfJL "72/" ~^ = U = 3-46^ (130). Whence substituting from equation (128) for n w a N - = 10-xl-97^-, (131), and from the tabular Nos. for L' = 100. N=100, /' = -36 - = 10~ 4 x 5'5 (inch) n 7 .(132). - = 10-*xlO(inch) n ' These are the extreme cases; for intermediate velocities intermediate values of - are found. n In order to be sure that these are the values of - , which result from the n application of the equations, it is necessary (since the approximate equations only have been used) to see that the squares of c may be neglected. Substituting from equation (124) in (96) which for L' = 100, N= 100, gives and for // = 100, ^=450, c = '033/i 2 c = -065n s .(133). So that the approximations hold, and, as already stated in Art. 30, this considerable increase in the value of a with the load constant suggests that the temperature of the film was not really 90. And as this point has been 294 ON THE THEORY OF LUBRICATION [52 considered in the last section, the equations of that section may be at once used to determine the law of this temperature, after which the values of - n may be determined with precision. 37. The Rise in Temperature of the Film owing to Friction. In order to determine the values of E, A, and B in equations (119) and (120), by substituting in these equations corresponding values of N and f for Z/=100, the tabular values of/' were somewhat rectified by plotting and drawing the curve N, f. These corrected values are in the second row, Table III. From these values, and the corresponding values of N, it was then found by trial that the equations (119) and (120) respectively . U e -(C+E)(T-T,) f a x + mL E c = '0221, are approximately satisfied for values of T T x = T m , if u, -01345 .(134) a x + m x 400 n A = -0223081 # = 0222 I (135). E = -95914 J TABLE III. Rise of Temperature in the Film of Oil caused by Friction, calculated by Equation (120) from Experiments with a Nominal Load 100 Ibs. (see Table I., Tower, p. 290). Nominal friction per square inch, as calculated by equation (119) from the rise of temperature. N Revolutions per minute . . 100 150 200 250 300 350 400 450 b <_, . sll o a f Nominal friction f^ ble L per square inch ] n O1 , ,' for olive oil 1 Correcte -8. 5-9 - 4-8 log -59 - log -48 From (136) Z+ ^ ^-$- . 10 -0443 x 10 x loge Therefore ^=35 ................................. (138). From this value of Z the values of f corresponding to those in Tower's Table IX. have been calculated and agree well with the experimental values. The smallest temperature of the oil bath recorded in Tower's Table IX. is 60 Fahr., therefore it is assumed that this was the normal temperature, whence T x = -35(^-60) + 60 ........................ (139). Hence it is concluded that the actual temperature of the oil film in all the experiments with the bath, at a temperature of 90 Fahr., is given by T=7Q-o + T m .............................. (140). By the formula for p, since T x = 70'5 = -000009974 = 00001 (approximately) .................. (141), and since, by equation (134), when L' = 100 ~ ~ T Ol345 = -0007413n = -00074w (approximately) ......... (142). This is the value of a x with a load of 100 Ibs. per square inch. 52] AND ITS APPLICATION TO MR B. TOWER'S EXPERIMENTS. 297 39. The Variation of a- with the Load. All Mr Tower's experiments, when the loads are moderate and the velocities high, show a diminution of resistance with an increased load. Since c increases with the load and the friction increases as c increases, a and fj. being constant, the diminution of friction with increased loads shows either that the load increases the temperature of the film and so diminishes the viscosity, or increases the radius of curvature of the brass as compared with that of the journal, i.e., increases a. These effects have been investigated by substituting the experimental values of/' and L' t obtained with the same velocity in equations (119) and (120). In this way, from equation (119) the value of m is determined, where, from equations (117) and (135), 7 'o> ........................ (143). And the equation (120) gives the effects of the load on the value of the constant A. After trial, however, it appears that the effects of the load upon the constant A are small so long as the loads are moderate, and that the diminution of the resistance with the increased load is explained by the value obtained for m from equation (119). From this equation, taking LSf' 1 , L 2 'f' 2 , simultaneous values of L' and/', and assuming T x independent of the load, f* (144) which gives the value of m. The slight irregularities in the experiments affect the values of m thus found to a considerable extent, and a mean has been taken, which is (145). Putting T x = 70-5, T = 60, a x = '00074, when L' = 100, from equation (143) a -i- ml' = -0005861w. Therefore a = '0004885/1 (146). The value for a thus obtained is therefore ci = -0004885/i(l +-002// l )e' oaB(r * +r ^ (147), and for the experiments with the bath at 90 Fahr. a = -0004885;i (1 + -002/7) ea , ,// for the experiments in Table I. (Tower), these being made with brass No. 1. Putting as in equation (124) ,, _ nM J ~ f) 7?2 ' 52] AND ITS APPLICATION TO MR B. TOWER'S EXPERIMENTS. 299 equation (94) gives and by equation (150) fl >=. 004658 .(152). From equation (152) the values of/' to a first approximation have been calculated, using the values of T m given in Table III. These are given as// in Table IV. t TABLE IV. Olive Oil, Brass No. 1. Length of the journal 6 inches Chord of the arc of contact of the brass 3-92 inches. Radius of the journal 2 inches. Temperature of the oil bath 90 Fahr. surrounding objects 60 Fahr. (assumed). Difference in radii of brass and journal at 60 0-0006 inch (deduced). Effect of necking or variations in radius to increase friction ...1-25. // the nominal load in Ibs. per square inch, being the total load divided by 24. N the number of revolutions per minute. /' the nominal friction in Ibs. per square inch from Table I. in Mr Tower's first Report (see Art. 34, p. 290). (//) the nominal friction calculated by complete approximation for c-5 (see Art. 40, equation (159) ). / 2 ' the nominal friction calculated to a second approximation, equation (154). // the nominal friction calculated to a first approximation, equation (152). c the ratio of the distance between the centres of the brass and journal to the difference in the radii, equation (153). a the difference in the radii of the brass and journal (see equation (157)). 0j the angular distance from the middle of the arc of contact of the point of maximum pressure, equation (91). p -* the angular distance from the middle of the arc of contact of the point of nearest approach (see equation (89)). 2' the rise in temperature of the film of oil owing to the friction, equation (120). N. 100 150 200 250 300 350 400 450 T m Fahr. 3-45 5-83 8-13 10-02 1177 13-26 14-46 15-37 498 580 622 705 787 870 995 P 300 (57) 360 (65) 414 460 504 544 / 2 '-108 589 c 67 578 590 487 457 -I3U 413 L =41.) a 00154 00162 0017 00178 00182 00187 00191 00194 -*" -420'0" - 42 0' 0" ... ... \ r * 2 TABLE IV. Olive Oil, Brass No. 1 continued. N. 100 150 200 250 300 350 400 450 r m Fahr. 3-45 5-83 8-13 10-02 11-77 13-26 14-46 15-37 f 472 580 616 689 725 798 907 ... (-498) / 2 '-920 J\ 233 314 378 434 482 526 573 616 , c 520 462 408 380 357 340 312 ~ 1 00151 00161 00168 00171 00177 00181 00183 Ui ... - 7 0' 0" ... ... ... U-l - 42 0' 0" ... ... f * 464 526 588 650 684 742 835 (-370) A '765 805 845 A 249 336 404 464 517 582 610 660 c 510 392 337 305 284 266 254 234 // =310 a 00138 00144 00151 00158 00161 00166 00172 00172 0i - 7 0' 0" ... ... ... , IT n - 42 0' 0" TO 2 /' 361 436 514 592 644 669 747 789 /i 62 670 712 760 810 X' 266 358 431 495 550 600 650 707 c 377 287 242 224 200 195 180 171 // = 253 a 00128 00135 00141 00148 00151 00156 00159 00161 f/ 368 430 512 572 613 675 736 818 A 380 457 530 595 65 701 755 810 /i' 285 385 464 534 592 646 700 755 ^ c 255 195 165 152 L41 132 126 118 ./^ == ^vjo a 00119 00126 00131 00138 00240 00145 00148 00135 0i ... ... - 113'0" - 1 5'G ^-f ... ... -60 O'O" -62 O'O r/ 351 458 535 611 672 718 764 871 / ' / 9 352 458 530 601 665 717 778 840 A' 307 414 498 574 638 695 753 813 C 165 126 107 098 089 086 082 075 x, =152 { a 00111 00116 00122 00128 00130 00134 00137 00139 (I). ... - 113'0" - 10'0" - 057'0" - 050'0" - 044'0" - 038'0" - 031'0 <7)| * ... -61 O'O" -65 O'O" - 27 20' 0" -69 O'O" -6940'0" -7020'0" -72 O'O // 360 450 550 630 690 770 820 890 A' 352 465 555 637 708 770 831 897 A 336 463 546 628 697 760 823 890 c 090 0691 0600 0541 0492 0471 0460 0420 // =100 < a 00101 00106 00112 00116 00120 00123 00125 00127 0i - 043'0" - 026'0" - 017'0" - 010'0" - 4'0" - 02'30" - O'O" + 5'0 ^ "" l*o-2 -6830'0" -7320'0" -7520'0" -7630'0" -7720'0" -7740'0" - 078'0" - 78 40' 52] ON THE THEORY OF LUBRICATION AND ITS APPLICATION, ETC. 301 As compared with the experimental values/' given in the Table IV., it is seen that the agreement holds as long as c is less than - 06, after jvhich, as c increases, the values of// become too small, or while the values of// continue to diminish as the load and a increase, the experimental values of/' after diminishing till c is about '1 or '15 begin to increase again. In order to see how far this law of variation was explained by the theory, it was necessary to find /' the values of /' to a second approximation, and before this to obtain the values of c. Putting, as in equation (124), L = 4-84Z', equation (94) gives c=- 2-059^; and by equation (150) 2 T ' c = '031 16(1+ -1002 L') 2 ^ e 0665 ^ (153). Equation (153) gives the values of c w 2 ' s* To obtain the value of n from the experiments, these values of are substituted in the equation for /', retaining the squares of c, which obtained from (.-filiations (85) and (89), is //=//(l + 5c 2 ) (154), whence, substituting the values of obtained from (153) we have /'-/' + '( Therefore, choosing any experimental values of /', and subtracting the corre- sponding value of// in Table IV., n is given by 5 - 1 >/ .^..,.(158), aod since n = T25, this gives for c = '5 /' = 1-585/V. If, therefore, any of the approximate values of c were exactly '540, the complete value of f would be 1*585 times the value of f^. This does not happen, the nearest approximate values of c being "578, '520, '520, '510. Multiplying the corresponding values of f^ by 1*585, the results are as follows : Tabular, c Tabular. A' 1-605 /,' Experimen- tal. / Difference. 578 36 57 58 01 520 414 656 65 -005 520 314 498 472 -026 510 249 394 It thus appears that the approximation is very close, the calculated values for the first, in which c is greater than *540, being too small, and for the rest, in which c was smaller than *540, too large, which is exactly what was to be expected. These corrected values of // have been introduced in Table IV. in brackets. As they occur with different loads and different velocities, they afford a very severe test of the correctness of the conclusions arrived at as to the variations of A and T with the load and temperature, also as to the condition expressed by n. Had the values of c and f been completely calculated as for the case of c=*5, there would have been close agreement for all the calculated and experimental values of /'. This close agreement strongly implies, what was hardly to be expected, namely, that the surfaces, in altering their form under increasing loads, preserve their circular shape so exactly that the thickness of the oil film is everywhere approximately a(l + c sin (0 -)). A still more severe test of this is, however, furnished by the pressure experiments with brass No. 2 in Mr Tower's second report. 304 ON THE THEORY OF LUBRICATION [52 41. The Velocity of Maximum Carrying Power. The limits to the carrying powers are not very clearly brought out in these recorded experiments of Mr Tower, as indeed it was impossible they should be, as each time the limit is reached the brass and journal require refitting. But it appears from Table I. and all the similar tables with the oil bath in Mr Tower's reports, that the limit was not reached in any case in which the load and velocity were such as to make c less than '5. In many cases they were such as to make c considerably greater than this, but in such cases there seems to have been occasional seizing. There seems, however, to have been one exception to this case, in which the journal was run at 20 revolutions per minute with a nominal load of 443 Ibs. per square inch with brass No. 2 without seizing, in which case c, as determined either by the friction or load, becomes nearly '9. It does not appear that any case is mentioned of seizing having occurred at high speeds, so that the experiments show no evidence of a maximum carrying power at a particular velocity. This is so far in accordance with the conclusions of Art. (33a), for, substi- tuting the values of ABCE, as determined Art. (37), it appears by equation (123 B) that the maximum would not be reached until T m , the rise of tem- perature due to friction, reached 72 Fahr., which, seeing that at a velocity of 450 revolutions T, n is less than 17, implies that the maximum carrying power would not be reached until the speed was 1500 or 2000 revolutions; notwithstanding that -jjj (c constant) is very small at 450 revolutions. This is with the rise of temperature due to legitimate friction with perfect lubrication. But if, owing to inequalities of the surfaces, there is excessive friction without corresponding carrying power, i.e., if /?, the effect of necking, is as large as 3 or 4, which it is with new brasses, then the maxi- mum carrying power might be reached at comparatively small velocities ; thus suppose T= 13 when N= 100, U = 21, equation (123 B) gives ^ = dU or the maximum carrying power would be reached ; all which seems to be in strict accordance with experience, particularly with new brasses. 42. Application of the Equations to Mr Towers Experiments with Brass No. 2 to determine the Oil Pressure round the Journal. The approximate equation (74) is available to determine the pressure at any part of the journal, i.e., for any value of 6 so long as c is small, but these 52] AND ITS APPLICATION TO MR B. TOWER'S EXPERIMENTS. 305 approximations fail for much smaller values of c than for others ; for this reason, together with the fact that the only case in which the pressure has been measured c is large, the pressures have only been calculated for c = '5, in which the approximations have been carried to the extreme extent. These are obtained directly from equation (107), and the pressures divided by K,c are given in Table II., Section VI. The results of Mr Tower's experiments with brass No. 2 are given in Table XII., Art. 34. Had the friction been recorded in the experiments in which Mr Tower measured the pressures with brass No. 2 as with brass No. 1, the values of c might have been obtained as in the case of brass No. 1. But as this was not done the value of c for these experiments with brass No. 2 could only be inferred from the agreement of the relative oil pressures measured in different parts of the journal, those calculated for the same parts with a particular value of c. This was a matter of trial, and as it was found that the agreement was very close when c = -5, further attempts were not made. With the section at the middle of the brass the calculated and experi- mental results are shown in Table V. TABLE V. Comparison of Relative Pressures, calculated by Equation (107) when c = '5, with the Pressures measured by Mr Tower, see Table XII., Art. 34, Brass No. 2. The values of measured from middle of arc at which pressures were measured Pressure measured at the middle of the journal. Table XII., Tower P ~ l> calculated. Table II. Kelative values, experimental llelative values, calculated -* -20 20 20 500 7923 800 781 639 000 625 1-0150 1-000 1-000 615 29 20 20 370 6609 592 651 560 This agreement, although not exact, is, considering the nature of the test, very close. The divergence seems to show that in the experiments c was somewhat more than '5, but it is doubtful if the agreement would have been exact, as, owing to the journal having been run in one direction only, it seems probable that the radius of the brass was probably slightly greatest on the on side. o. R. ii. 20 306 ON THE THEORY OF LUBRICATION [52 Deducing the value of K& by dividing the experimental pressure by the calculated values of -- the values given in the last column are found. An alteration in the value of c would but slightly have altered the middle value of - ,_ in the same direction as the alteration of c ; hence taking Kc this value, and making c = '520, as being nearer the real value, K,c = - 640 (159). In these experiments N = 150, 77 = 333 (160). From equation (104) L = 2'5504 x Kc, = 408 (161), therefore s = jy? = 1-21 (162). To find a K, = - 1230, by equation (150) ft-A-'0665r m #, whence a x 2 = '000001 Q88e~' WG5Tm .............. . ...... (163) and taking T m the same as with brass No. 1, and olive oil at JV=180, i.e., 5-83 Fahr., with brass No. 2, at 70'5 Fahr. a x =-00086 instead of with brass No. 1 = -0007 7 (164). The difference in the radii of curvature of the two brasses, the one deduced from the measured friction, the other deduced from the measured differences of pressure at different positions round the journal, come out equal within j^^th part of an inch, and the values of a differing only by 11 per cent. Had the frictions been given with brass No. 2, this agreement would have afforded an independent comparison of the values of a. As it is, the only probability of equality in these two brasses arises from the probability of their having been bedded in the same way. In deducing the value a for brass No. 2, it has been assumed that the oil, which was mineral, had the same law of viscosity as the olive oil. Both these oils were used with brass No. 1, and the results are nearly the same, 52] AND ITS APPLICATION TO MR B. TOWER'S EXPERIMENTS. 307 the mean resistances, as given by Mr Tower, are as - 623 to 0'6o4, or the viscosity of the mineral oil being 0'95 that for olive oil ; had this-been taken into account, the value of a x for brass No. 2 would have been still nearer that for brass No. 1, being '00084 as against '00077. As the radii of the two brasses seem to be so near, and as the resistance was measured for brass No. 1 under circumstances closely resembling those of the experiment with No. 2, a further test of the exactness of the theory is furnished by comparing the calculated friction with brass No. 2 with that measured with brass No. 1, with the same oil, the same speed, and nearly the same load. As in equation (158) -/' = 2-3773^ + 1 -37 (w- l)K z .................. (165) Whence, taking account of the values of /* for mineral and olive oils, and the values of a for brass No. 1 and No. 2 for mineral oil and brass No 2, K z has 0'87 1 of the value in equation (150) 871 x -00346- - (1 + -002/7) n 6) ' which, when T=5-83, iV=150, /7 = 337, n = l'25, being substituted in the equations #a = - 01665 /' = 446 .................................... (167). In Mr Tower's Table IV., it appears that with brass No. 1, mineral oil, ^=150 /7 = 310 /'=4-4 77 = 415 /' = -51, whence interpolating for L = 337 /' = 4-58 ................................. (168). This agreement is very close, for taking account of the difference of radius, the calculated friction for brass No. 2 should have been about '95 of the measured friction with brass No. 2. In order to show the agreement between the calculated pressures and those of Mr Tower, the values of ^^- for c = '5 have been plotted, and are K& shown in Figs. 18 and 19 (page 280), the crosses indicating the experiments with brass No. 2, as in Table VII. (Tower). 202 308 ON THE THEORY OF LUBRICATION [52 43. Conclusions. The experiments to which the theory has been definitely applied may be taken to include all Mr Tower's experiments with the 4-inch journal and oil bath, in which the number of revolutions per minute was between 100 and 450, and the nominal loads in Ibs. per sq. inch between 100 and 415. The other experiments with the oil bath were with loads from 415, till the journal seized at 520, 573, or 625 ; and a set of experiments with brass No. 2 at 20 revolutions per minute. All these experiments were under extreme conditions, for which, by the theory, c was so great as to render lubrication incomplete, and preclude the application of the theory without further integrations. The theory has, therefore, been tested by experiments throughout the entire range of circumstances to which the particular integrations under- taken are applicable. And the results, which in many cases check one another, are consistent throughout. The agreement of the experimental results with the particular equations obtained on the assumption that the brass, as well as the journal, are truly circular, must be attributed to the same causes as the great regularity presented by the experimental results themselves. Fundamental amongst these causes is, as Mr Tower has pointed out, the perfect supply of lubricant obtained with the oil bath. But scarcely less important must have been the truth with which the brasses were first fitted to the journal, the smallness of the subsequent wear, and the variety of the conditions as to magnitude of load, speed, and direction of motion. That a brass in continuous use should preserve a circular section with a constant radius requires either that there should be no wear at all, or that the wear at any point P should be proportional to sin (90 -POH). Experience shows that there is wear in ordinary practice, and even in Mr Tower's experiments there seems to have been some wear. In these experiments, however, there is every reason to suppose that the wear would have been approximately proportional to c sin (< -0) = c sin (90 - POH), because this represents the approach of the brass to the journal within the mean distance a, for all points, except those at which it is negative; at these there would be no wear. So long then as the journal ran in one direction only, the wear would tend to preserve the radius and true circular form of that portion of the arc from C to F (Fig. 17, p. 266), altering the radius at F, and enlarging it from F to D. On reversal, however, C and F change sides, and hence alternate motion in both directions would 52] AND ITS APPLICATION TO MR B. TOWER'S EXPERIMENTS. 309 preserve the radius constant all over the brass. The experience, emphasized by Mr Tower, that the journal after running for some time in one -direction would not run at first in the other, strongly bears out this conclusion. Hence it follows that had the journal been continuously run in one direction, the condition of lubrication, as shown by the distribution of oil pressure round the journal, would have been modified, the pressure falling between and B on the on side of the journal, a conclusion which is borne out by the fact that in the experiments with brass No. 2, which was run for some time continuously in one direction, the pressure measured on the on side is somewhat below that calculated on the assumption of circular form, although the agreement is close for the other four points (see Fig. 18, page 280). When the surfaces are completely separated by oil it is difficult to see what can cause wear. But there is generally metallic contact at starting, and hence abrasion, which will introduce metallic particles into the oil (blacken it) ; these particles will be more or less carried round and round, causing wear and increasing the number of particles and the viscosity of the oil. Thus the rate of wear would depend on the impurities in the oil, the values of c, I/a and the velocity of the journal, and hence would render the greatest velocity at which the maximum load could be carried with a large value of c small. A conclusion which seems to be confirmed by Mr Tower's experiments at twenty revolutions per minute. In cases such as engine bearings, the wear causes the radius of curvature of the brass continually to increase, and hence a and c must continually increase with wear. But in order to apply the theory to such cases the changes in the direction of the load (or E7i and Fj) have to be taken into account. That the circumstances of Mr Tower's experiments are not those of ordinary practice, and hence that the particular equations deduced in order to apply the theory to these experiments do not apply to ordinary cases, does not show that the general theory, as given in equations (15), (18), and (19) could not be applied to ordinary cases were the conditions sufficiently known. These experiments of Mr Tower have afforded the means of verifying the theory for a particular case, and hence have established its truth as applicable to all cases for which the integrations can be effected. The circumstances expressed by ^ 77' R' ' ^' ^ l ' n ' m ' G> ^' E ' B which are shown by the theory to be the principal circumstances on which 310 ON THE THEORY OF LUBRICATION, ETC. [52 lubrication depends, although not the same in other cases, will still be the principal circumstances, and indicate the conditions to be fulfilled in order to secure good lubrication. The verification of the equations for viscous fluids, under such extreme circumstances, affords a severe test of the truth and completeness of the assumptions on which these equations were founded. While the result of the whole research is to point to a conclusion (important in Natural Philosophy) that not only in cases of intentional lubrication, but wherever hard surfaces under pressure slide over each other without abrasion, they are separated by a film of some foreign matter, whether perceivable or not. And that the question as to whether this action can be continuous or not, turns on whether the action tends to preserve the matter between the surfaces at the points of pressure, as in the apparently unique case of the revolving journal, or tends to sweep it to one side, as is the result of all backwards and forwards rubbing with continuous pressure. The fact that a little grease will enable almost any surfaces to slide for a time, has tended doubtless to obscure the action of the revolving journal to maintain the oil between the surfaces at the point of pressure. And yet, although only now understood, it is this action that has alone rendered our machines, and even our carriages possible. The only other self-acting system of lubrication is that of reciprocating joints with alternate pressure on and separation (drawing the oil back or a fresh supply) of the surfaces. This plays an important part in certain machines, as in the steam-engine, and is as fundamental to animal mechanics as the lubricating action of the journal is to mechanical contrivances. 53. ON THE FLOW OF GASES. [From the " Philosophical Magazine," March, 1886.] (Read before the " Manchester Literary and Philosophical Society," November 17, 1885.) 1. AMONGST the results of Mr Wilde's experiments on the flow of gas, one. to which attention is particularly called, is that when gas is flowing from a discharging vessel through an orifice into a receiving vessel, the rate at which the pressure falls in the discharging vessel is independent of the pressure in the receiving vessel, until this becomes greater than about five- tenths the pressure in the discharging vessel. This fact is shown in Tables IV. and V. in Mr Wilde's paper; thus, the fall of pressure from 135 Ibs. (9 atmospheres) in the discharging vessel is 5 Ibs. in 7 '5 seconds for pressures in the receiving vessel, ranging from one half-pound to nearly 5 or 6 atmo- spheres. With smaller pressures in the discharging vessel, the times occupied by the pressure in falling a proportional distance are nearly the same, until the pressure in the receiving vessel reaches about the same relative height. What the exact relation between the two pressures is when the change in rate of flow occurs, is not determined in these experiments. For as the change comes on slowly, it is at first too small to be appreciable in such short intervals as 7\5 and 8 seconds. But an examination of Mr Wilde's Table VI. shows that it lies between '5 and '53. This very remarkable fact, to which Mr Wilde has recalled attention, excited considerable interest fifteen or twenty years ago. Graham does not appear to have noticed it, although On reference to Graham's experiments it appears that these also show it in the most conclusive manner (see Table IV., Phil Trans. 1840, Vol. IV. pp. 573632; also Reprint, p. 106). These 312 ON THE FLOW OF GASES. [53 experiments also show that the change comes on when the ratio of the pressures is between '483 and '531. R. D. Napier appears to have been the first to make the discovery*. He found, by his own experiments on steam, that the change came on when the ratio of pressures fell to '5 (see Encyc. Brit. Vol. xn. p. 481). Zeuner, Fliegner, and Hirn have also investigated the subject. At the time when Graham wrote, a theory of gaseous motion did not exist. But after the discovery of the mechanical equivalent of heat and thermodynamics, a theory became possible, and was given with apparent mathematical completeness in 185(5. This theory appeared to agree well with experiments until the particular fact under discussion was discovered. This fact, however, directly controverts the theory. For on applying the equations giving the rate of flow through an orifice to such experiments as Mr Wilde's, it appears that there is a marked disagreement between the calculated and experimental results. The calculated results are even more remarkable than the experimental ; for while the experiments only show that diminishing the pressure in the receiving vessel below a certain limit does not increase the flow, the equations show that by such diminution of pressure the flow is actually reduced and eventually stopped altogether. In one important respect, however, the equations agree with the experi- ments. This is in the limit at which diminution of pressure in the receiving vessel ceases to increase the flow, which limit by the equations is reached when the pressure in the receiving vessel is '527 of the pressure in the discharging vessel. The equations referred to are based on the laws of thermodynamics, or the laws of Boyle, Charles, and that of the mechanical equivalence of heat. They were investigated by Thomson and Joule (see Proc. Roy. Soc., May 1856), and by Prof. Julius Weisbach (see Civilingenieur, 1856); they were given by Rankine (articles 637, 637 A, Applied Mechanics), and have since been adopted in all works on the theory of motion of fluids. Although discussed by the various writers, the theory appears to have stood the discussion without having revealed the cause of its failure ; indeed, Hirn, in a late work, has described the theory as mathematically satisfactory. Having passed such an ordeal, it was certain that if there were a fault, it would not be on the surface. But that by diminishing the pressure on the receiving side of the orifice the flow should be reduced and eventually * The account of E. D. Napier's experiments is contained in letters in the Engineer, 1867, vol. xxiii. January 4 and 25. They were made with steam generated in the boiler of a small screw-steamer and discharged into an iron bucket, the results being calculated from the heat imparted to a constant volume of water in the bucket in which the steam was condensed. 53] ON THE FLOW OF GASES. 313 stopped, is a conclusion too contrary to common sense to be allowed to pass when once it is realized; even without the direct experimental evidence in contradiction, and in consequence of Mr Wilde's experiments, the author was led to re-examine the theory. 2. On examining the equations, it appears that they contain one assump- tion which is not part of the laws of thermodynamics, or of the general theory of fluid motion. And although commonly made and found to agree with experiments in applying the laws of hydrodynamics, it has no founda- tion as generally true. To avoid this assumption, it is necessary to perform for gases integrations of the fundamental equations of fluid motion which have already been accomplished for liquids. These integrations being effected, it appears that the assumption above referred to has been the cause of the discrepancy between the theoretical and experimental results, which are brought into complete agreement, both as regards the law of discharge and the actual quantity discharged. The integrations also show certain facts of general interest as regards the motion of gases. When gas flows from a reservoir sufficiently large, and initially (before flow commences) at the same pressure and temperature, then, gas being a nonconductor of heat when the flow is steady, a first integration of the equation of motion shows that the energy of equal elementary weights of the gas is constant. This energy is made up of two parts, the energy of motion and the intrinsic energy. As the gas acquires energy of motion, it loses intrinsic energy to exactly the same extent. Hence we have an equation between the energy of motion, i.&-. the velocity of the gas, and its intrinsic energy. The laws of thermodynamics afford relations between the pressure, temperature, density, and intrinsic energy of the gas at any point. Substi- tuting in the equation of energy, we obtain equations between the velocity and either pressure, temperature, or density of the gas. The equation thus obtained between the velocity and pressure is that given by Thomson and Joule ; this equation holds at all points in the vessel or the effluent stream. If, then, the pressure at the orifice is known, as well as the pressure well within the vessel where the gas has no energy of motion, we have the velocity of gas at the orifice ; and obtaining the density at the orifice from the thermodynamic relation between density and pressure, we have the weight discharged per second by multiplying the product of velocity with density by the effective area of the orifice. This is Thomson and Joule's equation for the flow through an orifice. And so far the logic is perfect, and there are no assumptions but those involved in the general theories of thermodynamics and of fluid motion. But in order to apply this equation, it is necessary to know the pressure 314 ON THE FLOW OF GASES. [53 at the orifice ; and here comes the assumption that has been tacitly made : that the pressure at the orifice is the pressure in the receiving vessel at a distance from the orifice. 3. The origin of this assumption is that it holds, when a denser liquid like water flows into a light fluid like air, and approximately when water flows into water. Taking no account of friction, the equations of hydrodynamics show that this is the only condition under which the ideal liquid can flow steadily from a drowned orifice. But they have not been hitherto integrated so far as to show whether or not this would be the case with an elastic fluid. In the case of an elastic fluid, the difficulty of integration is enhanced. But on examination it appears that there is an important circumstance connected with the steady motion of gases which does not exist in the case of liquid. This circumstance, which may be inferred from integrations already effected, determines the pressure at the orifice irrespective of the pressure in the receiving vessel when this is below a certain point. 4. To understand this circumstance, it is necessary to consider a steady narrow stream of fluid in which the pressure falls and the velocity increases continuously in one direction. Since the stream is steady, equal weights of the fluid must pass each section in the same time ; or, if u be the velocity, p the density, and A the area of the stream, the joint product upA is constant all along the stream, so that W A = , gpu where is the mass of fluid which passes any section per second. U In the case of a liquid p is constant, so that the area of the section of the stream is inversely proportional to the velocity, and therefore the stream will continuously contract in section in the direction in which the velocity increases and the pressure falls, as in Fig. 1, also Fig. 2 A. Fig. 1. In the case of a gas, however, p diminishes as the velocity increases and the pressure falls ; so that the area of the section will not be inversely 53] ON THE FLOW OF GASES. 315 proportional to u, but to u x p, and will contract or increase according to whether u increases faster or slower than p diminishes. As already described, the value of pu may be expressed in terms of the pressure. Making this substitution, it appears that pu increases from zero as p diminishes from a definite value p^ until p = 5%7p 1 ; after this pu diminishes to zero as p diminishes to zero. A varies inversely as pu, and therefore diminishes from infinity as p diminishes from p l till p = '52 t jp l ; then A has a minimum value and increases to infinity as p diminishes to zero, as in Fig. 2. Fig. 2. The equations contain the definite law of this variation, which, for a particular fall of pressure, is shown in Fig. 2 A. Fig. 2 A. For the present argument it is sufficient to notice that A has a minimum value when p=-527jt>,; since this fact determines the pressure at the orifice when the pressure in the receiving vessel is less than '527^, that being the pressure in the discharging vessel. 316 ON THE FLOW OF GASES. [53 5. If, instead of an orifice in a thin plate, the fluid escaped through a pipe which gradually contracted to a nozzle, then it would follow at once, Fig. 3. from what has been already said, that when p z was less than '52*7 p lt the naiTowest portion of the stream would be at N, for since the stream converges to N the pressure above N can be nowhere less than '527^ ; and since emerging into the smaller surrounding pressure p 2 the stream would expand laterally, N would be the minimum breadth on the stream, and hence the pressure at N would be '527p 1 . In a broad view we may in the same way look on an orifice in the wall of a vessel as the neck of a stream. But if we begin to look into the argument, it is not so clear, on account of the curva- ture of the paths in which some of the particles approach the orifice. Since the motion with which the fluid approaches the orifice is steady, the whole stream, which is bounded all round by the wall, may be considered to consist of a number of elementary streams, each conveying the same quantity of fluid. Each of these elementary streams is bounded by the neighbouring streams, but as the boundaries do not change their position they may be considered as fixed. The figure (4) shows approximately the arrangement of such stream. But for the mathematical difficulty of integrating the equations of motion, the exact form of these streams might be drawn. We should then be able to determine exactly the necks of each of these streams. Without complete integration, however, the process may be carried far enough to show that the lines bounding the streams are continuous curves which have for asymptotes on the discharging-vessel side lines radiating from the middle of the orifice at equal angles, and, further, that these lines ail curve round the nearest edge of the orifice, and that the curvature of the streams diminishes as the distance of the stream from the edge increases. These conclusions would be definitely deducible from the theory of fluid motion could the integrations be effected, but they are also obvious from the figure and easily verified experimentally by drawing smoky air through a small orifice. From the foregoing conclusions it follows, that if a curve be drawn from 53] ON THE FLOW OF GASES. 317 A to B, cutting all the streams at right angles, the streams will all be converging at the points where this line cuts them, hence the jiecks of the streams will be on the outflow side of this curve. The exact position of Fig. 4. these necks is difficult to determine, but they must be nearly as shown in the figure by cross lines. The sum of the areas of these necks must be less than the area of the orifice, since, where they are not in the straight line AB the breadth occupied on this line is greater than that of the neck. The sum of the areas of the necks may be taken as the effective area of the orifice ; and, since all the streams have the same velocity at the neck, the ratio which this aggregate area bears to the area of the orifice may be put equal to K, a coefficient of contraction. If the pressure in the vessel on the outflow side of the orifice is less than 527_/) 1 , this is the lowest pressure possible at the necks, as has already been pointed out, and on emerging the streams will expand again, as shown in the Fig. 4, the pressure falling and the velocity increasing, until the pressure in the streams is equal to p. 2 , when in all probability the motion will become unsteady. If 7> 2 is greater than '527/),, the only possible form of motion requires the pressure in the necks to be p 2 , at which point the streams become parallel until they are broken up by eddying into the surrounding fluid (Fig. 5). 318 ON THE FLOW OF GASES. [53 6. There is another way of looking at the problem, which is the first that presented itself to the author. Fig. 5. Suppose a parallel stream flowing along a straight tube with a velocity u, and take a for the velocity with which sound would travel in the same gas at rest, the velocity with which a wave of sound or any disturbance would move along the tube in an opposite direction to the gas would be a u. If then a = u, no disturbance could flow back along the tube against the motion of the gas ; so that, however much the pressure might be suddenly diminished at any point in the tube, it would not affect the pressure at points on the side from which the fluid is flowing. Thus, suppose the gas to be steam and this to be suddenly condensed at one point of the tube, the fall of pressure would move back against the motion, increasing the motion till u = a, but not further ; just as in the Bunsen's burner the flame cannot flow back into the tube so long as the velocity of the explosive mixture is greater than the velocity at which the flame travels in the mixture. 53] ON THE FLOW OF GASES. 319 According to this view, the limit of flow through an orifice should be the velocity of sound in gas, in the condition as regards pressure, Density, and temperature, of that in the orifice; and this is precisely what it is found to be on examining the equations. 7. The following is the definite expression of the foregoing argument. The adiabatic laws for gas are : p being pressure, p density, r absolute temperature, and 7 the ratio of the specific heats, y-l ' 1 m The equation of motion, u being the velocity and x the direction of motion, is du _ dp P dx dx' u? [P dp or - ~ +G ( 2 >- Substituting from equations (1), JP dp _ 7 JP O r_ Jo p 7- 1 /3 r ' 7 - l p T O ( v^j ^ y^ \p, Hence along a steady stream, since W is constant, equation (5) gives a relation that must hold between A and p. Differentiating A with respect to p and making -^-- zero, it appears Y-I r -i .............................. (6), __ / 2 V 1 or 1: = - ^ P'or air 7 = 1'408. /. - = -527 ...(8). 320 ON THE FLOW OF GASES. [53 It thus appears that as long as p falls, the section continuously dimin- ishes to a minimum value when p = -o2lp 1 , and then increases again. Substituting this value of p in equation (3), -/ v. - 1 , i.e. the pressure may either be a maximum or a minimum at G. In this respect gas differs entirely from a liquid, with which the pressure can only be a maximum at C. 54. ON METHODS OF INVESTIGATING THE QUALITIES OF LIFEBOATS. [From the " Proceedings of the Manchester Literary and Philosophical Society," Vol. xxvi.] (Read December 14, 1886.) THE lamentable accidents to the St Anne's and Southport lifeboats on the 9th inst. seem likely to lead to steps being taken to obtain a more systematic investigation as to the qualities of these boats than has yet been undertaken. It seems, therefore, a proper time to direct attention to certain facts and general considerations, the importance of which have impressed themselves upon me during many years' investigation. Before entering upon this, it may be remarked that there is probably no class of boats, on the design and construction of which more attention and skill have been spent, than on lifeboats, or of which the qualities are so well adapted to the circumstances, taken all round. If we compare the results of the use of these boats with the results obtained in the use of the navies of this or any other country, it will, without a moment's hesitation, be admitted that the designers of lifeboats and lifeboat paraphernalia have arrived much nearer perfection than the designers of war vessels and their armaments. That the high standard already obtained by these boats has not been the result of scientific investigation, or the theoretical application of any known principles of equilibrium, does not render the method less scientific, for the base of all science is observation and experiment, and these boats o. R. ii. 21 322 ON METHODS OF INVESTIGATING THE QUALITIES OF LIFEBOATS. [54 are the result of such a course of direct experiments and experimental observation as has not been expended on any other modern structure, nor is this method of arriving at the best form peculiar to lifeboats. With the exception of the large modern steamers and ironclads, the peculiar construction of boats of all sizes is the result of a prolonged process of trial and failure, and that, although certain general principles, connecting the qualities of ships with their shapes, have been discovered and recognised during the last thirty years, still, the recognition of these principles has not resulted in the suggestion of any considerable improvement to be effected in what were before high class vessels, such as yachts and fast sailing vessels, but rather have confirmed the form previously arrived at in these as the best, and led to their being copied in larger vessels. The discovery and recognition of principles have undoubtedly been of immense service in improving the types of our large modern vessels. But this is mainly because with large ships there is not the same opportunity for trial and failure as with the small, the number is so much smaller, and experiments are so much slower and more costly ; but the main reason is, that the circumstances which call out the highest qualities of the large vessels become so extremely rare. There is no doubt that many large vessels pass through their lives without meeting weather which tests their sea-going qualities in the way in which those of a fishing boat are tested many times every winter. It was, therefore, an immense step in the way to study the resistance qualities of large ships, when the late Mr Fronde brought into practice the rules connecting the resistance of the full-sized vessel with that of an exact model to scale. By means of a tank 200 feet long, and models on scales of 1 to 50, or 1 to 20, the resistance and rolling qualities of all Her Majesty's ships have since been verified before they are constructed. And the same is now done by manufacturers of mercantile vessels, like Mr Denny, who have tanks of their own. The qualities of ships thus tested were originally limited to those of resistance and of rolling, and so far as I know, no extension has taken place; for although in 1876 it was pointed out by the author before Section 9 of the British Association, that by constructing models of our war ships on a scale large enough to enable them to be used as launches, say 1 to 16, and supplying these launches with power as the cube of their dimensions then the manoeuvering qualities would be similar if conducted on scales proportional to their lengths, the time occupied by the launches in executing a particular evolution, as compared with that occupied by the ships, being as the square root of their lengths. So that with such models the officers and seamen could be instructed in the handling of their ships without cost or risk. This has not been done. The Admiralty replying, 54] ON METHODS OF INVESTIGATING THE QUALITIES OF LIFEBOATS. 323 so far as they did reply, that their officers were continually experimenting with the launches disregarding the fact that the launches in ijse_were in no sense models of the ships, and were supplied with power five or six times too great in proportion thus ignoring the point of the suggestion, namely, that the experience gained by the models might be applicable to the ships, which with their present launches it is not, and only tends to mislead those who attempt a comparison. Since making this suggestion, I have been much engaged in experiments with water, which have enabled me to extend this law of similarity, until I find it is possible now to lay down the conditions under which to test the seaworthy qualities of a vessel from those of its model. Certain conditions have to be observed, but, in general, it may be asserted that provided the models are to scale, that the height and length of the waves are to the same scale, the velocity of the wind being as the square root of the scale, or in other words, the corresponding depressions of the barometer in the same scale as the models the behaviour of the model would be similar to that of the boat. Thus, the behaviour of a model three feet long in waves two feet high, and with a wind twenty miles an hour, would correspond with that of a boat twenty-seven feet long in waves eighteen feet high, and a velocity of the wind sixty miles an hour. The main object of this communication is to point out that this similarity in the behaviour of models and larger boats under circumstances as regards the stress of weather, corresponding in scale to that of the models and boats, affords an opportunity of testing the seaworthy qualities of the lifeboats in a degree that they cannot otherwise be tested. For, although the size of the boats does not preclude the possibility of their qualities being actually tested under any circumstances of sufficiently common occurrence to afford opportunities, yet the circumstances which call for the highest qualities in these boats, and in which the boats are most needed, are of extremely rare occurrence ; this appears at once, when it is considered that it is years since anything approaching such a storm as wrecked the two boats has been experienced, and that in order to submit any modified boat to a similar test, it may be years before there will be another chance, even if it could be made available when it did come. To make satisfactory tests on the full-sized boats, command is wanted of the extreme circumstances, and this cannot be had ; while on the other hand, to test the same qualities in their models, these extreme circumstances, modified to scale, are all that is wanted, and these are of such common occurrence as to afford ample opportunity, even if they cannot be commanded by artificial means. 212 324 ON METHODS OF INVESTIGATING THE QUALITIES OF LIFEBOATS. [54 If the qualities to be tested involved the handling of the boats, then the models must be large enough to carry a crew ; that is to say, they would have to be small lifeboats. Even with such, much experience can be and has been gained, which could not be obtained with larger boats, for bad weather for the smaller is only moderate for the larger, and is of compara- tively common occurrence compared with that which affords a similar test for the larger boats. It is, however, the self-righting qualities of these boats that is for the moment in question ; this requires no crew, or at most a dummy crew, so that there is no limit to the smallness of the models, except what arises from the conditions of dynamical similarity, and these would admit of models as small as two or three feet. It may be well to say one word as to the powers of self-righting, and the question as to how far these powers may be affected by the wind and waves. I do not know that it has ever been suggested that wind and wave have any such effect. But it is equally certain, that there is no d priori reason why they should not, and. short of actual experience, it cannot be said that any boat which would right itself in calm water would do so equally well in any storm that might blow. On the other hand there are reasons why wind and waves must, individually and collectively, affect the stability of an upturned boat. In the first place, the wind will keep such a boat broadside on, which will be in the trough of the sea raised by the wind, although the swell may, of course, be running in another direction. The wind, acting on the bottom, will further drive the boat broadside on through the water. This horizontal thrust of the wind, acting on the part of the boat above water, and balanced by the resistance of the water on the submerged portion, will tend to right the boat by turning her keel to leeward, and so far it would seem that the wind would help to right her, but owing to the shape of the bottom of the boat when broadside on, there will be a vertical force resulting from the wind as well as the horizontal, and this vertical force will bear down that side of the boat toward the wind, and this effect will be enhanced by the weight of the waves breaking on this side of the boat tending to right her by turning her keel to windward, or in direct opposition to the horizontal effect ; and more than this, the vertical effect of the wind and waves to turn the keel to windward will be greatest when the windward side of the boat's bottom has some definite inclination to the horizontal, while the horizontal effect to turn the keel to leeward will continually increase as the keel turns to windward, so that it is possible that in a particular wind and sea there may be a position of very stable equilibrium, towards which, if the keel is to leeward, the vertical effect of the wind and the waves predominating 54] ON METHODS OF INVESTIGATING THE QUALITIES OF LIFEBOATS. 325 over the horizontal effect, will bring it back, and vice versa ; if the keel is turned to windward, the horizontal effect predominating will alsa tend to bring it back. The fact that two boats were found stranded bottom upwards, with part of their crews underneath, and that one of these is known to have upset in comparatively deep water, and to have remained in that position during a long time while drifting into shallow water, seems altogether inconsistent with the supposition that these upturned boats were in their normal condition of instability, as when in calm water. For although in a calm sea the effect of three or four men hanging on to each side of the boat might prevent the initial motion of turning, before the weight of the iron keel and ballast obtained sufficient leverage to lift the weight of the men and so keep the boat stable, this could hardly be the case in a rough sea, when the waves would be continually altering the balance of the boat. These are questions which can only be set at rest by experiments, and the method of models thus affords a means of testing the righting qualities of these boats under circumstances as severe or more severe than any to which they will ever be subjected, and this without waiting and without danger; while with full-sized boats such tests are impossible, for even should an extreme storm occur opportunely for making the experiment, the danger involved with full-sized boats would preclude the possibility of their being undertaken. It is this last consideration which has led to these suggestions, and not the idea that the experiments on models would be more satisfactory ; while the fact that the experiments on models could be made at much smaller cost, is too small a matter to be considered, when, as in this case, the lives of some of the most heroic of our fellow countrymen, and the sentiments of the entire nation, are involved. 55. ON CERTAIN LAWS RELATING TO THE REGIME OF RIVERS AND ESTUARIES, AND ON THE POSSIBILITY OF EXPERI- MENTS ON A SMALL SCALE. [From the " Report of the British Association," 1887.] 1. THE object of this communication is to bring before Section G certain results and conclusions with respect to the action of water to arrange loose granular material over which it may be flowing. These results and con- clusions were in the first instance arrived at during a long-continued in- vestigation, undertaken with a view to bring the general theory of hydro- dynamics into accord with experience, rather than with any special reference to the subject in hand, but have since been to some extent made the subject of special investigation. 2. A systematic study of the regime of rivers naturally divides itself under three heads, which may be stated as follows : (1) The more general facts observed as regards the regimen of the beds. (2) The movements of sand consistent with these observed facts. (3) The necessary actions of the water to produce these movements in the material of the beds. Observed facts. Amongst the most general facts to be observed as to the arrangement of the material forming the beds of estuaries are (1) The general stability or steadiness of these beds, so far as is shown by their outline or figure, while, at the same time, as is shown by the obliteration of all footprints and markings casually placed upon them, also by the ripple mark, the material at the surface of these beds is being continually shifted. 55] ON CERTAIN LAWS RELATING TO RIVERS AND ESTUARIES, ETC. 327 (2) The almost absolute steadiness in figure of some of these beds. (3) The gradual changes in the position and form of -others the growth or accumulation of sand-banks *in some places, and the wasting of banks or removal of sand in others. Movement of sand. As regards the movement of sand consistent with these changes, in the first place the movement, whatever it may be, is one of the surface, and not one in bulk ; and in the next place such movement of the surface must be continually going on, whether it produces any change in the figure of the banks or not. The invariable obliteration of footprints and marks which may have been left on the sand at low water, as well as the ripple marks, are absolute evidence of a general disturbance of the surface, and it requires but little observation to show that the disturbance is of the character of a drift of sand, in whatever direction the water may be moving. Uniform drift. Where the outline of the banks is not altered, this drift or motion of the sand must be uniform, as much sand being deposited at each point as is removed from that point. Although there may be a general flow of the sand in some direction, if the drift is uniform this movement will not alter the figure of the bed, which, like the balance in another kind of bank, does not depend on the rate of deposit and with- drawal, but on the excess of one of these over the other. The gradual accumulation or diminution of sand at any point is clearly not due to a simple action of deposit or removal, as it is always attended with the same evidence of the drifting of the surface, and is clearly the result of a difference in the quantities of sand deposited or removed by the drift. Movement of water. The manner in which a current of water acts on the granular material forming the bed of the current has been the subject of an investigation by various experimenters. It has been found that the primary action is not so much to drag the grains along the bottom, but to pick them up, hold them in a kind of eddying suspension, at a greater or less height above the bed, for a certain distance and then drop them, so that, when the water is drifting the sand, there is a layer of water adjacent to the bottom, of a greater or less thickness, charged to a greater or less extent with sand. The faster the current and the finer the sand the greater will be the thickness of the charged layer, as well as the denser is the charge in the layer. A certain definite velocity, according to the size and weight of the grains, is required before the water will raise the grains from the bottom, and for all velocities above the minimum necessary to raise the sand the suspended charge increases with the velocity, and the rate of drift or the quantity 328 ON CERTAIN LAWS RELATING TO RIVERS AND ESTUARIES, [55 of sand which passes a particular section increases much faster than the velocity. Attempts have been made, with greater or less success, to deter- mine exact laws connecting the minimum velocities at which the sand begins to drift, with the weight of the grains and other circumstances ; also to determine the exact law of rate of increase of the drift with the velocity. For my present purpose, however, it is not necessary to enter upon such considerations. From the facts already mentioned, it will appear that the effect of a uniform current of water over a uniform bed of sand will not be to raise or lower the bed ; for, as the charge of sand in the water remains uniform, it must drop as many particles as it raises everywhere on the bed. This is the action of the water in causing a uniform drift. It is also evident that, if the charge in the water as it comes to any particular place is less than the full charge due to its velocity, it will pick up from that place more sand than it drops, and so increase its charge at the expense of the bed, which will there be scoured or lowered. And conversely, if the water as it arrives at any place is overcharged, it will relieve itself by depositing more than it picks up, and so raise or silt up the bed. As regards the circumstances which can cause the water to be charged to a greater or less extent than that which it would just maintain with such velocity as it has, the most important are (1) An increasing or diminishing velocity. When the water is moving in a stream from a point where the velocity is less to one where it is greater, the velocity of the actual water as it moves along is increasing, as will also be its normal charge of sand ; hence it must be continually picking up more than it deposits. And conversely, when moving from a point of greater velocity to one of less, its normal charge will be continually diminishing through deposits on the bed. (2) Another circumstance which affects the charge of sand with which the water may arrive at a particular point is a variation in the character of the bed. If, for instance, water flows from a rocky bed on to sand, it may arrive on the sand without charge, and immediately charges itself at the expense of the bed. Or again, where water flows from a sandy bottom on to a clean or grassy rocky bottom, it gradually loses its charge, silting up the bottom. The direction in which the sand is moved by the water is sensibly in the direction in which the water which holds the charge is moving. But, as was first pointed out by Dr James Thomson as affording an explanation of the 55] AND ON THE POSSIBILITY OF EXPERIMENTS ON A SMALL SCALE. 329 generally observed fact that the beds of rivers are scoured on their convex sides and silted on their concave, the layers of water adjacent tn the bed do not always move in the general direction of the stream. There are often steady cross currents at the bottom, as in the case mentioned, though such cross currents do not exist except under circumstances which may be readily distinguished. The most important of these is that pointed out by Dr Thomson curvature in the general direction of the stream, in which case the centrifugal force of the more rapidly moving water above over- balances that of the water retarded by the bottom, and forces the latter back towards the centre of the curve. This action is universal, where even the lateral boundaries are such as to require the water to move in curved streams ; the drift at the bottom does not follow the general direction of the stream, but sets towards the centre of the curve. The result of the foregoing consideration is to lead to the conclusion that the regime of each part of the bed as to maintenance in steady condition, lowering or raising it any time, depends solely on the character of the motion of the water, which if straight and uniform, neither acquiring nor losing velocity, causes a uniform drift in the direction of the stream, which main- tains the condition steady. If losing velocity, causes a depositing drift and raises the bed ; if gaining velocity, causes a scouring drift and lowers the bed ; while if curved, the direction of the drift is diverted towards the centre of the curve, with its attendant effect to lower the convex side and raise the concave side of the bed. This conclusion seems to be of the utmost im- portance in dealing with this subject. For if it is correct, not only can the character of the action going on at the bed be inferred from the observed motion of the water, and vice versa, but since, according to this conclusion, the character of the action is independent of the magnitude or velocity of the stream, the results will be the same on a small scale as on a large one, provided only that the character of the motion of the water is the same at all points. In this latter respect this conclusion affords an explanation of a fact that cannot fail to have struck every one who has observed the sand- beds of the streams running over sands which have been left by the tide, viz., what an almost exact resemblance they bear to each other, whether having the size of a moderate river or of the smallest rivulet. On the large scale of actual estuaries we can only test the conclusion by actual observation, but on a small scale we can experimentalise in whatever condition of motion we want to test, and readily observe the effects pro- duced ; a possibility of which great use has been made in this investigation, and which will be again referred to. As applied to a non-tidal river, in which the direction of the motion 330 ON CERTAIN LAWS RELATING TO RIVERS AND ESTUARIES, [55 is always the same, the foregoing conclusion would lead us to expect that the regime would be steady except at the bends, the sources, and the mouth, which is exactly what is observed, so that the conclusion so far agrees with experience. The most striking feature about rivers is the way they wriggle about in the alluvial valleys ; a phenomenon pointed to by Lyeli as one of those causes still in progress which had produced the present conditions of the valleys, and which, as already stated, was explained by Dr Thomson. From the source of the river, as the rain-water acquires the velocity, it charges itself with deposit, which charge it maintains with continual taxes and drawbacks until it reaches the ocean or lake, when its water in again losing its velocity deposits its charge, continually carrying forward the bar and extending its delta. In non-tidal rivers, whether large or small, fast or slow, the characters of these actions are invariable, however much they may differ in intensity. The case of tidal estuaries is, however, by no means so simple. Here we have not, as in a river, a continuous progression of the same character of action at the same point. On the contrary, at every point the action is changed twice a day. For the change in the tidal current does not merely change or reverse the direction of the sand-drift at each part of the bed, but it changes and often reverses the character of this drift, changing what has been a scouring drift during the ebb-tide into a depositing drift during the flood ; so that the question as to whether the regime is stable, depositing, or scouring is not simply a question as to whether the current at this point is uniform, accelerated, or retarded, but whether the action of the ebb to cause, say, scour is equal to, less than, or greater than the action of the flood to cause deposit. As there is no likelihood that the resultant effect as regards the general regime of two opposing influences will resemble what would have been the simple effect of either of the influences acting alone, this dual control affords abundant reason why the configuration of the beds of these tidal estuaries should differ in character from the configuration of the sand-beds of continuous streams. There is, however, another and an equally important difference between the general motion of the water in rivers and tidal estuaries. The function of the estuary is by no means that of a simple channel to conduct the tidal water up and down. It equally discharges the function of a reservoir or basin, to be filled and emptied by each tide. In consequence of this action as a reservoir, the directions of the motions of the water during flood and ebb, and particularly towards the top of the flood and commencement of the ebb, are generally very different from what 55] AND ON THE POSSIBILITY OF EXPERIMENTS ON A SMALL SCALE. 331 they would be were the estuary acting the simple part of a channel conduct- ing the water from one place to another. When a vessel is filled by a stream entering on one side, the forward motion of the water is stopped before reaching the opposite side. But if, as is always the case, the motion which the water has on entering is more than sufficient to carry it as far as is necessary, the remaining momentum is spent in setting up eddies, or a general circulation in the water, so that when the vessel is full the water within it is not by any means at rest, but may be circulating round or have any other motion. If, then, the water is allowed to flow out, the initial motion will not be a steady movement towards the outlet from all parts of the vessel, but those portions of the water which are moving towards the outlet will have their motion accelerated, while those which are moving in the opposite direction will have first to be stopped before they begin to approach the outlet. And thus the ebb will begin earlier at some points in the vessel than at others. It was the observation of such an effect as this in one of our largest estuaries that first directed my attention to the subject of this paper. Having investigated this point sufficiently for my own satisfaction nothing further was done until 1885, when my attention was directed to the inner estuary of the Mersey. This estuary may be described as a crescent-shaped shallow pan, eleven miles long by three broad, lying north-west and south-east, having its upper horn pointing east and its lower horn north ; the northern horn, being prolonged for five miles into a narrow deep channel, runs north to the outer estuary or sandy bay of the sea. One of the most marked features presented by the configuration of the bed of this inner estuary is the invariable prefer- ence of the low-tide channels for the concave or Lancashire side ; whereas, were the estuary acting merely the part of a river, whether during flood or ebb, it would be expected to follow the usual law, and have the deepest water on the convex or Cheshire side. That this prevalence of the deepest water on the concave side must be the result of the momentum left in the water by the flood at once seemed to me probable ; for if the bottom were level or deepest on the Lancashire side the effect of the curved shape would be to cause the flood entering at the northern horn to follow the south-eastern or Cheshire shore, and the momentum of this water would tend to carry it round the head of the estuary and back along the Lancashire side ; would, in fact, tend to set up a circulation before the top of the flood was reached; so that on the Lancashire side the water would be moving down the estuary before the ebb commenced; whence, considering that the flood tends to raise the bottom and the ebb to 332 ON CERTAIN LAWS RELATING TO RIVERS AND ESTUARIES, [55 lower it (for the reasons already pointed out), it seems that the stronger flood on the Cheshire side would raise this side, while the stronger ebb on the Lancashire side would lower this. This is supposing the bottom to be level. In order to verify these conclusions a vessel was constructed having a flat bottom and a vertical boundary of the same shape as the high-tide line of the inner estuary from the rock to the same distance above Runcorn. The horizontal scale was 2" to a mile, and the vertical scale 1 inch to 80 feet, A shallow tin pan was hinged on to the otherwise open channel at the rock, by raising and lowering which, when full of water, the motion of the tide could be produced throughout the model through the narrows ; the true form of the bed of the channel was given to the model by means of paraffin. And in order to obtain approximately the proportional depth in the inner estuary, sand was placed level on the bottom so that the high-tide depth was reduced to the equivalent of about twenty feet. The idea in making this model was not so much to obtain a shifting of the sand, as to show the circulation of the water as resulting from the flood tide with a level bottom. In the first instance the tide pan was raised and lowered by hand, but as at the first trial it became evident that the model was not only going to show the expected circulation, but was also capable of showing, by the change in the position of the sand, the effect of this circulation on the configuration of the estuary and other important effects, it was arranged that the model should be worked from a continuously running shaft. The working of the model by hand at once showed that there was only one period of working at which the motion of the water in the model would imitate the motions of the actual tide in the Mersey, which period was found to be about forty seconds ; a result that might have been foreseen from the theory of wave motions, since the scale of velocities varies as the square roots of the scales of wave heights, so that the velocities in the model which would correspond to the velocities in the channel would be as the square roots of the vertical scales about 3^ and the ratios of the periods would be the ratio of horizontal scales divided by this ratio of velocities, or 33 J. 31800 " 950 ' Hence, taking 11 '25 hours or 40,700 seconds as the tidal period, the period of the model 40700 = -jpr:- = 42 seconds (about). you This period was adopted for working the model from the shaft. It was then found that the circulation at the top of the flood, which was 55] AND ON THE POSSIBILITY OF EXPERIMENTS ON A SMALL SCALE. 333 very evident while the bottom was flat, caused a general rise of the sand on the Cheshire side and lowering on the Lancashire, which went ,_ ' ib z ' so r - i,6.V Prri Ic ftit ZH,. Iron, tl,r Hot-Jim Fig. 15. ofi] ON TRIPLE-EXPANSION ENGINES AND ENGINE -TRIALS. 37l THE CONDENSATION, PRIMING AND LEAKAGE OF STEAM IN_ THE CYLINDERS, AS SHOWN IN THE DIAGRAMS. There are two quantities which it is almost impossible to separate by the inherent evidence of the diagrams. The missing quantity, to use Mr Willans' expression, which is here shown by the horizontal breadth of the black band, may equally well arise from the steam having escaped by the piston, or having been temporarily converted into water. This much, however, is evident from the diagrams, that with steam in the jackets, in whatever manner the steam has vanished in the high-pressure engine, it has all reappeared before the end of the stroke in the inter- mediate engine, and though some of it has disappeared at the cut-off in the low-pressure cylinder, it has reappeared again before the end of the stroke. Hence it seems that there is no escape of steam by the pistons of these two engines. The question remains, however, as to whether steam has not escaped by the pistons of the high- pressure engine, and through the valves, during expansion into the cylinders of the intermediate and low-pressure engines. Certain differences in the diagrams taken from No. II engine, when \vorking with different cuts-off, suggested that the rider valves were held somewhat off the back of the main valve by the spindle, so that they leaked steam until the pressure in the cylinder was sufficiently lower than that in the steam -chest to spring the spindle and force the valves home. It became particularly evident in the fifty-fifth trial, and then the cover was removed and the conclusion verified. This source of error was put right, and the fifty-sixth trial, as compared with the earlier ones, shows what has been the effect of leakage in these, namely, the breadth of the black band towards the tops of the diagram from No. II engine. When the covers were last put on, in August, 1888, the cylinders and valve-faces were all in equally good condition, and there has been no leak from the jackets, while the engines were standing with full pressure in the jackets. The regulators opening into the intermediate receivers were made tight in August, 1888, and were not again opened till after the forty- sixth trial. There was then occasion to open them, and as the engines were standing preparatory to starting the fifty-sixth trial, it was seen that steam was leaking into No. II receiver, probably about -lb. per minute; as the valve was found to be shut, there was nothing to be done, so the trial was run ; and, as was to be expected, the diagrams from No. I engine show what, considering the circumstances, is an unusually large black band. 242 372 ON TRIPLE-EXPANSION ENGINES AND ENGINE-TRIALS. [56 In the absence of definite evidence of leakage, the Author concludes that the missing quantity shown by the black band is everywhere due to con- densation. It is not the intention in this Paper to endeavour to establish a complete theory of cylinder-condensation. Though it may be well to state that, before designing the engines, the theory was carefully considered and formulated, leaving only the arbitrary constants to be determined from the experiments. For anything like a complete determination of these constants, the experi- ments have not sufficiently advanced ; but this is not necessary to show that in the case of a series of cylinders, all jacketed up to boiler-pressure, the law of condensation would be precisely that which is shown in the diagrams. Whenever the bounding surfaces are colder than the steam adjacent to them condensation occurs. To prevent condensation it is therefore necessary to maintain all parts of the cylinder surfaces, and port passage surfaces, at a temperature at least as high as that of the initial steam. To do this, in the case of expansion, it is not sufficient (as seems to be commonly assumed) to keep the outside of the metal constituting the walls and covers, merely at the temperature of the initial steam. That, of course, would be sufficient if there were no condensation other than what results from the temperature of the surfaces. Forty years ago no such other cause of condensation was known. It was revealed, however, by the discoverers, Rankine and Clausius, in 1849, that the expansion of steam reduces its temperature below that corresponding to saturation unless some of the steam is condensed. The manner of action of this supersaturation, caused by expansion, in absorbing heat from the walls of the cylinder maintained at a higher temperature than the steam, does not appear to have been yet ascertained with any degree of certainty ; but it is certain that steam in this state of supersaturation does absorb heat with immense rapidity, when the walls are at a higher temperature than the expanded steam. Also the amount of heat necessary to prevent supersatura- tion is definitely known, though it is, perhaps, well to recall the fact that it is not, even approximately, the heat equivalent of the work done by the steam during expansion. If the walls of the cylinders are maintained at the temperature of the initial steam, the expanding steam will absorb heat. This heat must pass through the walls ; and as heat only flows through metal down the gradient of temperature, the temperature on the outside must be greater than that on the inside. Hence it follows that either the steam in the jackets must be hotter than the initial temperature of the steam in the cylinder, or the mean temperature of the internal surface of the cylinder will be below that of the initial steam, in which case there will be cylinder-condensation. 5b'] ON TlilPLE-EXPANSION ENGINES AND ENGINE-TRIALS. 373 How important this degradation of temperature through the walls is, will, perhaps, best be rendered apparent by stating an actual case. In expanding 1 Ib. of steam from a pressure of 203 Ibs. to a pressure of 79 '3 Ibs., the heat per Ib. necessary to prevent supersaturation is 551 T.U. or about 5 per cent, of the total heat in the initial steam In a cylinder passing 600 Ibs. of steam per hour, to prevent supersaturation there should pass through the walls of the cylinder 33,060 T.U. Now the jacketed surface of the cylinder of the H.-P. engine is less than 1*5 square foot, and the thickness of the metal is more than 0*4 inch. Hence the heat would have to flow through this thickness of metal at a rate of 2'2,OQO T.U. per square foot per hour. From the known laws of conductivity of iron, this would require a difference of temperature of 38 Fahrenheit. Thus it appears that, to prevent supersaturation, the temperature of the steam in the jackets of No. I engine must be 38 higher than the mean temperature of the internal steam ; or, in other words, that the mean tem- perature of the internal surfaces will be 38 lower than that of the initial steam, which is at the same temperature as that in the jackets. What amount of surface-condensation this difference of temperature would cause may be, to some extent, inferred by comparison with the differ- ence between the mean temperature of the surfaces and that of the initial steam when the jackets are empty. Here the initial temperature is about 383, and that of the exhaust, 302 ; the mean, 342 ; difference of mean and initial, 41; so that in this engine the mean temperature of the walls would only be affected to the extent of about 3 Fahrenheit by the jackets, suppos- ing the whole of the heat to prevent supersaturation were supplied by the jackets. But this would not be quite the case, as some heat is obtained from the difference in the heat given up and absorbed by the cylinder-con- densation ; and there is no proof that the steam may not be discharged with a certain degree of supersaturation. However, the reasoning leads to the conclusion that, with steam at initial pressure in them, the jackets would produce a comparatively small difference on the cylinder-condensation in this engine when passing 10 Ibs. of steam per minute. In No. II, the intermediate engine, the case is different. Here the heat which has to flow into the cylinder through the walls is nearly the same as in 374 ON TRIFLE-EXPANSION ENGINES AND ENGINE-TRIALS. [56 No. I ; but the surfaces are double as large and of the same thickness, so that the fall of temperature would be about one-half, or 20. The tempera- ture of the steam in the jackets is 81 above that of the initial steam, and the internal walls would still be 60 above the initial temperature. Hence there should be no condensation on those surfaces which are jacketed. Still there are in this engine, although much less than is usual in jacketed engines, portions of the surfaces which are not, so to speak, jacketed, mainly the surface of the ports and of the piston ; and though these derive heat from the jackets, it is through a much greater thickness of metal, and hence would require a much greater difference of temperature to prevent condensa- tion. Thus, even with the jackets at a temperature of 60 above that of the steam, there should probably be some initial condensation. In No. Ill engine the jackets have a temperature of 140 above the steam, hence the initial condensation should probably be much less than in No. II The diagrams (Fig. 15, p. 370) show that this is the case. They exhibit a little condensation, which seems to increase from cut-off until the expansion reaches about V5 or 2, and it then diminishes to zero. The increase after cut-off may be owing to the inertia of the indicator piston depressing the curve, as the springs used have always been as weak as possible on account of the low pressure. They also demonstrate conclusively, with such jacketing as there is in these cylinders, that a temperature of 140 in the jackets above the initial temperature is sufficient to prevent sensible cylinder-condensation with as much as 720 Ibs. of water per hour passing through the cylinders. The diagrams for the trials 41, 35, 40, show the condensation when the jackets are empty. These three diagrams are from trials as nearly as practicable corresponding in power with those with the jackets on. They are reduced to show the volume per Ib. of water through each engine, and the outside curve is the saturation-curve for 1 Ib. of steam ; the horizontal breadth of the black band, therefore, represents volume of steam missing. This includes the volume missing on account of the condensation resulting from expansion in each cylinder as well as on account of cylinder-condensa- tion. It is to be noticed, however, that the steam probably entered each steam-chest dry, so that the only water in excess of cylinder- condensation is that resulting from expansion in that cylinder. This would be represented by a curve draAvn from the points in the saturation-curve corresponding in pressure to the points of cut-off, and gradually diverging inwards from the saturation-curve, until at release the horizontal divergence should be about 5 per cent, of the horizontal breadth of the white diagram at that pressure. 50] OX TKII'LE-EXI'AXSION ENGINES AND ENGINE-TRIALS. 375 The great excess of condensation in the intermediate cylinder over the high-pressure, and in the low-pressure cylinder over the intermediate, is very apparent. This fully explains the difference in the relative speeds of the engines with and without the jackets already mentioned, the speed of No. Ill compared with No. I being as 1*5 with jackets to 1 without jackets. The distributions of condensation are very similar in the three cylinders. The ratios which the steam condensed bears to the steam passing through the engines at cut-off, middle stroke, and release, are shown in Appendix, Table IV. The testing of the boiler was carried only so far as was necessary to check the results of the trials. No chemical tests were taken of the air or coal. The coal used was Nixon's Navigation mixture, weighed as it came from the heap in the boiler-house. In most of the trials the feed was carefully measured, with the result, already mentioned, that it was from 5 to 10 per cent, greater than the discharge from the hot-well. Taking the feed, these trials show that the boiler generally evaporated 10'4 Ibs. of water per Ib. of coal with the pressure 195 Ibs. and the feed at 130. This, if all the water were evaporated, would give 11,350 units of heat per Ib. of coal. The temperature at which the gases left the boiler was 500, and after passing the water heater 250, the rise of temperature in the water-heater being about 100. The source of the loss of water was not discoverable, so that it was not possible to determine whether it escaped as water or steam ; and until this point could be determined it was impossible to say from observations on the boiler what the quantity of heat obtained in the boiler might be. The results in Table I are therefore confined to the steam received by the engines. - - CO OS 50 .S S O O O 50 "0 CD t i i i O X CD O O O rH CD O T i X O5 O O O CD O CO "* O r-pressure OgjOl ^ Tt* ^ n i- dc5 SrHOq -S CO r CO lO CM M 1-1 CD >0 ID CO rH o o o l~ CO O1 O 1^ rH o o o Ol Ol O5 CO 1^ Ol |i 51 - S *' "dl- i> >-5 iO Tj( !>. i 1 i i O O CO X 1^ o o o CO 01 rH rH O5 I R 1 fe 8 * S .^ id ^ B,<** o (M 05 00 J o * O O O X iO OJ O1 1- rH O o o o CD O O1 O X CO O O CO rH 10 O 01 O .S'3 uV T* of"* 3 ti 2 I-H e_ * X *? 3 <**$ rn O O5 >-0 i r-H O O5 CD 01 C-) rH CO Oi CM J>- (M rH o o 01 i i CO "O rH i I rH X OS iO O Ol O CO "^ CO X rH rH . : : < : i 1 1 i .iiiiii 'odd ^frfrfr h - fe 'o > S r-l 1 t .0 XKK 0) 3 '> s R 3 ^'S S R- ^2 +2 O fl i indicated in ) >5 55 | i S s s s % I " r-l r-l .2 n R R ^ - 'o ^ R S O O O KKK g C _s S c 1 <0 I Q ' r-l r- 1 rH rH |_| HH _c ' ^3 -u steam-jackets 'S 'S S3 -543 - 9 s j a .. ^ ^ I" i 1 . . . Tg a s 1 rC " S O .S ^ , ^ p, , "^ *^ O *' 9 s . 1, \ I-H r-l rH 1 1 rH d c o ^^^; 0) d d d KKK 1 E R a ^ ' s $ rH ,_, rH rH rH rH I 1 o' d d ^^^ 3 >+ o 1 -4-i 00 Number of 1 Date of the Time of tria a J ; r 40 BJ R 8 5 Revolutions 55 55 'So - - 1 o PH' rd R E rH ^ a s M X A- c " p., ri o 56 " K M I-H (N CO Tf 10 W J> 00 OS O rH N ^ iO CO t X 05 O rH 01 CO * o o OS -H X 1- X I- t- l-H go op O X O Tf O ~ i- X O o o o l~ O o CO 'M -t O X X Q >O Ob O iS 2 -t (M I-" ^""i CO 1- 'M I 1 T ~^' r ^, t 1 1 1 (M X CO CO . T*< O O O O O O O O O OS O O O I-H x o I-H r- x o r 1 CD >~ of O CO N O5 m oo i~ i- 666 01 CO OS CO 6 Tj< co CM OS 1- 6 CC M O X O O O O Tt< i i co i o i i CD O X CO CM CD o X 6 CO x p p p p p O (N Ol X 1> 1 O s^ ; s S :.g III I 1 ~ ~ '^S i i ( 1 1 i o* o" c fcfcfc o '1 . s 3 =0 1 V r. *s "3 1 1 s . . TO * ,fl O a *"^ ste h-i 33 Mechanical efficiency of engines I, II, III with ii Lbs. per minute of water passing through the co Degrees Fahrenheit initial temperature final temperature T.U. per minute token up by T.U. discharged as I.H.-P. per minute T.U. per minute discharged as I.H.-P. and in co Thermal efficiency as given by heat discharged ii Lbs. water discharged from hot-well per minute Degrees Fahrenheit temperature in the hot-well v a T.U. i^er minute discharged from the hot- well... by radiation totcil per minute discharged from engines T.U. necessary to evaporate 1 Ib. of feed to econ< per minute received as dry steam ( = lbs. di received from each Ib. of water condensed a Lbs. per minute of water from the jackets, &c. T.U. per minute received from the water from tl Total T.U. received per minute by the engines Degrees Fahrenheit temperature of feed to the e< Degrees added to temperature of feed in econom Ll. ]>er hour of water discharged from the hot-\ T.U received by water -t >~ --T T CD to 32 1 h SSfg* O M 8 ^ - x co O co r- o >O OC CD i i J~ O r 77 "M ^C> O_ r-T X~ 1~ 71 1 1 Ci 1 i r O O5 X 00 ofcT I i O ~ i~ i-r co x Oi iO ^* CD"I-~ i 1 1 i C '~ 'T CD CO OS O rH i * 01 u 00 00 O O O 0-1 O O o o o o 999 o o o 'T i- O CD CD O CD CD rH 10 CD ^s 9 1 * .--5 s . ^"* ^ . CD goi^ P i- re -35 >o CM o rH 00 r- 1 O i i O rH SO CM X O X i-T-nf 00 1 r-T CO oo r-i n ^f >0 O iO CD I-H t- cTcT i 1 r i O X X rH CD t^- r^l- QO i-TcD"l^ r- >~ CD iO O O 1^ X O rH rH "5 S-i S-, V <0 OS s O s O O O 00 O O O O O o o o -* >T X * ^ Tt< X Tt< Ol O2 CD O O 1- X r-l'S 'Pi ow 3*3* *". o I** 1-5 CS CO 00 IN PI O OS 00 rH CD r- 1 O CO CM O O rH r-Tc? i^ >o o o oc i^T CO 00 r- O5 O5 CD O X O5 l^ of o~ 1 1 CD Oi 10 l~ ZD <* O5 35^Oi_ aces i i i i O CD 1- i 1 rH CM -t l- iO O O rH rH OS X o O O O TH o o o O O O ^ CO X X X CD r)< X CM OS ^ CO O (M CO 1 X s PLi i & o5 <1 r~ co t-~ t- * o co oo o t~ 1-- o cs os^ OS 1,048,700 O5 O X O5 r-ToTcT i i X t- iO L- CD -^ C5 iO O ro"^" i i i i O i i O CS X O CM CO (M t-- O5 i i o CD CD CM * CC X CD * O So o CM (N o w oo 1-1 rH M 1^ I- Tj< CD 1^ 6 fl rH 3 S ' 8 Oi ^ Q oo i-^ i-- CD" (M 00 O ^H i I i i i I CO * 1 1 I 1 ffe cs 00 g CM O O J O O O O O O o o o O (N CM O O O rH CJ O (M CM iT .S 9 '>> ^ (N -^ OH' ^8- CO CC Tf CO I~ O CO 00 TjH rH 00 O i i cc CM 10 os ro oc O CD 5O 1^ X CD i i X CD O CD r-H CM CO CM i CO CM iN ^ 1^ CO O "* >O O rH rH a<3 1*3 h o T5 O iO m O 01 o O *-i fN CD CD ^3 M -* iO CD SO CD CD I- O CD CD 1^ rH CM CO -t O CO 56] ON TRIPLE-EXPANSION ENGINES AND ENGINE-TRIALS. 379 TABLE II. Jackets at Boiler-Pressure Jackets Empty Number of the trial 44 33 56 41 35 40 Thermal units j Calculated from the con- 1 denser per Ibt-j Measured 1,011 1,075 1,014 1,043 1,011 1,042 1,014 1,065 1,009 1,001 990 978 the hot- well... I Differences -64 -29 -31 -51 8 12 TABLE III. RELATIVE AREAS OF DIAGRAMS PER LB. OF STEAM THROUGH THE ENGINES, AND THERMAL EFFICIENCIES OF ENGINES. Number of trial 44 33 56 41 35 40 1 2 3 Theoretical area, ft. & 11 >. .Mc.isured area Percentage of theoretical} area 238,645 188,096 79-0 233,545 192,067 82-0 228,420 192,000 84-0 235,500 127,545 54-0 233,000 139,546 60-0 221,860 144,350 65-0 4 5 6 7 8 !) 10 Theoretical efficiency, p.c. Measured efficiency, p.c. Percentage of theoretical} efficiency 5 23-3 18-5 79-4 23-2 19-2 82-6 22-7 19-4 85-4 23-3 13-8 59-2 23-2 15-3 65-9 22-4 15-5 69-4 TABLE IV. CONDENSATION WITHOUT JACKETS. Number of the Trial Revolutions per Minute Ratio of Expansion Proportion of Total Steam condensed at Cut-off Mid-Stroke Release ( 41 146 27 0-40 0-39 0-30 Engine No. I ... 1 35 40 229 322 23 2-0 0-29 0-22 0-27 0-21 0-22 0-17 ( 41 127 2-4 0-41 0-345 0-29 Engine No. II... ^ S5 40 215 320 2-4 2-2 0-38 0-30 0-34 0-27 0-26 0-14 ( 41 109 2-7 0-51 0-48 0-37 Engine No. lit... I 86 40 184 276 3-05 2-6 0-48 0-32 0-47 0-36 0-33 0-23 57. REPORT OF THE COMMITTEE APPOINTED TO INVESTIGATE THE ACTION OF WAVES AND CURRENTS ON THE BEDS AND FORESHORES OF ESTUARIES BY MEANS OF WORKING MODELS. [From the " British Association Report," 1889.] THE Committee held its first meeting in the Central Institution of the City and Guilds of London Institute. It was then resolved that the Committee should avail itself of the permission of the Council of the Owens College, and conduct its experiments in the Whitworth Engineering Laboratory. At the suggestion of Prof. Reynolds it was arranged that the first experiments should be directed to determine in what respects, and to what extent, the distribution of sand in the beds of model estuaries of similar lateral configuration is affected by the horizontal and vertical dimensions, and the relation which these bear to one another and to the tide period so as to place the laws of similarity on which the practical applications of the method depend, on as firm an experimental basis as possible. It was suggested provisionally that two working tanks should be con- structed, one as large as the circumstances of the laboratory would admit, and one of half the linear dimensions of the larger tank. Prof. Reynolds was empowered to appoint an assistant to make the necessary observations. At a second Committee, held at Owens College, Manchesoer, the models constructed were examined, and it was arranged that Prof. Reynolds should draw up a report on the results so far obtained. 57] ON THE ACTION OF WAVES AND CURRENTS. 381 On Model Estuaries. By Professor Osborne Reynolds, Having carefully considered and sketched out designs for the tanks and appliances in accordance with the resolutions of the Committee on February 6, I obtained the assistance of Mr H. Bamford, B.Sc., from Easter up to the date of the meeting at Newcastle. The working drawings for the appliances were commenced immediately after Easter, and the work put in hand, the experiments being commenced in each tank as soon as it was ready. The General Design of the Appliances. A great object in designing the tanks was to make the most of the facilities in the Whitworth Engineering Laboratory, Owens College, in respect of a continuously running shaft, a supply of town's water and wastes, also a water supply (13,000 gallons) from a storage tank at 116 feet above the floor of the laboratory, and a discharge into a similar tank below the floor, with pumping power to raise the water back when required, also floor space. The available floor space, although very conveniently placed with respect to the water and power, was strictly limited by resisting structures to 10 feet wide and 22 feet long. This admitted of an extreme length for the larger tank of 16 feet, and an extreme width of 4' 8", leaving 2' 6" for the width of the smaller tank, the remainder of the space being the least possible that would admit of access to all parts of the tanks. The internal dimensions of the tanks as designed are : TANK A. Length Breadth Height Fixed rectangular tray having one end open From laboratory Moor of the tray . . ... 11'IOJ" 3' 9" 2' 6" Sides and end above the bottom 9" Tide generator, one end open 3' 10" 3' 9A" Sides at open end 9" At closed end 1'7" TANK B. Half the dimensions of A. The proportions of the tide-generators and fixed pans were determined, so that in tank A the greatest rise of tide over the whole tank should be 2"; which was double the tide used in my previous experiments, and that con- sistently with this the generators should be as short as possible. This tide in tank A required that the generator should displace 10 cubic feet, and as 382 ON THE ACTION OF WAVES AND CURRENTS. [57 the greatest rise and fall that could be conveniently obtained for the end of the generator was 16", giving a mean rise of 8", the area required was 15 square feet. A period of 30 seconds was adopted for tank A as the shortest period likely to be required, and the gearing arranged accordingly. With this period, and a 2" tide, the horizontal scale would be 1 in 20,000 of that of a tank with a 30- foot tide, and a period of 12 hours 20 minutes. So that the 12-foot pan would represent 45 miles. Provision was made for the production of waves with periods ^th the tidal period. Provision was also made for the introduction of land water into the tank at any point that might be required ; also for scumming the water by an adjustable weir, which would serve to keep the level of low water constant, water being supplied into the generator when no land water was required. The drawings (fig. 1, p. 401) show the tanks and apparatus as they have been constructed. The pans and tide generators are of pine-boards fastened with screws. The former rest in a fixed cradle formed by six legs with cross- bearers, bottom ties, and braces. The floor boards of the pan are screwed to the cross-bearers, but are left free to expand, the joints being made with marine glue, after the manner of the decks of ships. The sides are screwed to the floor only ; they receive lateral support against the pressure of the water from the prolongations of the legs upwards. The pans are lined with calico saturated with marine glue, and put down with hot irons, then covered with a coat of paraffin. The pans of the generators are constructed in the same way as the others, only instead of the cross-bearers being attached to legs, they are suspended from two side levers, which are supported on cast-iron knife-edges resting in cast-iron grooves on the top of the legs at the end of the pan. These knife-edges are at the exact level of the top of the floor of the pan, and in line with the joint in the floor between the pan and the generator, so that there is no opening and closing of this joint. This joint is, however, covered with indiarubber, which extends up the sides, and by stretching allows for the opening and closing of these joints. In tank A these side levers extend 4 feet along the sides of the pan, beyond the joint, and to their ends is attached a large box for holding balance weights. These weights are considerably below the knife-edges, and consequently their moment diminishes as the box descends, i.e. as the tide rises, but this diminution by no means compensates the diminution of the water in the generator. If, therefore, sufficient weight were put into the box to balance the generator when the tide is low, it would much overbalance it when the tide is high. To meet this the weights in the box are used mainly to balance :~>7] ON THK ACTION OF WAVES AND CURRENTS. 383 the dead weight of the generator, which requires about 300 Ibs., and a varying balance is arranged for the water. This varying balance consists, in tank A, of a cast-iron cylinder of 500 Ibs. weight, suspended by links from the side levers across and under the tank. The cylinder is also suspended by two links from the frame, and this second suspension is so arranged that when the generator is down the links from the levers are vertical, and when the generator is up they are horizontal. In this way a varying balance is obtained, which as far as possible effects a complete balance in all particulars. In tank B, arrangements which have the same effect have been carried out in a somewhat different manner, which will be clear from the drawings. The glass covering for tank A consists of eight glazed frames, each having two panes of sheet glass 3' 10" x 10", with |" bearing on the frame all round ; the external dimensions of the frames are 4' x 2', so that they are easily handled. The glass is let in flush with the top of the wood, and each pane is fixed by four small brass clips screwed to the frame. In this way, except for the clips, the top of the glass cover over the pan presents a level surface. The frames over the tide generator are connected with those over the pan by a hinge joint, made of two strips of pine hinged to each other and to the frames. A somewhat similar arrangement exists in tank B, except that there are only four frames each with a single pane 2' x 2'. In both tanks the glass frames are fastened by screws to the sides, which screws have to be taken out before the frames can be removed. The gearing, which is arranged to be driven either from a small water- engine or the running shafting, is shown in the drawings. The crank is adjustable so as to give any required tide up to the maximum. In tank A, the pulley driven by the belt from the motor or shaft makes 700 revolutions for one of the crank, and has a fly-wheel which considerably helps the motor over any little irregularities in the balance. The gearing in tank B is driven either direct from the motor or from a pulley on the second shaft in the gearing of A, in which way a fixed relation in speed is obtained when the tanks are working together. The motor was obtained from Alderman Bailey, Albion Works, Salford ; it is a double- acting oscillating water-engine with a " piston and 4" stroke. The available pressure of water is 50 Ibs. steady pressure ; the consumption is about 1 gallon per 100 revolutions. At the highest speed, 2 tides a minute, the motor only makes about 200 revolutions per minute, so that the 13,000 gallons will keep it going over three days, and has done so from Saturday till Tuesday, Monday being Bank Holiday. It has run day and night and 384 ON THE ACTION OF WAVES AND CURRENTS. [57 Sunday, since starting in June, without once stopping, making over 12,000,000 revolutions, and is none the worse. If it used the full pressure it would, when run at 100 revolutions, do about '044 horse-power. Owing to the careful balance of the tanks and the use of spur instead of worm gearing, the work required is not more than '008 horse-power, so that five- sixths of the pressure is spent in overcoming the fluid resistance, which, increasing as the square of the speed, affords a very important means of regulating the speed, which, indeed, is thus rendered very regular. Surveying Appliances. Since the configuration of the sand produced under different circumstances can only be compared by means of records such as charts or sections, the practicability of the investigation depended on the finding of some means by which the sections or contour-lines on the sand could be rapidly and accurately surveyed. The floor of the estuary was made flat and carefully levelled, so that the depth of sand at any point could be at once ascertained by sinking a fine scale through it to the bottom ; and for this purpose scales were constructed of strips of sheet brass - 01' broad and '01" thick. On these the alternate 01' were painted white, and the intermediate spaces in the first 0' 1 were painted red, in the second O''l black, and so on, the scales being then varnished with paraffin. These scales would stand in the sand edgeways to the current, and so be made into permanent sand-gauges, which could be read periodically without removing the glass or stopping the tide. For tank B the scales were half the size of those for tank A. The resistance which a few such thin obstructions offered to the water would be very small, but if the gauges were numerous the resistance would be a serious matter, so that a more general method of taking a final survey was necessary. The ease and simplicity with which the contour-line could be found when the tides were not running, by adjusting the level of the still water and observing its boundary on the sand, reduced the question of making a contour survey to the providing of the means 1. Of adjusting the level of still water to any required height. 2. Of rapidly and accurately determining the horizontal position of points on the edge of the water. The tide-gauge, shown in the drawing on the top of the tank, which would stand on the glass which gave a level surface, answered well to determine the level of the water. For the purpose of surveying the contours a system of horizontal survey- 57] ON THE ACTION OF WAVES AND CURRENTS. 385 lines were set out in the covering frames, consisting of black thread stretched immediately beneath the glass in the frames. The lines are 6" apart ; those parallel with sides are called lines, and those at right angles sections. The first section is 3" from the end of the tank* and the lines are so placed that one of them bisects the tank. These survey-lines divide the entire surface of the fixed tray into equal squares. They are, however, 11" from the bottom and about 8 from the sand ; besides, they are six inches apart, so that to make accurate use of them for surveying the sand it was necessary to use some means of projecting a point vertically up to the level of the glass and scale its distance from a line and a section. This is accomplished by a little instrument, which may be called a projector, shown on the top of tank A. It has a foot which consists of two scales placed at right angles, so that the zero- lines on both, if produced, would meet in a point about half an inch from the edge of the scale. About this point there is a hole through the foot, with cross-wires so placed that they intersect in the point of intersection of the zero-lines. Vertically above this is a horizontal plate with a pin-hole, so that, when placed on the horizontal glass, any point below, seen through the pin-hole on the cross-wires, is vertically below the intersection of the zero-line of the scales ; and hence if these scales are parallel to the lines and sections, the distances of the point from these are read at once on the scales. This method of surveying lends itself readily to plotting on section paper. This may be done directly, the glass cover of the tank serving for a table ; each point may be plotted as it is observed ; and in this way Mr Bamford is now able to survey and plot a complete contour-line in from fifteen to thirty minutes, and requires only about five hours to make a complete survey plotting the charts. One very great desideratum has been a graphic recording tide-gauge. So much depends on the manner of rise and fall of the tide that it does not seem sufficient to know that it is produced by a simple harmonic motion ; the curve should be recorded for each experiment at different parts of the tank. The want of means and time have prevented any attempt to supply such a gauge. Curves have been obtained for most of the experiments by means of the simple tide-gauge. The crank-wheel being divided into sixteen equal arcs, one observer observes the wheel and another the gauge. When a particular number comes to the index the observer at the wheel calls, and the other observer reads the gauge, and then shifts the sliding pointer to the point at which the tide-index was, so that on the next revolution, when the call * This somewhat awkward arrangement was necessary on account of the wood in the frames. o. K. ii. 25 386 ON THE ACTION OF WAVES AND CURRENTS. [57 comes again, he can observe if the pointer coincides exactly with the index or requires adjustment. Having brought about coincidence, he then proceeds to the next number. In this way it takes about half an hour to read the curve. Time, however, is not the only objection, a greater one being that any irregularities in the motion of the wheel do not appear in the curve. The motion of the wheel has been as far as possible checked by the clock, but still there is room for important errors, which a chronograph would obviate. The Selection of Sand. Sir James Douglas having informed me that clean shell sand could be obtained, and having sent me samples which, from the tests to which I subjected them, seemed to be quite as readily moved by the water as the finest Calais sand, I asked him to procure a quantity fifteen bushels of Huna Bay shell sand and in the meantime I procured a similar quantity of Calais sand, so that I might be prepared with whichever showed itself best in actual experiments. Selection of the Experiments. It having been decided that in the first instance the purpose of the experiments should be the comparison of the distributions of sand produced under particular lateral configurations, and with different relations between the vertical and horizontal scales in the same model, and with similar relations in these scales in the two models, the only matters left for selection in starting these experiments were the scales and particular configurations. There was apparently no reason for attempting the very difficult operation of modelling any actual estuary, and, setting this aside, the question of choice mainly turned on whether it was best to begin with complex or simple circumstances. There was considerable temptation to commence with complex, i.e. boldly irregular boundaries, so that the influence of the boundaries might predominate over such other influences as exist ; in which case the influence of the boundaries would be tested by the similarity of the distributions produced with different ratios of horizontal and vertical scales. On the other hand, however, it appeared that as the main object of these researches is to differentiate and examine the various circumstances which influence the distribution of the sand, it was desirable, in starting, to simplify as much as possible all the circumstances directly under control, and so afford an opportunity for other more occult causes to reveal themselves through their effects, and to determine the laws of similarity of these effects. The simplest of all circumstances would be that of no lateral boundaries whatever a straight foreshore of unlimited length with a shelving sandy beach, up and down which the tide runs until it has brought the beach to a state of equilibrium. 57] ON THE ACTION OF WAVES AND CURRENTS. 387 This being an impossibility, the nearest approach to it is that of a beach or estuary with vertical lateral boundaries parallel to the direction- of flow of the tide. And the broader such estuary is in proportion to its length the less would be the effect of the lateral boundaries. The effect of the tide running straight up and down such an estuary might tend to shift the sand up or down according to the slope at each point, and the period and height of the tide, or until some definite relation between these three quantities was attained. If such a relation exists, its elucidation would seem to be fundamental to a full understanding of the regime of estuaries. Further, there was the very important question how far such a tidal action would leave the bed beach-like, with uniform slope and straight contours, or groove it with low-water channels as in the mouths of estuaries, i.e. whether a parallel estuary without land water, having a uniform slope and straight contours, would be stable under the action of a tide of which the general motion was straight up and down. Considering that the new rectangular tanks with their clean paraffined sides were admirably adapted for such experiments, and that any internal modelling would have required further time, which was already very short, if a report was to be presented at the Newcastle meeting, it was decided to commence with a series of experiments on the general slope and configura- tion of the sand with parallel vertical sides, after making some preliminary experiments while the tank A was having a preliminary run to test the working of the motor. Following is an abstract account of these experiments and the results obtained. It has not been thought desirable to introduce into this report a complete copy of the note-book. The initial conditions of each experiment are given, together with the date, the number of tides run, and the mean period of the tide; also notes made during the running on circumstances which are likely to have affected the general results. The final results are contained in the charts (or plans as they are headed), the longitudinal and cross sections, which have been taken from the charts, and the diagram of mean slope obtained from the areas of contours. These are all appended to the report. Preliinlintri/ Experiments with Balls. Little balls of paraffin the size of IK ;is were prepared, colouring matter having been first mixed with the paraffin to distinguish the balls, and to so load some that they would just sink while others floated. Then, before the motor was started, the water being quite still, the balls were placed in rows across the tank at definite distances down the tank, and from the centre line one set of balls on the bottom and another set floating above. The motor was then started, and the change in position of the balls noted. 252 388 ON THE ACTION OF WAVES AND CURRENTS. [57 It was supposed that the floating balls would move with the water and show by any change of their mean position if there was any circulation in the water. This was what they did when the surface of the water was perfectly clean, but the slightest scum very greatly diminished the range of the motion of the floating balls. This matter of scum, if it can be so called, when it is entirely unperceivable by the eye, is very important in these model experiments ; for, however slight it is, it tends to prevent the horizontal motion of the immediate surface, and indirectly to modify the internal motion of the water ; the only test of perfect freedom from surface impurity is that small drops caused by a splash falling on the surface float along. When the surface was in such a state the floating paraffin balls oscillated up and down with the water, and kept the position for many oscillations both up and down and across the channel. The sinking balls are subject to the constant resistance of the bottom, so that their motion is not equal or proportional to the motion of the water for a sufficiently slow motion of the water the ball would not move ; so that if the ebb were just not sufficient to move the ball, and the flood were stronger, the ball would be moved up each tide, or vice versa, and the same resultant motion would follow, even though the ball might be moved some- what by both ebb and flood ; the strongest would carry the ball farthest. In this way they furnished a very delicate test as to the symmetry of the tides and the sufficiency of the balancing. Experiments on the Movement and Equilibrium of Sand in a Tide Way. Series 1. Tide running in a uniform rectangular pan with vertical sides and end, and a level bottom for the sand to rest on. Experiment 1 (tank A), commenced June 22. Three cubic inches of Calais sand was placed in a heap on section 17 and line l r and 3 cubic inches of Huna Bay shell sand similarly on section 17 and line l r , the tank being otherwise clean and empty. Then, with low water O083 of a foot and high water '23' from the bottom, the tide was set running with a period of 55 sees. After 3000 tides, the white sand spread upwards from section 16'5 to 12 7 in 7 ripples, having just painted the bottom 1 grain thick down to section 22. The Huna Bay shell sand spread upwards from section 18 to 14'25 in 4 ripples, also having painted the floor down. Experiment 2 (tank A), commenced June 24. Calais sand was introduced as a uniform bank across the channel. Experiment 3 (tank A). The Calais sand was arranged exactly as for Experiment 2. 57] ON THE ACTION OF WAVES AND CURRENTS. 389 Experiment 4 (tank A). A repetition of Experiment 3, observations being directed more closely to the motion of the sand after starting. Experiment 5 (tank A), July 5 (Plans III. to VI.). Calais sand passed through a sieve into the tank, in which there was sufficient water to cover the sand until there was enough sand to fill the tank from the upper end to section 18, to a depth of 25 foot, terminating in a natural slope from section 18 to the floor. Then the sand, which was in excess at the upper end, was carefully levelled by means of a wooden float guided on the sides of the tank, and having its straight edge completely across the tank 0'25 foot from the bottom. The scummer was then adjusted to keep the low water at 0'181 from the floor, and the crank adjusted to give a rise of 0'166 over the whole tank. The actual rise at starting, owing to the sand above low water, was 0*2' over the whole surface. The tank was then started, and ran 12,607 tides at a period of 53 sees., when the speed was increased to 50 sees, and continued for 3589 tides ; then, as the condition seemed very steady, the survey for Plan I. was made. At the starting of the tank careful note was taken of the progressive appearances, and during the interval of running, which occupied from July 5 to July 15, sand gauges were introduced and read daily, as well as other notes of progress made. The periods of rise and fall of the tide were ckecked, and a curve taken which showed the rise and fall at the generator to be symmetrical and nearly harmonic. The sand was found to descend down the tank towards the generator in a steadily diminishing manner, while at the same time it rose towards the head of the tank at a steadily diminishing rate, until both these changes ceased to be observable. The configuration of the surface also changed at a steadily diminishing rate. The chief features in this configuration were the banks, which gradually formed at the head of the tank in a very sym- metrical form, and then extended down the tank past low-water level, losing their symmetry as the low-water channels between them began to take effect. These banks and channels appear clearly in the plan. The minor features were numerous minor channels caused by the water running off the banks. These covered the surface with very beautiful detail, which, however, it is quite impossible to record. Also ripple bars across the channel below low water. After the first survey was taken the tank was started again July 17 at a somewhat slower period, viz., 6C'7 seconds, and ran for 7815 tides, when the second survey was made. The daily observation taken as before showed considerable changes of detail so much so that it was a matter of surprise to find that Plan II. corresponded almost exactly with Plan I., the only 390 ON THE ACTION OF WAVES AND CURRENTS. [57 difference being slight divergences and deepenings of the depressions and raising of the banks. The tank was again started on July 25 at a period of G0'6, to keep the sand in motion until the agitator for producing waves could be put in action. This was accomplished after 7780 more tides, from which no considerable change was observed. The agitator made 200 beats in the tide generator for every tide. At first the agitating bar was straight, 3' 6" long, with a section 6" broad and 1|-" deep. This caused parallel waves '06' high, '8 long. The effect of the waves was at once apparent in the destruction of all the beautiful tracery on the banks, which soon presented a smooth washed-out appearance. After 4000 tides a V-shaped agitator, as shown in the drawing, was substituted for the first. This sent oblique waves of much the same size as before. The waves were kept going during the day and stopped at night ; and after 6000 further tides the tank was stopped to survey for Plan III. This shows that at low water the waves had to some extent levelled the sand ; they had also washed off the ridges of the banks and filled the narrower channels ; yet on the whole the depressions are deeper, and at the head of the estuary there is a marked change in the arrangement of the left side. The model was (August 8) set to run at 33" with the wave agitator going during the day. On the 19th it was found that so much additional sand had come down into the generator as to disturb the balance so as to require 100 Ibs. additional weight to equalise the period, this was added, and again on Monday the 1 2th it was found that more sand had come down, requiring 50 Ibs. more weight to re-establish the balance. On the 13th the sand was removed from the generator and the balance restored by removing the 150 Ibs. It had also been found that from some cause the speed fell otf considerably ; at one time the speed was 70". The cause of this was not at first perceived, but somehow the speed was restored, though it was subsequently found that the belt on the motor was slipping ; this having been put right, the running at 33" was continued till 12,705 tides had been run since this speed was commenced. Survey IV. was then taken. It was found that the mean period over the interval had been 43'2" instead of 33". It being uncertain how far the irregular speed and disturbed balance had affected the results, the model was started again August 16 to run at 33" with a mean speed of 33", and after it had run 17,289 additional tides a partial survey was again taken and plotted in dotted lines on the plan showing Survey IV. The dotted contours show a slight change, chiefly in the retreat of the low-water contour up the estuary, and a change in the distribution of the sand at the head of the estuary. These changes, however, are very slight 57] ON THE ACTION OF WAVES AND CURRENTS. 391 compared with the great difference presented between Plan IV. and all the previous plans. In Plan IV. the contour '004 lies between" sections 6 and 11, while in Plan III. it lies between 12 and 13, which shows an increase from 1 to T47 in the general slope. The low-tide channel on the left has also increased in magnitude and length, extending nearly across the head of the estuary. Experiment 6 (tank A), August 28. In this experiment the initial conditions aimed at were precisely the same as in Experiment 5. The sand which had been removed was returned, and all the sand well washed in the tank and then placed as before, the float having been examined and straightened on the surface plate. After the sand was laid the water in the tank was brought as near as possible to the level of the sand, and was allowed to stand for twelve hours, when it was found that the sand looked drier on the right side than on the left. The generator was then raised by turning the pinion which, in the position the crank was, would raise the generator about '008 of an inch for one revolution ; this, considering the surface of water exposed, would raise the water '005 inch, in which way it was found that the sand was something like '01 of an inch higher on the right all along the tank. The model was then started at the same speed as in Experiment 5, and the development carefully watched. In all respects it appeared to be the same as in the previous experiment, and the daily observations showed the same rate of progress; not only did the sand gauges and the descent of the sand agree, but the surface of the sand presented the same general appearance. The experiment was stopped after 8686 tides, before it had reached the stage of the first survey, Experiment 5, so no survey was taken. Experiment 1 (tank B), August 5. In starting this experiment it was intended that the circumstances should be in every respect homologous to those of Experiment 5 (tank A). The sand was introduced in the same way, and brought to the same figure. The tank was started with a period of 36'5 seconds, that of A having been 53 seconds, which numbers are as the square roots of the dimensions of the tanks. The progressive appearances accorded identically with those noted in tank A, except in one apparently minor particular. And for the first 1200 tides the downward progress of the sand was nearly the same (a trifle less) About this stage an appearance presented itself which had not been noticed in the previous experiment. The arrangement of the sand appeared 392 ON THE ACTION OF WAVES AND CURRENTS. [57 to show a greater rate of downward progression, at the middle of the tank, towards the generator than at the sides, and this was followed by a somewhat more rapid descent of the lower edge of the sand, which after 5000 tides began to accumulate in the generator, from which about seven pounds was removed. From this stage the lower end of the tank B differed considerably from that of tank A in the same stage. At the upper end the appearances were almost identical, and the reading of the sand gauges agreed well. As the experiment progressed, the sand, instead of having a nearly uniform downward slope from the head to the generator, had a uniform slope down the middle of the tank, with two large banks extending from section 8 to section 17 on each side, that on the right being longer. The experiment was continued for 11,013 tides, when it was found that the water was much too low, owing to misadjustment of the scummer; then as there was no possibility of saying how long this had been going on, the experiment was stopped. Experiment 2 (tank B, Fig. 7, p. 407), August 28. Plan 1. In this the conditions of Experiment 1 were repeated, the edge of the float having been replaned. The results from starting were almost identical with those observed in Experiment 1. The sand again came down fastest in the middle, and faster than in tank A. Seven pounds were removed from the generator, and subsequently the condition of the model as regards the lateral banks was nearly the same, except that the longer bank was on the left. The experiment was continued with speeds exactly corresponding to those of Experiment 5, tank A, until 16,344 tides had been run; then Plan I. was taken. The tank was then set running again at 35'5 seconds and continued for 6757 tides, when considerable changes had taken place towards the lower end of the tank. A partial survey was then made and recorded, and the experiment stopped. Experiment 3 (tank A, Fig. 8, p. 408, and tank B, Fig. 9, p. 409), Sept. 2. The sand in both tanks was arranged as before, a new float straightened to a surface plate being used for B, and the level of the sand in both tanks tested by water, as in Experiment 6 A, which tests showed that the sand in A was perhaps -01" highest on the left, while in B it was to something like the same extent highest on the right. The tanks were coupled, A being driven from the motor and B from A. Both were set to low tide at starting, and the start made at full speed, 33 seconds tank A. The progressive appearances simultaneously observed were identical, with the same exception as before noted. Immediately after starting, the periods of rise and fall of the generator of A were observed, and the fall being slightly the larger, 25 Ibs. was removed from the balance 57] ON THE ACTION OF WAVES AND CURRENTS. 393 weight, which restored the equality. After 77 tides it was observed that the sand in A was coming down much faster than in B, and had already begun to come into the generator; the periods of rise and fall were noted, and it was found that the rise was 17 seconds and the fall 15 seconds. The weight was replaced, the tanks stopped, and 56 Ibs. of sand removed from the generator and lower end of the trough of A which left the end of the sand the same in both tanks. The tanks were then started, and the rise and fall in A were equal. It may be well to remark that though the tank B is driven from A, the periods do not synchronise, so that the unequal motion caused by imperfect balance of A eventually affects all stages of the tide in B equally, while the resistance of B is so small compared with that of A, that any want of balance hardly affects the motor when driving both tanks. In starting there would be the same disturbance of balance in both tanks owing to the slow descent of the water, from the flat sand, but it would be only that of A that would affect the balance. After running 1653 tides, tank A, it was seen that the sand had come into the generators of both tanks, so a stop was made, and all sand below section 20 again removed from both tanks 120 Ibs. from A and 12 Ibs. from B, making altogether 176 Ibs. from A against 12 Ibs. from B. Considering that 1 Ib. in B is equivalent to 8 Ibs. in A, and that altogether in A there would be 1100 Ibs., B was left with about 7 per cent, more sand, in proportion, than A. The experiment was then continued, the sand coming down in both tanks, but not so as to get into the generators. The motion of the sand in the two tanks followed almost exactly the same course, B gradually taking the lead. In this case there was not the least sign of the middle channel in B, the sand keeping level across and following the same course as had previously been observed in Experiments 5 and 6 A. When B had run 16,570 tides it was stopped for surveying, while A was allowed to run on to make up the number of tides. Surveys were then made. DISCUSSION OF THE RESULTS. OBTAINED. Since the experiments have been arranged in accordance with the law of kinetic similarity, followed in rny previous experiments, it may be well to restate this law before proceeding to discuss the results. If h be the depth of water in a uniform trough, it is well known that the velocity of a wave, of which the length L compared with h is great, 394 ON THE ACTION OF WAVES AND CURRENTS. [57 and of which the height is proportional to h, varies as the square root of h. For geometrical similarity at any instant the lengths of the troughs must be proportional to L. The period of rise and fall, p, will thus be inversely proportional to Hence for the law of kinetic similarity, P^ ....................................... (1), has a constant value for all scales. This law takes no account of the resistance of the bed, for a first approxi- mation to which the law would be (2), constant, where A and B are constants to be determined by experiments. Since the comparative periods of the two tanks have been made propor- tional to the square roots of their dimensions, e.g. the period of tank A, \/2 times the period of tank B, the bottom resistances produce dynamically similar results. In comparing the results obtained with the same values of k in the same tank with different periods, the bottom resistances would be different, and this difference should appear in the results unless too small to be appreciable, in which case the results would compare with the simple period. There are four other sources of possible divergence from the simple dynamic law, which will become larger as the periods become slower and the tide lower : 1. The drainage of the sand after the tide has left it supplies the low- water channels with a constant stream at low water; the velocity of this stream will depend on the slope and quantity supplied, and supposing the quantity to be proportional to hL", the depth of the water in the low- water channels (not the depth of the channels) will be proportional to the cube root of the slope ; 2. The size of the grains of sand, which require a certain velocity before they move ; 57] ON THE ACTION OF WAVES AND CURRENTS. 395 3. The fouling of the sand by growth, &c., which increases as the shifting of the sand diminishes ; and 4. The viscosity of the water, which causes a definite change in the internal motion of the water when the velocity falls below a point which is inversely proportional to the dimensions of the channel. The effect of 1 would be confined to the channels ; 2 and 3 would tend to diminish the rate of action ; the 4th might seriously alter the rate of action at different parts of the estuary, and would also affect the appearance of the sand surface. The ground so far covered by the experiments has been confined to one initial arrangement and to one height of tide in each tank, these being similar. Two periods have been tried in each tank, the relation between the periods in the different tanks being as the square roots of the dimensions. Six experiments have been started: 2 in tank A with a period of 53 sees. 1 ?. 33 ,, 2 in tank B 36'5 1 M 23-3 Of the two experiments started at 53 seconds in tank A the first was continued for 12,097 tides, and then for 3589 tides at a period of 50 sees., und a survey made (Fig. 3, p. 403). It was then continued 7815 tides at 65'1 sees., and the plan marked, Fig. 4, p. 404; it was then continued 17,750 tides with intermittent waves at a period of 6(V6 sees., and a survey made (Fig. 5, p. 405). It was then continued for 12,705 tides at periods varying from 33 sees., having a mean 43, and Plan 4 (Fig. 6, p. 406) made, then continued at a period of 33'3 sees, with intermittent waves, when it was re-surveyed (dotted on Plan 4). The second experiment at 53 sees., tank A, was continued to 8700 tides with the same results as the first. Of the two experiments in tank B the first was continued to 11,013 tides as in A, then stopped. The second was continued to 12058 tides at 36'8secs. ; then at 4280 tides at 36 sees., and surveyed (Plan 1 B, Fig. 7, p. 407) ; then continued to 6769 more tides at 36, and again surveyed. The experiments started at 33 sees., tank A, and 23'3 sees., tank B, were continued to 16',603 tides and then surveyed (Figs. 8 and 9, pp. 408, 409). In all these six experiments the manner in which the water commenced and proceeded to redistribute the sand was essentially the same, the general appearances of the surface being, with the exception of one or two particulars, ON THE ACTION OF WAVES AND CURRENTS. [57 the same at the same number of tides up to 1200. After this the two low- speed experiments in B began to present more noticeable differences from the other experiments, which continued to present similar appearances at corre- sponding tides to the end. It thus appeared : 1. That the rate of action was proportional to the number of tides ; 2. That the first result of the tide-way was to arrange the sand in a continuous slope, gradually diminishing from high water to a depth about equal to the tide below low water ; 3. That the second action was to groove this beach into banks and low- water channels, which attained certain general proportions (plans 5 and 7 A and 2 B, and cross sections, Figs. 5, 8 and 7); 4. That the slope arrived at after 16,000 tides was the same at the high speed in both models working at corresponding periods, \/2 to 1 (sections, Figs. 8 and 9) ; 5. That in both models the steepness of the actual slope increased as the tidal period diminished (sections, Figs. 5, 8, 7 and 9). Owing to the grooving of the surface, the exact slopes at the various speeds cannot be exactly compared. One way of effecting a comparison has been to take the highest points on each cross section down the slope, and plot them as a longitudinal section, and in the same way to take the lowest points and plot them as another. These are shown in the two longitudinal sections which accompany each plan. The increase of the slope with the diminution of the tidal period, both as regards the banks and channels, is thus rendered apparent ; but these sections do not admit of an accurate comparison. Some definite and accurate method of comparing these slopes was essential before any definite conclusions could be arrived at respecting the laws of similarity. To meet this the areas above the successive contours have been taken out. These areas respectively divided by the breadth of the plan give the mean distance of the respective contours from the head of the estuary, and the heights of these contours plotted to this mean distance give a definite mean slope of the sand. There are certain minor objections to this method, but it is eminently definite and practical, and admits of great accuracy, the areas being readily taken out with a planimeter with very great accuracy even for the most complicated contours. The slopes thus taken out are more readily compared if plotted to scales such that the vertical distances between high and low water are all equal, the horizontal scales being determined so that the vertical exaggeration is the same in all cases. 57] ON THE ACTION OF WAVES AND CURRENTS. 397 The slopes thus taken out from 5 A (Figs. 3 and 6), 7 A (Fig. 8), and from 3 B (Fig. 9) are shown in (1) Fig. 2. They present a great degree of regularity ; and it is seen at once that the result of corresponding periods (33 sees, tank A, and 23 sees, tank B, Figs. 8 and 9) agree very closely. In order to compare the slopes with the conditions of kinetic similarity, all that is necessary is to reduce the horizontal distances in the inverse ratio of the periods, when the slopes should become identical. In doing this the horizontal distances have all been reduced to represent (according to the kinetic law) a 30-feet tide with the natural period 44,400 seconds, namely, the ratio of the lengths of the estuaries made equal to the ratio of the periods multiplied by the square root of the ratio of the heights. The actual rise and fall of the tide in the models being taken : The horizontal and vertical scales for the five experiments as thus reduced to a 30-feet tide are given in Table I. TABLE I. Reference Period in Seconds Horizontal Scale Inches to Mile Vertical Scale Rise of Tide V. A, Plan 1 ...(3) 50 f -0000862 ) U in 1 1,600 ; 5-45 f -00587 \ 1 1 in 170 / 176 V. A, Plan 4 ...(6) 33-3 f -000055 } tl in 18,200) 3-49 f -00533 | 1 1 in 187 ) 161 VII. A, Plan 1 (8) 33-6 ( -000056 | (1 in 17,900)" 3-55 f -0055 ) ( 1 in 182 / 165 II. B, Plan 1 ...(7) 35-4 j -0000431 | |1 in 23,200)" 2-72 f -00293 \ 1 1 in 341 j 088 III. B, Plan l...(9) 237 ( -0000299 ) |l in 33,400)" 1-895 ( -00313 1 1 1 in 317 I 094 Table II. shows the measured height from low water for each of the contours, together with its mean distance from the contour at the height which, reduced, is 30 feet above low water. Also the corresponding heights of the contours of the 30-feet natural tide, and the corresponding mean distances of the contours measured in miles, from which the curve of reduced mean slopes shown in (2) Fig. 2, have been plotted. Considering the character of the investigation, the agreement between the slopes is quite as close as could be expected, and there is nothing to argue from the divergences, except that the effect of the bottom resistances has here been too small to affect the results. TABLE II. TANK A GO d Experiment V Experiment VII o fi to Plan 1 Plan 4 Plan 1 ^ SBt CO 3 S ^ ix S 9> S fe -Ik* CO ,-, s 3 > 00 . 3t^ 0) -g 6D fl o . SJ* a 56 Mays ffUTf jnvfuqy yip ito svjittry 58. SECOND REPORT OF THE COMMITTEE APPOINTED TO IN- VESTIGATE THE ACTION OF WAVES AND CURRENTS ON THE BEDS AND FORESHORES OF ESTUARIES BY MEANS OF WORKING MODELS. [From the " British Association Report," 1890.] THE Committee held a meeting in the City and Guilds of London Institute and considered the results obtained since the last report and the proposals of Professor Reynolds for the continuation of the investigation, which were approved. At a second meeting, held at the Owens College, Manchester, it was arranged that Professor Reynolds should draw up a report on the results obtained. At a third meeting, held in the committee room, Section G, at Leeds, the report submitted by Professor Reynolds was considered and adopted. On Model Estuaries. By Professor Osborne Reynolds, F.R.S., M. Inst. C.E. I. INTRODUCTION. 1. In accordance with the suggestion in the report read at the Newcastle- upon-Tyne meeting of the British Association, 1889, the investigation has been continued with a view (1) to complete the first series of experiments by determining the smallest vertical exaggeration at which similar results can be obtained with tides ranging upwards from half an inch in rectangular estuaries, and so determine the law of the limits ; (2) to determine how far similar effects can be obtained with land water acting on such slopes as had 58] ON THE ACTION OF WAVES AND CURRENTS. 411 been already obtained in rectangular estuaries ; and (3) to investigate the character and similarity of the results which may be obtained with Y-shaped estuaries. 2. The two models, subject to such modifications as were required for the various experiments, have been continuously occupied in this investigation, running, driven by the water motor, at all times when they were not stopped for surveying or arranging a fresh experiment. They have thus run about five-sixths of the time day and night. In this way the large model has worked through in the twelve months 500,000 tides, corresponding to 700 years. These tides have been distributed over ten experiments in numbers from 32,000 to 100,000. The smaller model has run more tides than the larger, and these have been distributed over fourteen experiments. 3. The experiments have all been conducted on the same system as is described in last year's report (see p. 380). Initially, with two exceptions, the sand has been laid with its surface as nearly as possible horizontal at the level of half-tide, extending from the head of the estuary to Section 18, and in the later experiments to Section 17. The vertical sand gauges, distributed along the middle line of the estuary, . have been read and recorded once a day. Contour surveys have been made after the first 16,000 tides, and again after the first 32,000, and in the longer experiment further surveys have been made ; in all, fifty complete surveys have been made, and forty-four plans, showing contours at vertical intervals corresponding to 6 feet on a 30-foot tide, are given in his report. The general conditions of each experiment, together with the general results obtained, are given in Table I., p. 436, and a description of each experiment is given in IV. p. 423. The importance of a better means of recording the tide curves was mentioned in last year's report. Such means have been (see p. 422) obtained during this year, and automatic tide curves have been taken as nearly as practical at corresponding numbers of tides during the experiments, these curves being taken at several definite sections in each tank. Two series of these curves have been taken in the later experiments, one in which the paper is moved by a clock, the pencil being moved by a float; the other in which the paper is moved by the tide generator, by which means exactly similar motion for the paper is secured at all points of the estuary, so that differences in the phases of the tide at different parts of the estuary are brought out. These curves are shown on the plates. Mr H. Bamford has continued to conduct the experiments, but on account of the very great amount of detailed work the entire time of a second 412 ON THE ACTION OF WAVES AND CURRENTS. [58 assistant has been occupied. For this the services of Mr J. Heathcott, B.Sc., were obtained from October to February, when Mr Heathcott obtained an appointment in the office of the engineer to the L. & N.W.R. in Manchester. Mr Greenshields then applied for and obtained the post, and has continued the work with great patience and zeal. II. GENERAL RESULTS AND CONCLUSIONS. 4. The Limits to Similarity in Rectangular Estuaries. In the experi- ments of last year it was found (1) that as regards 1. Rate of action as measured by the number of tides run ; 2. Manner of action ; and 3. The final condition of equilibrium with tides of 0176 foot and periods of 50 and 35 seconds the results were f) V h similar, according to the hydrokinetic law -, constant ; (2) that, as regards rate and manner of action, the results obtained with tides of 0'094 foot and periods 23'7 seconds were similar to those with the tide of 0176; but the experiment had not proceeded to the final condition of equilibrium. It was also found that with tides of '088 foot and periods 35*4 seconds, the results obtained differed in a marked manner from the others as regards rate and manner of action, so much so as to render the attainment of a final state of equilibrium impracticable. These results seemed to indicate that for each rise of tide there exists some critical period such that for all smaller periods the results would be similar according to the simple hydrokinetic law, while for larger periods the results would be dissimilar in a greater or less degree to those obtained with periods smaller than the critical period. Whether or not the results obtained with periods greater than the critical periods would present a general similarity amongst themselves, or even similarity under particular relations among the conditions, were still open questions. The experiments, as shown in Table I., Table II., made this year, empha- tically confirm the conclusions (1) as to the existence for each rise of tide of a critical period at which the rate and manner of action begin to change, being similar for all smaller periods ; (2) these experiments also confirm the general similarity of the final states of equilibrium as regards slopes for periods smaller than the critical period, as shown in Table II. The experiments (Experiments IV. and VIII., B) this year, also show that with tides of 0'094 and 0'097 foot the periods 34'4 and 35'4 seconds are 58] ON THE ACTION OF WAVES AND CURRENTS. 413 greater than the critical periods, although the results show a nearer approach to similarity, as regards manner and rate of action, than the results -obtained last year in II. B, with the tide 088 foot and period 35 4 seconds, while the final conditions of similarity were approximately reached. With tides of 0'088 foot and periods 69'3 seconds the results in rate and manner of action are emphatically different from those with less than the critical period, and with tides of 0'042 foot and periods 5O5 seconds still greater differences are presented. On the other hand, it is found (V. B) with tides 0'042 foot and periods 50'5 seconds that if the sand be given a condition corresponding with the condition of final equilibrium, as if the period were above the critical period according to the simple hydrokinetic law, this is a state "of equilibrium ; and, further, that it is not a state of indifference is shown, since on diminishing the period the sand readily shifted so as to bring it nearer the theoretical slope for the new period. This shows that the state of equilibrium follows the simple hydrokinetic law for periods greater as well as less than the critical period, which is thus shown to be critical only as regards rate and manner of action in reducing the sand from the initial level state to the final condition. The experiments carefully considered suggest that there is some relation between the rise of tide and critical period. They do not, however, cover sufficient range to indicate what this relation is with any exactness. The critical period diminishes with the rise of tide, but much faster than the simple ratio. 5. Causes of the Change in Manner and Rate of Action. The change in the action, which sets in at the critical period, is the result of some action, of which no account is taken in the simple hydrokinetic law. A list of five such sources of possible divergence from the hydrokinetic law is included in last year's report (p. 39G), and with a view to obtain an indication of some relation between the rise of tide and period (or vertical exaggeration, as compared with the standard tide of 30 feet, by the kinetic law), which relation would be a criterion of the limiting conditions under which the simple kinetic law may be taken as approximately accurate, these five discarded actions were carefully considered. The fouling of the sand by the water, although it comes in as preventing further action, cannot take any part in imposing these limits, since it is at the immediate starting of the experiments that the action is observed to fail. For the same reason the limits cannot be in any way due to the drainage from the banks, as these banks have not appeared above water. Again the limit cannot be due to the size of the grains of sand because it would then occur at particular velocities, whereas this is not the case. 414 ON THE ACTION OF WAVES AND CURRENTS. [58 The other actions are the bottom resistances and the viscosity of the water, which causes a definite* change in the internal motion of the water as the velocity falls below a point which is inversely proportional to the dimensions of the channel. That this last source of divergence from the simple kinetic law must make itself felt at some stage appeared to be certain. But the critical velocity at which the motion of the water changes from the 'sinuous' or eddying to the direct is inversely proportional to the depth, and by the kinetic law the homologous velocities in these experiments are proportional to the square roots of the depths only ; hence this action would seem to place a limit, if it were a limit, to the least tide at which the kinetic law would hold independently of the period, and this is not the case. Observa- tion of the action of the water above and below the critical periods, however, confirmed the view that the limit was in some way determined by this critical condition of the water. For when water is running in an open channel above the critical velocity, the eddies, of which it is full, create distortions in the evenness of the surface which distort the reflections, creating what is called swirl in the appearance of the surface. Now it was noticed and confirmed by careful observation, that in the cases where similarity failed, the swirl was absent at the commencement of the experi- ment, while it was easily apparent, particularly on the ebb, in the other experiments. Subsequently it appeared that the velocity of the water, particularly during the latter part of the ebb, which has great effect in the early stages, might be much affected by the bottom resistances, and hence not follow exactly the kinetic law. 6. Theoretical Criterion of Similar Action. The velocities of the water running uniformly in an open channel, i being the slope of the surface and m the hydraulic mean depth, is given by v = A \/im, where A is constant. If, then, i is proportional to e (the exaggeration of scale) and m propor- tional to h, since at the critical velocity v is inversely proportional to h, at this velocity h*e has a constant value. The function h 3 e=C is thus a criterion of the conditions under which similarity in the rate and manner of action of the water on the sand ceases. 7. The Critical Values of the Criterion for Rectangular Tanks. Taking * Reynolds on the Two Manners of Motion of Water, Phil. Trans., 1883, pt. iii. (see page 51). 58] ON THE ACTION OF WAVES AND CURRENTS. 415 h to represent the rise of tide in feet, and e to be the vertical exaggeration as compared with a 30-foot natural tide by the simple hydrokinetic law, the values of this criterion have been calculated for each of the experiments and are given in Table I. Experiments I. and II., B, First Report, C'=0'046, showed marked slug- gishness and local action; IV., B, (7 = 0'058 and VIII., B, (7 = 0-064 showed less, but still a certain amount of sluggishness and local action*, while in III., B, (7=0-083, the rate of action was good and the action similar to the experiments with values for C higher than 0'087*, whence it would seem that the critical value of the criterion is about 0*087, and it may provisionally be assumed that (7=0-09 indicated the limits of the conditions of similar action *. 8. The value of the Criterion for V-shaped Estuaries. This critical value of C deduced from the experiments in rectangular tanks appears to correspond very well with the results of the experiments in the V-shaped estuaries. In the experiments Table I. with V-shaped estuaries in the small tank, the value of (7 is in no case far from the critical value '09 on either side. In Experiment IX., B, however, the value of C at starting was only 0'046 as in I., B, and in consequence of the observed sluggishness and local character of the action in the lower estuary, the rise of tide was increased from 0'088 to O'll, which remedied the action and raised the criterion to O'lOl, and in Experiments X. and XII., B, and in I., D, the values are between 0'095 and 0'084. In Experiments II., D, F, and F', owing to the falling off in the tide in consequence of the addition of the river, the criterion is as low as 073. In these experiments signs of slug- gishness and local action in the lower estuary were observed at starting, and the difference in the action of the upper estuary as compared with Tank E in respect of closing up the tidal river may have been due to the low value of the criterion. In the experiments in the large tanks the values of (7 are all well above the critical value: the nearest are the experiments in Tank E, (7=0'17, which is only double the critical value, and the action was as quick and general as in the case where C = 0'5. It may be noticed that the range through which the value of C as a criterion has been tested is small. Had the form of criterion been appre- hended sooner this might have been somewhat extended, though considerable adaptation of the apparatus would be required to carry it far. * In both these experiments, IV. and VIII., B, the mean level of the tide was above the initial level of the sand, which would naturally increase the value of the criterion. 416 ON THE ACTION OF WAVES AND CURRENTS. [58 9. If C =0-08 With a tide 0-1 ft. the greatest period is 32 sees. and least exaggeration 80. 0-12 ft. 60 sees. 47. 0-14 ft. 102 sees. 30. 0-2 ft. ., 6 mins. 9 sees. 10. 0-43 ft. 1 h. 33 m. 48 s. 1. From which the size of tanks and length of periods necessary to verify this law for exaggerations of less than thirty can be seen. 10. The General Distribution of Sand in V-shaped Estuaries. The experiments all show that with sufficiently high values of the criterion, as in the rectangular tanks so in those of symmetrical V-shape, the sand arrives at a definite general state of equilibrium after a definite number of tides. This state in the rectangular tanks was a general slope which corresponded to a definite curve, twelve miles long as reduced by the kinetic law to a 30-foot tide, between the contours at high and low water in the generator. This slope was furrowed by 3 or 4 shallow channels at distances of some two miles, commencing very gradually at the top and dying out at some distance below low water. In the V-shaped estuaries the state of equilibrium differs from that in the rectangular tanks in a very systematic manner; it consists in a main low- water channel commencing at the end and extending all the way down the V out into the parallel portion of the tank. If this channel is in the middle it is the only channel, but if, as is as often as not the case, it takes one side of the estuary, then at the lower end there is on the other side a second channel starting at some distance down the estuary. The height of the banks above the bottom of the main low-water channel towards the lower end of the V is much greater than in the rectangular estuaries. No general method of comparing the general slope or distribution of the sand in the V-shaped estuaries has been suggested other than that of comparing the contoured plans and the longitudinal section taken down the highest banks and lowest channels, together with the cross sections which have been plotted on the plans. These are very similar for the similar tanks and corresponding periods. They show that the slope in the channels down to low water is nearly the same as in the rectangular tanks, the level of low water being reached at distances from the head of the estuary a little greater than in the rect- angular tank, and a little greater in the long V than in the short. Below low water the slope in the channels is less than in the rectangular estuaries, which is, doubtless, a consequence of lateral spreading. The slope of the banks is much less than in the rectangular tanks, and these extend from two to three times as far from the top of the estuary, according to the angle of the V. The range of observations on V-shaped estuaries has necessarily been 5V THE ACTION OF WAVES AND CURRENTS. 419 in the model, the model being then nearly ready, but the experiments had not commenced. M. Mengin paid several visits to the laboratory and carefully examined the apparatus and experiments, for which all facilities were placed at his disposal. 15. Recommendations fur further Experiments. Although the immediate objects proposed for investigation this year have been fairly accomplished, there remain several general points on which further information is very important, besides the further verification of the criterion of similarity and the determination of the final conditions of equilibrium with tidal rivers, already mentioned. It seems very desirable to determine the effect of tides in the generators diverging from the simple harmonic tides so far used, simple harmonic tides being the exception at the mouths of actual estuaries. It would also be desirable before concluding these experiments that they should include the comparative effects of tides varying from spring to neap. III. MODIFICATIONS OF THE APPARATUS. 16. General Working of the Apparatus. The apparatus has worked perfectly in all respects except that of the driving cord connecting the water motor with the gearing. For this cord hemp was first used, as it was liable to be wet. This hemp cord wore out with inconvenient rapidity. A continuous cord made of soft indiarubber was then tried, and, after several attempts, has been made to answer well. The only other failure was the small pinion, which was fairly worn out, and had to be replaced. 17. Extensions. For carrying out the experiments on the V-shaped estuaries the original tanks had to be increased in length. To do this it was necessary to remove temporarily part of the glass partition dividing the engine room of the laboratory, in which the tanks are placed, from the testing room. This being done, the tanks were then extended, as shown (Fig. 1, page 439), the first extension being an addition of a trough 6 feet long and 2 feet wide to Tank A, and a similar extension of half the size to Tank B, the new tanks being thence called C and D. 18. Extensions for Tidal Rivers. The second extension consisted of a trough 19 feet long and a foot wide to the end of C, the new tank being thence called E. The corresponding extension to D was not at first made in the same way, because to do so would require the removal not only of a panel of the glass partition, but also of a fixed bench, which was a much more serious matter, or else the extension would have closed up an important passage. The extension was therefore made, as shown in Figs. 46 and 47, 272 420 ON THE ACTION OF WAVES AND CURRENTS. [58 page 479, which admitted of the tidal river being the corresponding length to that in E, but required a bend of 180, which was effected by two sharp corners. This tank was thence called F'. This was the best that could be done during the time the students were in the laboratory. It was not certain that the corners would produce any sensible effect, whereas if the results obtained in F' were not similar to those in E no time would have been lost, since the straight extension could not be made till the end of June. As the results in F' were not similar to those in E in a way which might be explained by the bends, as soon as possible the straight extension was made similar to E, and the tank called F. All these tanks were constructed in the same manner as the original tanks, and covered with glass at the same level as A and B, under which glass survey lines, conforming to those on A and B, were set out. 19. The Numbering of the Cross Section. The extension of the tanks raised the question as to how the new cross sections should be numbered : the numbering of A and B ran from the ends of the tanks, and it seemed best to run the numbers in C and D from the ends of these tanks, con- tinuing this new numbering to the generators. On the other hand, as the long, narrow extensions in E and F were more in the nature of a tidal river than an estuary, the numbers in these were carried backwards 1, &c., from the ends of C and D, in which the cross sections preserved the same numbers as before. 20. Appliances for Land Water. The introduction of land water, besides the extension of the pipes for its introduction, required certain arrangements for its regular supply in definite quantities. The water was to be taken from the town's mains. And in first laying down the pipes, it had been anticipated that it would be sufficient to regulate the supply by cocks against the pressure in the mains. Fresh water, regulated in this way, had been from the first supplied in small quantities into the generators, to ensure the level being kept properly. The experience thus gained showed that it was impossible to obtain even approximate regularity in this way, as the nearly closed cocks always got choked even within twenty-four hours. To meet this it was arranged to supply the water through thin-lipped circular orifices under a small but constant head of water, which head can be regulated to the quantity required. The head of water in the tank from which the orifices discharge is regulated by a ball cock, which only differs from an ordinary ball cock in that the ball is not fastened directly on to the arm of the cock, but is suspended from it by a rod so arranged that the distance of the ball below the arm can be adjusted at pleasure. This arrangement has answered well. The cylinder in which the ball cock works 58] ON THE ACTION OF WAVES AND CURRENTS. 421 is made of sheet copper, with a water gauge in the form of a vertical glass tube, with a scale behind to show the height of water above the -orifices, which are made in the bottoms of two lateral projections from the sides of the cylinder. One of these orifices feeds the large, and the other the small tank. The streams from the orifices descend freely in the air for about 4 inches, and are then caught in funnels on the tops of lead pipes leading to the respective tanks. The cylinder is fixed against a wall about 8 feet above the floor, and conveniently near the tanks. Any obstruction in the pipes conveying the water to the tanks would be at once shown by the overflow of the funnel. The orifices are made with areas in proportion to the quantities to be supplied to their respective tanks. Then the supply cock connecting the ball cock with the main is fully opened, and the ball is adjusted till the quantity supplied to one of the tanks is correct. The other is then measured ; if this is not found correct, one of the holes is slightly enlarged until the proportions are correct. This having once been done for an experiment, no further regulation is required except to test the quantities and wipe the edges of the orifice. When the tanks are stopped for surveying, the water is shut off from the main and simply turned on again on restarting. 21. The Tide Gauges. In the experiments made last year a tide gauge was used. This gauge consisted of a small tin saucer with a central depression in its bottom, in which a vertical wire rested, restraining any lateral motion in the float, the wire being guided vertically by a frame made to stand on the level surface of the class covers, while the wire passed down between two of the covers opened for the purpose, the frame carrying a vertical scale. This gauge was used, both to adjust the levels of the water and to obtain tide curves, by observing the heights of the tide at definite times, and then plotting the curves with the heights of the tide as ordinates and the times as abscissae. For the earlier experiments this year the same gauge was used for both purposes, and it has been used all through for the purpose of adjusting the levels of the water, automatic arrangements being used for drawing the tide curves. In devising these automatic arrangements several difficulties presented themselves, besides those inherent in all chronographic apparatus. Anything in the nature of standing apparatus was inadmissible, as it would interfere with the working and adjusting of the tanks. The apparatus must be such as could be put up and taken down with facility, and hence could not admit of complicated arrangements. A pencil worked direct by a float with a drum turning about a vertical axis by a clock, all to stand on the level glass surface, appeared the most drsirable arrangement. In the first instance, 422 ON THE ACTION OF WAVES AND CURRENTS. [58 a clock driving a detached vertical cylinder with a cord was kindly lent by Dr Stirling from the Physiological Laboratory of Owens College, and an arrangement of float and stand was constructed by Mr Bamford. The loan of this clock was temporary, and experience gained with it led to the purchase of an ordinary Morse clock from Latimer, Clark, & Co. at comparatively small cost. A pulley was fitted so that the clock would drive the borrowed cylinder. This clock did its work quite as well as the more costly instrument. Its rate of action varied considerably with the resistance of the apparatus to be driven, so much so that the curves taken at different times from the same experiment could not be compared by superposition. Still the action of the clock during the individual observations was sufficiently regular to give a fairly true tide curve, and it became obvious that it would be impossible to obtain any independent clock-driven apparatus that would give absolutely constant speeds such as would admit of the comparison of the curves taken from different parts of the estuary by direct superposition. To obtain such comparison it would be necessary to move the paper by the gearing which moved the generator. 22. Compound Harmonic Tide Curves. On considering how best this might be done, it appeared that if the paper had a horizontal motion corresponding to the rise and fall of the generator while the pencil had a vertical motion corresponding to the rise and fall of the tide at any point in the tank, then, if the tide were in the same phase as the generator, the curve would be a straight line or an ellipse of infinite eccentricity, with a slope (tan 0) equal to the rise of tide divided by the horizontal motion imparted to the paper, while any deviation of phase would be shown by the character of the ellipse or closed curve described by the pencil, and that to obtain the time-tidal curve from such "curves would be easy by projecting on to a circle, while for the purpose of comparison, and bringing out any difference of phase or deviation from the harmonic curves, such compound harmonic curves would be much more definite than the harmonic curves. This plan was therefore adopted with the happiest results, for, although it may take some study to become familiar with the curves, the obvious differences in these curves taken at different parts of the tanks, and at the same part at different stages of the progress towards a state of equilibrium are clearly brought out. The method also shows the similarity of the curves taken in the two tanks, or in different experiments at the corresponding places and corresponding numbers of tides run, as well as in the final states of equilibrium. The tide curves (Fig. 48, page 481) bring out emphatically the inter-dependence of the character of the tide on the arrangement of the sand, and the coincidence of a state of equilibrium of the sand with a particular tide curve at each part of the estuary. In these experiments the balance of the tanks has been adjusted so as 58] ON THE ACTION OF WAVES AND CURRENTS. 423 to make the time intervals of rise and fall of the generator equal, i.e. to make the motion of the generator harmonic, so that these compound har- monic curves are at all parts of the tank comparable with a simple harmonic motion. But it is important to notice that they are not essentially so, being merely compai'able with the motion of the generator, so that if the generator were given a compound harmonic motion, such as that of the tide in the mouths of most estuaries, these curves would have a different dynamic significance. These curves would still be valuable as showing the state of progress and final similarity of the tidal motion at the same parts of the estuaries, but to bring out their dynamical significance it would be necessary to substitute a simple harmonic motion with the same period as that of the generator. IV. DESCRIPTION OF THE EXPERIMENTS ON THE MOVEMENT OF SAND IN A TIDEWAY FROM SEPTEMBER 9, 1889, TO SEPTEMBER 1, 1890*. 23. Continuation of Experiments VII., Tank A, and III., B, (see Figs. 4, 5, 6, pages 44-1,...) September 7 to October 11. These experiments were in progress at the time of the Newcastle Meeting of the British Association, and had so far advanced that tracings of the first surveys were exhibited and included in the First Report. So far as they went, they took an important place in the conclusions arrived at in that report, showing that with a vertical exaggeration of 100, the results obtained in the small tank (B) with rectangular estuaries, without land water, as to rate and general distribution of the sand, were closely similar to those obtained in A, and that the mean slopes, reduced to a 30-foot tide, in these experiments agreed with those obtained in A, with vertical exaggerations of 64. It was desirable to continue these experiments to see how far a state of equilibrium had been arrived at. This was accomplished by the assistance of Mr Foster, who kindly looked after the running of the tanks till the return of the author and Mr Bamford in October, and thus enabled a month, which would other- wise have been wasted, to be utilised, in obtaining an experience of the effect of about 100,000 tides after apparent equilibrium had been obtained in each tank. Daily records of the counters were taken, and, although there were several stops, the intervals of running gave the periods very constant. The plans show but little alteration, except that the sand, particularly in B, had shifted upwards and accumulated somewhat at the head of the estuary, leaving the slope the same ; a circumstance which would be ac- counted for by a difference in the level of the water, and which is also * In the published report of these experiments it is not thought desirable to give the daily records of progress in the notebook. 424 ON THE ACTION OF WAVES AND CURRENTS. [58 indicated by the mean slope reduced to a 30-foot tide shown in Figures 2 and 3, page 440. The agreement of the slopes here shown as compared with the mean slope in the case of Experiment V., A, which has been introduced in this diagram for the sake of comparison, is quite as great as could be expected, considering the difficulties of the experiments, and affords very valuable evidence of the permanence of these slopes when once a condition of equilibrium has been attained. In respect of the ripple the two tanks presented a very different appear- ance, which is clearly shown in the plans and sections. While the ripple in A was comparatively small and shallow, in B it was larger and deeper than anything previously noticed ; that this was a symptom of the condition of B being on the verge of dissimilarity seemed probable, and to test this the period of B was increased from 23'85 to 26'5 seconds, and it was allowed to run on 16,000 more tides and again surveyed. Plan 3, page 443, shows the result; the ripple has increased in breadth though rather diminished in depth. 24. Experiments to find the Limits to Similarity. Experiment IV., B, Fig. 7, page 444, October 22 to November 27. In this the rise of tide was 0'094 foot, and the vertical exaggeration as compared with a 30-foot tide 71. In Experiments I. and II., B, with a rise of tide 0'088 and a vertical exag- geration 68, described in the First Report, it had been found that the rate and manner of distribution of the sand did not correspond with that in the corresponding experiment in the larger tank, indicating that with an exag- geration 68 the tide of '088 was somewhat below the limit of similarity. The determination of these limits being a primary object of the investigation, it appeared desirable to repeat these experiments with a slightly higher tide. In IV., B, the character of the action presented the same peculiarities as previously observed, but in a smaller degree, and the final state, as shown in the plans and in the curve of slopes (Figs. 2 and 3), is a much nearer approach to the general law, the conclusion being that in IV., B, the con- ditions were still below the limit, but nearer than in I. and II., B. Experiment VIII., A, October 22 to November 14. This was an experiment to determine the manner of action with the same horizontal scale as the first part of Experiment V., A, but half the rise of tide. Experiments I. and II., B, with a rise of tide of "088 foot and a period of 36 seconds, being a vertical exaggeration of 68, had indicated that with this rise of tide a change in the manner of action had already set in, but it was none the less desirable to see what would be the character of the action and the final state of equilibrium well below this limit. The rise of tide in VIII., A. was 0'088 foot and the mean level 0'138 foot from the bottom, and the period 70 seconds, the sand being placed level at a 58] ON THE ACTION OF WAVES AND CURRENTS. 425 uniform depth of 1 inch to Section 18 as in the previous experiments. The vertical exaggeration would thus be only 34. The manner of action of the water on the sand was in this case essentially different from that in any previous experiments even in I. and II., B, although it presented characteristics which had been indicated in those experiments. Instead of the sand being in the first instance rippled over the whole surface a middle depression was formed, extending some way up the estuary, the bottom and sides of which were rippled ; the rest of the sand soon became set and yellow. After 16,000 tides a survey was made and the experiment continued to 24,000, when another partial survey was made, showing very small alterations, and those nearly confined to the rippled channels. It was, in fact, clear that the apparent equilibrium was owing to the sand having become set, and that to proceed till real equilibrium was established would take an almost indefinite time. As the setting of the sand, owing to the slow action of the water, appeared to play such an obstructive part, it seemed possible that better results could be obtained if the sand could be kept alive with waves. Accordingly the experiment was stopped, to be repeated with waves. Experiment IX., Tank A, Plans 1, 2, 3, Figs. 8, 9, 10 (with Intermittent Waves}, November 16 to January 4. The conditions were the same as in Experiment VIII., with the addition of the waves. This experiment presented the same characteristics as those observed in VIII., A. The rate of action did not fall off so rapidly or completely as in VIII., but was mainly confined to the channels; and, although the experiment was continued to 57,000 tides, the condition of equilibrium was far from being arrived at, owing to the setting of the sand. After the last survey a small stream of land water (one pint per minute) was admitted at the top of the estuary, without any perceivable effect for 1000 tides, where- upon the experiment was stopped. Experiment V., B, Plan 1, Fig. 11, p. 448, November 21 to December 2. This was the corresponding experiment in B to Experiment VIII. in A, the rise of tide being one-half inch ( - 042 foot), and the period 50 seconds, exaggeration 32. The characteristics were yet more definitely marked, rippling being entirely absent, and the action being entirely confined to the space between Sections 14 and 18. Experiment VI., B, December "> to December 9. In this experiment the conditions wi-iv exactly the same as in Experiment V., B, except that the sand, instead of being laid level, was laid with a slope of 1 in 124, the slope corresponding to the theoretical condition of equilibrium as in the previous 426 ON THE ACTION OF WAVES AND CURRENTS. [58 experiment. After 6757 tides with a mean period of 601 seconds the sand was not moved anywhere in the slightest degree. Experiment VII., B, Plans 1 and 2, Figs. 12 and 13, December 9 to January 3. This was a continuation of Experiment VI., with the tidal period diminished in the ratio 1 to V2 from 50 to 35'35. The effect of changing the period would be to increase the vertical exaggeration, so that the slope of 1 in 124 would not be the theoretical mean slope of equilibrium as previously determined, which would be 1 in 87, so that any sensitiveness to the condition of equilibrium would be shown by the shifting up of the sand. This commenced at once and continued until the mean slope was about 1 in 100 above Section 13. The absolute quiescence of the sand in Experiment VI., B, when laid with the mean slope of equilibrium corresponding to the period, together with the increase of the slope with the increase of period in Experiment VII., B, indicates that, although, as shown in Experiment V., the limiting conditions under which the water could redistribute the sand from the level condition had been long passed, the conditions of equilibrium remained the same ; or, in other words, that for a half-inch tide, with a period of 50 seconds i.e., an exaggeration of 32 with the sand originally distributed according to the theoretical slope of equilibrium, the sand will be in equilibrium, while if the sand be laid with a smaller slope the water will shift it, tending to institute the slope of equilibrium. 25. Rectangular Estuaries with Land Water. Experiments X., A, and VIII. , B, Figs. 14, 15, 16, and 17, January 7 to March 10. The conditions in Tank A were the same as in Experiment V., Plan 1. The sand lay 0'25 foot deep, height of mean tide 0'256, rise 0'176, tidal period 50'2 seconds. A tin saucer was placed on the sand under Section 1 in the middle of the estuary, and a stream of water (one quart per minute, about 1/170 of the tidal capacity of the estuary per tide) run into the pan. During the early distribution of the sand the land water produced no apparent effect, but as the sand approached a condition of equilibrium, the effect of the fresh water in keeping a channel full of water at low tide, from the source all down the estuary, was very marked. The effect of this river in distributing the sand at the top of the estuary was also marked. The channel did not remain in one place ; it gradually shifted from the middle towards one or other of the sides, cutting away high sandbanks until it followed along the end of the tank into the corner, and then flowed back diagonally into the middle. Then, after some 10,000 tides, a fresh channel would open out suddenly towards the middle of the estuary, and then 58] OX THE ACTION OF WAVES AND CURRENTS. 427 proceed in the same gradual manner perhaps to the other side. This happened more than once during the progress of the experiment, which was carried to 85,000 tides. The different positions of the channels are apparent in the plans 1, 2, and 3 (Figs. 14 to 19). The comparison of these plans, and the accompanying sections with Plan 1, Experiment V., in the last report (Fig. 3, p. 403), shows but slight general effect of the land water so slight, indeed, that it might pass almost unnoticed. This shows that the land water does not alter the greatest height of the banks or the lowest depth of the channels. It will be noticed, however, in the plans, that the land water has lowered the general level of the sand in the middle of the estuary at the top, and raised it towards low water. This effect comes out in the mean reduced slopes shown in Figs. 2 and 3, p. 440. From these it appears that the effect of the land water, by continually ploughing up the banks at the top of the estuary, has been to disturb the previous state of equilibrium, lowering the sand near the top, and raising it further down the estuary. In Experiment VIII., B, the conditions at starting were the same as those in IV., B, and one quart of land water in 2 '8 minutes was admitted in the same manner as in X., A, the period being 35'4 seconds. The quantity of land water per tide was one-fourth the quantity in A, while the capacities of the estuaries are as 1 to 8, or the percentage of land water in B was 1'8 that of the tidal capacity at starting. After running GOO tides the rise of tide was increased from 0'094 to 097 foot without any alteration in the period. The experiment was then continued to 91,184 tides (Fig. 19). The apparent effects of the land water observed were exactly the same in character as in A, but were decidedly greater on account of the larger quantity. The curves agree fairly with those in A. 26. Experiments in short V-shaped Estuaries with and without Land Water. In the tanks A and B inner vertical partitions were introduced so as to form the upper end of the tank A into a symmetrical V, of length 6 feet and greatest breadth 4 feet ; while that of tank B was formed in a similar manner into a V of length 3 feet and breadth 2 feet. The lengths of the tanks were thus unaltered, the tidal capacity being reduced to three- quarters of what it was before. The sand was arranged in a similar manner to that previously adopted, except that the initial depth of the sand was 4 inches (0'33 foot in A) instead of 3 inches, and the scummers raised so as to maintain the water higher in a corresponding degree. Experiment* XL, A, and X., B, Fiyn. 20 t<> 23, March 18 to April 29. In tank A the rise of tide was 176 and the period 47 20. The experiments were first started without land water. The observed character of the action 423 ON THE ACTION OF WAVES AND CURRENTS. [58 was much the same as with the rectangular estuaries, being more intense towards the top of the V, and quieter at and below the broad end. The first attempt in Tank B showed that, owing to the diminished capacity of the estuaries, the sand would not come down even so well as in corresponding experiments with rectangular estuaries. This led to the abandonment of Experiment IX., B, and starting X., with a rise of tide O'llO, without, however, altering the level of the sand. The experiments were continued in both tanks without land water until about 40,000 tides had been run, and Plans 1 and 2 had been taken. These plans show the similarity of the effects in the two tanks. They also show decidedly the character of the distribution of the sand in the V-shaped estuary. It will be seen that the extreme positions of the contours up the estuary are much the same as in the rectangular estuaries, while the extreme positions down the estuaries are very much increased. The low-water contours extend from Section 11 to Section 19, while in Experiment V., A, Plan 1, it extends from Section 11 to Section 13. The low- water channels are nearly the same depth at corresponding points all down the estuary in both experiments, while in the V estuaries the banks extend 6 to 7 miles (reduced to a 30-foot tide) further down. After Experiments XL, A, and X., B, had proceeded to about 40,000 tides, corresponding quantities of land water were introduced at the tops of the estuaries, one quart in one minute in A, about 1/140 of the tidal capacity; in B one quart in 5'68 minutes, or about 1/140 of the tidal capacity. The tanks were then run on for 12,000 tides, and surveys for the plans 3 made (Figs. 24 and 25). The general effect of this land water, as shown in these experiments, is, as before, to lower the sand at the tops of the estuaries and slightly to raise it at the bottom. They were not, however, continued long enough to show a state of equilibrium. As in the rectangular estuaries, the detailed effects of the land water were much more observable than those shown in the surveys. The land water continually ploughed up the sand at the top of the estuary and kept the banks down, but owing to the narrowness of the estuary the general effects of this were not so striking as in the rectangular estuaries. Experiments XII., A, and XII., B, with Land Water, Figs. 26 to 29, April 29 to May 19. These were under conditions precisely similar to XI., A, and X., B ; XI., B, with land water, was started, but owing to an accident it was restarted as XII., B. Both experiments were run about 16,000 tides and then surveyed, and then run on about 16,000 more tides and surveyed again. The plans are all very similar, and show but little difference from the plans 3 with land water in the previous experiments. 58] ON THE ACTION OF WAVES AND CURRENTS. 429 27. Experiments in long V-shaped Estuaries without and with Land Water in Tanks C and D. Tank C was formed by extending AJay adding a rectangular trough to the top, and so as to admit of partitions forming a V extending from Section 23 (12 A), and D was formed by extending B in a similar manner. The lengths of the tanks were thus extended 6 feet and 3 feet greater than A and B, while the capacities were the same as the original capacity of A and B. The sand in C (A extended) was laid 4 inches deep from the top of the V to Section 28'5 C (17 -5 A). The sand in D (B extended) was laid 2 inches deep from the top of the V to Section 28'5 D (17'5 B). Experiments I., C and D, Figs. 30 to 33, May 24 to June 16, without Land Water. In C the tide was 0162 foot, and the scummer was placed so that the mean tide when running was O'OOS foot above the initial level of the sand ; this was riot observed at the time, being a consequence of the land water raising the level of low water by the necessity of getting over the weir. In D the tide was 0105 foot and the mean tide was '010 foot below the initial level of the sand. Thus reduced to a 30-foot tide, the initial depth of the sand was 5 feet higher in G than in B. The experiments were run for about 16,030 tides and surveyed, then restarted, when the level of water in C fell owing to a leak in the scummer. This lowered the sand at the lower end of the estuary, and a partial survey was made, and then the experiment continued until both tanks had exceeded 30,000 tides. The results, as shown in the plans, are very much alike, considering the very considerable differences in the initial quantities of sand. Owing to the much higher level of the sand in D, the top of the V was much more silted up in the early part of the experiment, and the sandbanks were higher towards the bottom of the estuary. Otherwise both tanks show the same characteristics. The highest point of the contour low water in the generator is still at Section 15, while the highest point of the contour at high water in the generator is at Section 4, so that the distance between the highest points of these sections was still about 11 miles, while the banks at low water extended down to Section 26. Experiments II. , Tanks G and D, with Land Water, Figs. 34 to 37, June 17 to July 8. The conditions in these experiments were the same as in Experiments I., Tanks C and D, except that the scummer in D was altered, until the mean tide level was only '003 foot above the initial height of the 430 ON THE ACTION OF WAVES AND CURRENTS. [58 sand, and in Tank A 002 foot above, while the rise of tide in A was slightly greater and that in B slightly less. Surveys were taken at about 16,000 and 32,000 tides respectively ; they are very similar, and the effects of the land water are, as before, to slightly raise the lower sand and lower the upper. At low water there was still water in the channels right up to the top of the estuary, and at high water there was what would correspond in a 30-foot tide with 10 or 12 feet of water at the top in the low-water channels. 28. Experiments in long V-shaped Estuaries with straight tidal Rivers extending up from the top of the V with and without Water in Tanks E, F', and F. Tank E was formed by opening out the partition boards in Tank C at the end of the V to a distance of 4 inches. That portion of the V below Section 12 remained as in Tank C, the position of the partition boards not being altered. At a section, 12 - 5, a small angle was formed, so that while the boards above the section remained straight their ends stood apart 4 inches instead of closing up to form a V. Tank C was extended by a trough 19 feet long, in which partition walls were constructed continuing the partitions in the lower portion up to a section, 38, above the zero in Tank C ; these were straight, vertical boards, the distance between them contracting from 4 inches at the lower end to 1 inch at the end of the river. Tank F' was formed in a similar manner, except that the upper extension was bent through two sharp right angles so as to return along the side of the tank ; and subsequently tank F was formed exactly similar to Tank E with half the dimensions. Experiment with Land Water, I. and II., Tanks E and F, Figs. 38 to 47, July 11 to July 31. In Tank E the sand was laid to a depth of 4 inches, the same as in C, from the upper end of the river, Section 38 down to Section 28. The rise of tide was 0*140 foot, and the mean level of the tide about "016 foot above the level of the sand. The period 49 sees, and water 1 quart a minute, or 1/200 the tidal capacity per tide, was introduced at the upper end of the river. In Tank F' the sand was laid similar to that in Tank E, the rise of tide 01 foot, and the mean tide O'OOG foot above the level of the sand. The period being 30'04, land water, 1/200 the capacity of the estuary, was introduced at the top of the river. In starting these experiments the effect of the tidal river was very marked. After the first tide in Tank E, some depth of water remained in the river, and a long way down the estuary, at low water, and the tide came up with a bore increasing in height all the way to the top of the river, and then returned with a bore to the lower end of the river. The 58] ON THE ACTION OF WAVES AND CURRENTS. 431 bore, as before, soon died out over the greater part of the estuary, as the sand at the bottom became lower. And the bore gradually died out in the top of the V. until, as the number of tides approached 16,000, the bore only began to show at about Section 4, and ran up the river very much diminished from what it was originally. Owing to the indraught and outflow of the river, the velocity of the water and its action on the sand was greater at the top of the V and the mouth of the river than at any part of the estuary, while for some way up the river, and all the way down the estuary, there was a large volume of water running at low water. The top of the river was ninety miles (reduced to a 30-foot tide) from the bottom of the estuary, and the tide did not commence to fall at the top of the river until after low water at the mouth, so that nearly all the tidal water in the river ran over the estuary during the low water. The delay in the return of the water from the river obviously played a most important part in the effects produced. At the bottom of the estuary the sand came down much as usual, but it did not rise at the head of the estuary. For the first 10,000 tides the sand was all covered at low water and rippled with active ripples up to the end of the river, and it seemed as if no banks were going to appear. The sections of the sand appeared as nearly as possible horizontal. The level having lowered from the bottom of the estuary up to Section 15, from Section 15 to Section 3 it was somewhat raised, then from 3 upwards to 7 it was lowered, and thence up to the top of the river it was raised in a gradual slope. At about 12,000 tides two small banks appeared at low water, one on each side of the estuary at Section 13. Everything was perfectly symmetrical so far, but from this time the bank on the right of the estuary extended downwards, while that on the left extended upwards and a depression or channel formed between them extending across the estuary in a diagonal manner. This was the condition when at 16,000 tides the first survey was made. As the running continued these banks continued to rise, that on the right downwards, that on the left upwards, until a distinct channel was formed from the mouth of the river down to Section 20, as shown in the second survey at 32,000 tides. The level of the sand at the mouth of the river altered very little, diminishing during the first 10,000 tides, and then reassuming its original height, but the sand passed upwards through the mouth, and gradually raised the level in the river above, until there was only about 0*02 foot in the shallowest places at low water (corresponding to 5 feet on a 30-foot tide) ; this level was first reached at the top of the river and then gradually extended down to Section 19, which point it had reached at 32,000 tides, 432 ON THE ACTION OF WAVES AND CURRENTS. [58 when the second survey was taken. In this condition the bore still reached the end of the river, raising the water 0'02 foot (5 feet on the 30-foot tide). Above Section 19 all motion of the sand had ceased, but below this the sand was still moving up when' the experiment stopped. The bore still formed at the mouth, but very much diminished, and was very slowly diminishing. The final condition of the estuary shows the contour at low water in the generators extending up to Section 9, and the contour at high water in the generator to Section 11. In tank F', with the sharp turns in the river, the action of the sand at the bottom of the tank was at first sluggish, as in Experiment IV. In the top of the estuary and river the appearance of things for the first 10,000 tides was much the same as in Tank E, except that the ripple of the sand did not extend more than half-way up the river, and deep holes were formed at the bends, banks being formed between them. The bore, however, ran up to the end of the river until some time after the first survey was taken, and the tide still rose very slightly when the second survey was made, though the river was barred by a bank between the bends, by which the flood just passed in small channels at the sides. The sand had risen in the top of the estuary until it virtually closed the mouth of the tidal river, and the condition of the estuary resembled that obtained in Tank D. This virtually ended the experiment, but opportunity was taken to try the effect of a larger quantity of land water, which was increased to one quart in two minutes i.e. nearly three times and the experiment continued for 20,000 more tides without any material effect. In Tank F the action at the lower end of the tank was again sluggish. At the top of the estuary and in the river the conditions of the sand were as near as possible similar to those in Tank E, but, as it came out, the mean level of the water, relative to the level of the sand, was some 5 feet (reduced to a 30-foot tide) lower in F than in E. The appearances for the first 16,000 tides were the same as far as was observed ; the ripple now extended up to the top of the river, and no banks formed at the mouth. Nevertheless, before the second survey was taken, the tide ceased to rise above the mouth of the river, proving that the previous failure to realise the same state in the small tank as in the larger, had riot been entirely due to the bends in the river. The question remained whether it might not be owing to the higher level of the sand relative to the mean level of the tide. This question brings into prominence a fact observed during all the experiments, but which had not, previous to the experiments on E and F, assumed a position of importance. This is the gradual diminution of the rise of tide owing to the lowering of the sand. 58] ON THE ACTION OF WAVES AND CURRENTS. 433 29. The rise of the tide depends not only upon the rise of the generator, but also upon the tidal capacity of the tank. This capacity is the product of the area of the surface at high water multiplied by the rise oFtfde, less the volume of sand and water above low water in the generator. Now in starting the experiments with the sand at the level of mean tide, not only is there much more sand above the level of low water in the generator than when the final condition of equilibrium is obtained, but the quantity of water retained on the top of the level sand is considerable, so that the tide rises considerably higher in the generator at starting than when the condition of equilibrium is obtained, which excess of rise gradually diminishes as the sand comes down at the lower end of the estuary. Although the foot of the sand comes down pretty rapidly at the com- mencement of the experiment, owing to the surface being rippled, the water runs off slowly, and it is not till the sand at the end of the estuary has been raised, and a slope formed, that the water runs down freely at low water, so that during the early part of the experiment not only is the rise of tide at the head of the estuary high, but also the low tide and the mean level of the tide. The result is that the mean level of the water at the head of the estuary is higher during the early part of the experiment. These changes in the tide at different parts of the estuary and at different stages of the tide are well shown by the automatic tide curves, page 481. As the sand is rising at the top of the estuary, the result of the high water is to raise the first banks above the level to which the tide finally rises. As these banks come out and the ripple is washed off, leaving smooth surfaces and channels, from which the water runs, and clean dry banks, the mean level as well as the rise of tide falls, leaving the tops of the bank, which were at first covered, high and dry. These effects were much greater in Experiments C and D than in A and B, and still more marked in E, F, and F'. In E, F, F', the plans 1 and 2, taken at 16,000 and 33,000 tides respectively, show the difference in the level of the sand at the mouths of the respective rivers. In Tank E the rise of tide at the mouth of the river was observed to be 0'02 higher at 16,000 than at 30,000 tides, and in Tanks F and F' at 16,000 tides there was a bore which ran up to the top of the river, while at 33,000 tides the sand at the mouth was not covered at high water. It thus seems that the condition of things which follows from starting with the sand level, and a constant height of low water, is to institute a distribution of sand at the top of the estuary, corresponding to a state of equilibrium with a higher tide than that which ultimately prevails ; and the greater the initial height of the sand relative to the mean level of the water the greater will be this effect. That this action tends to explain o. R. ii. 28 434 ON THE ACTION OF WAVES AND CURRENTS. [58 the closing of the mouths of the rivers in Tanks F' and F and not in E is clear. But it is not clear that this is the sole explanation ; the conditions in F' and F were not far removed from the limits of similarity obtained in the rectangular tanks, and it is not clear that these limits may not be somewhat different in the long estuaries with tidal rivers. This is a matter which requires further experimental examination, for which there has not been time. 30. Experiment II. in E and F, Figs. 42 to 45, without Land Water, August 5 to September 1. These experiments have been made under the same conditions as in I. E and F, except for the land water. The general appearance of the progress of the experiments was nearly the same, and Plan 1 shows little difference. But as the experiment in E proceeded, it became clear that the river was going to fill up gradually from the end. The bore no longer reaches the end at 16,000 tides, while it had ceased to exist and the tide had ceased to rise at Section 11 in the river at 32,000 tides, the end of the estuary also having filled up, the action in F being nearly the same. Thus we have evidence similarly shown by both estuaries that, although the fresh water produces little effect on the condition of equilibrium of a broad estuary, the existence of a long tidal river above the estuary does produce a great effect on the level of the low-water channels in the upper portions of the estuary, and that land water, even in such small quantities, is effective to keep open a long tidal river emptying into a sandy estuary or bay*. * For continuation, see p. 482. 282 436 ON THE ACTION OF WAVES AND CURRENTS. TABLE I. GENERAL CONDITIONS Shape of the Estu- ary Per- cent- age of Land Water References Period in seconds Horizoutal scales Vertical scale 1 in. Rise of tide in feet Vertical exagger- ation e. Ex- peri- ment Tank Plan Figure 1 in. Inches to a mile / o-o VII A 2 4 33-5 17,600 3-58 177 0-170 99-7 j III B 2 5 23-8 33,600 1-88 327 0-094 102-0 )> 3 6 23-8 33,600 1-88 327 0-094 102-0 IV 1 7 34-4 23,300 2-71 327 0-094 71-0 IX A 1 8 69-3 10,500 6-02 333 0-090 31-6 *j j> V B 1 11 50-5 23,600 2-68 720 0-042 32-0 3 IX A 2 9 69-3 12,400 5-08 379 0-080 32-0 bo w 3 10 67-3 12,600 5-02 366 0-082 34-5 3 > VII B 1 12 34-0 39,200 1-57 986 0-030 39-0 8 55 )) 2 13 34-0 39,200 1-57 986 0-030 39-0 PH 0-6 X A 1 14 50-2 11,500 5-49 171 0-176 67-0 1-2 VIII B 1 16 35-4 22,000 2-87 309 0-097 71-0 0-6 X A 2 15 48-6 11,900 5-30 171 0-176 69-0 1-2 VIII B 2 17 34-5 22,600 2-8 309 0-097 73-0 0-6 X A 3 18 48-6 11,900 5-30 171 0-176 69-0 v 1-2 VIII B 3 19 34-5 22,600 2-8 309 0-097 73-0 f o-o XI A 1 20 47-5 12,400 5-10 177 0-170 71-0 TS X B 1 22 35-4 20,700 3-05 273 0-110 75-8 9 & XI A 2 21 47-2 12,670 5-01 181 0-166 69-5 4 ,d X B 2 23 35-4 20,700 3-05 273 0-110 75-8 T 0-7 XI A 3 24 47-2 12,400 5-08 177 0-170 70-0 !> ' 0-7 X B 3 25 34-0 21,800 2-90 280 0-107 78-0 | 0-7 XII A 1 26 48-2 12,300 5-15 179 0-168 68-4 o J 0-7 XII B 1 28 34-2 21,700 2-91 280 0-107 77-6 CO 0-7 XII A 2 27 47-0 12,700 5-00 182 0-165 69-4 0-7 XII B 2 29 34-2 21,900 2-88 286 0-105 75-0 I C 1 30 49-8 12,100 5-22 185 0-162 65-4 I D 1 32 35-9 20,900 3-03 285 0-105 73-1 c3 ,3 I C 2 31 46-2 13,200 4-78 190 0-158 69-5 o3 I D 2 33 34-4 21,800 2-90 286 0-105 76-0 t> " 06 II C 1 34 48-4 12,500 5-04 188 0-160 66-8 be C 0-6 II D 1 36 34-6 22,200 2-85 300 o-ioo 74-1 3 0-6 II C 2 35 48-4 12,500 5-04 188 0-160 66-8 ^H 0-6 II D 2 37 34-6 22,200 2-85 300 o-ioo 74-1 t- s 0-5 I E 1 38 48-9 13,100 4-82 208 0-143 63-2 ^ 2 0-5 I F 1 3!) 30-0 25,800 2-45 313 0-096 82-5 h*i 0-5 I E 2 40 47-8 13,400 4-70 208 0-143 64-6 1 0-5 I F 2 41 30-0 24,700 2-56 313 0-096 82-5 H . o-o II E 1 42 47-9 13,500 4-67 214 0-140 63-4 > II F 1 43 31-5 25,400 2-49 327 0-091 77-8 +3 't> > II E 2 44 47-9 13,600 4-64 217 0-138 62-9 ^ bo II F 2 45 30-3 26,200 2-41 321 0-093 81-86 C 0-5 I F' 1 46 30-1 25,500 2-48 300 o-ioo 85-1 & 0-5 I F' 2 47 30-1 25,700 2-46 305 0-098 84-4 ON THE ACTION OF WAVES AND CURRENTS. * A *** 437 1 THI AND RESULTS OF .THE EXPERIMENTS. . Criterion of simi- larity / C=(tfe) / H.-itfht /of initial sand in feet Height of mean tide in feet Number of tides from the start Action of the water on the sand in forming the bed at the lower end of the estuary Manner Bate Final state 0-490 0-25 0-265 93,839 General Normal 0-083 0-125 0-140 99,388 General Normal Normal 0-083 0-125 0-140 130,176 Large ripple 0-058 0-125 0-130 16,344 Nearly normal Nearly normal Nearly normal 0-023 0-125 0-1325 13,078 Very partial Very slow 0-002 0-65 0-065 17,919 Zero 0-016 0-125 0-142 36,776 0-019 0-125 0-141 78,986 Not reached 0-001) Slope 1 (0-065 17,424 Zero o-ooi / in 124 1 39,727 Nearly normal 0-25S 0-25 0-256 19,437 Normal Normal 0-064 0-1 25 0-148 18,332 Nearly normal Nearly normal 0-362 0-25 0-256 42,820 Normal Normal 0-066 0-125 0-148 68,861 0-362 0-25 0-256 76,273 See description See description 0-066 0-125 0-148 91,184 See description See description 0-346 0-333 0-337 17,206 Normal Normal 0-101 0-166 0-17!) 17,879 Normal 0-320 0-333 0-348 39,809 Normal 0-101 0-166 0-169 40,268 Normal 0-343 0-333 0-348 60,243 Normal Normal Normal 0-095 0-166 0-169 57,024 Normal Normal Normal 0-327 0333 0-340 16,538 Normal Normal 0-095 0-166 0-168 15,981 Normal Normal 0-315 0-333 0-343 31,991 Normal Normal Normal 0-081 0-166 0-175 35,129 Normal Normal Normal 0-278 0-333 0-341 16,943 Normal Normal 0-084 0-187 0-179 16,383 Nearly normal Nearly normal 0-275 0-333 0-345 30,584 Normal Normal Normal 0-088 0-187 0-179 35,344 Nearly normal Nearly normal Nearly normal 0-274 0-333 0-344 16,90* Normal Normal 0-074 0-187 0-190 18,128 Nearly normal Nearly normal 0-274 0-333 0-335 31,127 Normal Normal Normal 0-074 0-187 0-190 31,928 Nearly normal Nearly normal Nearly normal 0-185 0-333 0-350 16,368 Normal Normal 0-073 0-187 0-191 16,577 Partial Sluggish 0-189 0-333 0-337 32,635 Normal Normal 0-073 0-187 0-191 32,880 Ripple large 0-174 0-333 0-349 15,871 Normal Normal 0-060 0-187 0-193 17,184 Partial Sluggish 0-163 0-333 0-349 32,501 Normal 0-066 0-187 0-192 29,!) 17 Ripple large 0-085 0-187 0-187 16,577 Partial Sluggish 0-080 0-187 0-187 32,677 Ripple large 438 TABLE II. MEAN SLOPES OF THE SAND IN RECTANGULAR TANKS. TANK A Measured Experiment V., Plan 4 Experiment VII., Plan 2 Heights of Contours shown on the Plan Height (re- duced to a 30-foot Tide) of Contours from L.W. Mean Horizon- tal Distance of Contours from the Contour at 30 feet above L.W. Horizontal Distances reduced to a 30-foot Tide Height (re- duced to a 30-foot Tide) of Contours from L.W. Mean Horizon- tal Distance of Contours from the Contour at 30 feet above L.W. Horizontal Distances reduced to a 30-foot Tide Feet Feet Unit 6 inches Miles Feet Unit 6 inches Miles 1 - 0-975 -1-65 -1-792 - 3-003 0-176 30-00 o-oo o-oo 30-000 o-ooo o-ooo 0-146 24-39 0-79 1-355 24-546 0-647 1-133 0-116 18-68 1-86 3-20 19-092 1-254 2-171 0-086 13-00 2-96 5-07 14-638 2-356 4-085 0-056 7-46 4-64 7-95 9-184 3-724 6-447 0-026 1-87 6-63 11-38 3-730 5-428 9-397 -0-004 - 3-74 8-43 14-50 - 1-724 7-467 12-930 - 0-034 - 9-35 10-30 17-80 - 7-178 9-283 16-070 - 0-064 -15-00 12-17 21-60 - 12-632 11-780 20-400 - 0-094 - 20-80 13-60 23-40 - 18-086 14-003 24-235 -0-124 - 26-20 15-88 27-30 Experiment X., Plan 1 Experiment X., Plan 2 Feet Feet Unit 6 inches Miles Feet Unit 6 inches Miles 1 - 0-690 -0-774 -1-302 -1-167 0-176 30-000 o-ooo o-ooo 30-000 o-ooo o-ooo 0-146 24-886 0-741 0-810 24-886 0-665 0-752 0-116 19-772 2-147 2-347 19-772 1-900 2-149 0-086 14-658 4-256 4-652 14-658 3-648 4-124 0-056 9-544 6-916 7-560 9-544 6-631 7-507 0-026 4-430 9-880 10-800 4-430 9-101 10-290 -0-004 - 0-684 11-533 12-606 - 0-684 11-227 12-594 - 0-034 - 5-798 13-737 15-013 TANK B Experiment III., Plan 2 Experiment IV., Plan 1 Feet Feet Unit 3 inches Miles Feet Unit 3 inches Miles 1 - 3-540 - 5-643 - 0-994 -1-124 0-094 30-000 o-ooo o-ooo 30-000 o-ooo o-ooo 0-079 25-213 0-760 1-240 25-213 0-665 0-760 0-064 20-426 1-330 2-163 20-426 1-558 1-773 0-049 15-639 2-052 3-340 15-639 2-185 2-487 0-034 10-852 3-249 5-290 10-852 4-142 4-714 0-019 6-065 4-332 7-044 6-065 6-859 7-806 0-004 1-278 6-061 9-854 1-278 9-766 11-120 -0-011 - 3-509 7-828 12-727 - 3-509 12-046 13-710 -0-026 - 8-296 9-291 15-110 - 8-296 -0-031 -13-083 11-341 18-430 Experiment VIII., Plan 1 Experiment VIII., Plan 2 Feet Feet Unit 3 inches Miles Feet Unit 3 inches Miles 1 - 0-595 -0-621 -1-925 - 2-062 0-097 30-000 o-ooo o-ooo 30-000 o-ooo o-ooo 0-082 25-360 0-608 0-634 25-360 0-988 1-060 0-067 20-720 2-090 2-181 20-720 1-672 1-792 0-052 16-080 3-268 3-410 16-080 2-983 3-197 0-037 11-440 5-224 5-472 11-440 5-168 5-538 0-022 6-800 8-987 9-378 6-800 8-398 9-000 0-007 2-160 11-400 11-896 2-160 11-285 12-100 -0-008 - 2-480 13-148 13-720 - 2-480 13-108 14-050 -0-023 -7-120 14-535 15-570 439 440 JKapnm f Actual -top* WSt Oiagrvmof ibpa miuotltoa,30 feetTUb ! y. ftax * Peried, 33 -3 Jos? R,se cf IMt a - ft ' 3z-a . . oies Fig. 2. DtAgroffi tfAf tnal Slept* tfieJi- ax. exG-ggtrution of Z9. of fte/Hs na&tted to a 30 f**>[. & 442 s o -O o e 443 S g I s o e O vjo Sftnunevut ^oaaj JWVM MOJ *urpq jo iMMfv firoj. Tip. Tijr s^uij jnonun vip -uo svjnfy vt^ 444 s o i-o e -2 O CO s o 445 J 3 o -o e 5^ & - ? s to - to u i~Ci _ 5 ^ - Ci o tt 3 i | i I 1 1 * ?l :i - - 446 & . S* o: u 2 - -2 ? 8 u a S u) 1 o CO 3 I / ' / f \ \ 447 8 rC> O e o o I I tJL 448 J * < e 449 o CO as 7014; vjo fpnuwp in ifiartJrraM. yftuviftp jimpn> Jtnfl Mays' rrurj jnajrvcy vifl -u O. B. II. 29 450 05 I o CO ja VAoqno yi - miff jnajvay 9>{t -ua sunSy yt/ e o 292 452 1 \ i i ! k ' i \ 1 | ). 1 ( i i f 453 1 o JC 454 o g I * o CO 8 O fcO 455 -f^ Tidf fit/re- 11 514 tides at 43 8 stand n ?. IT S 8 7 S ? S. <=. > > /r snni jnonioy yip 110 svjnfy ^ifj t 467 (M e o 302 468 o rO e S flf 469 o CO 5 5 *a I I 470 o e o cc u8 r 1 bS * % 471 s o 472 473 I i I 5- d o CO fit it 474 "" 475 i 476 : : ;JE!*S 477 R3 ' r 3 ^ HiliP p ) i / ( 1' \ P 5 c |;\ : & P it ( ! 1 I SB R fl ft '11 > 1 v 31 Su * P n ^ ^Jv > > | i , c R 1 * S c a a * 1 9 [ 4 4 \ 1 i 1 1 1 1 | I c 4 [ 478 r ss O I a 479 I r-t T" 1 AH =c *s 99 8 to 480 o CO 1 481 JTute. Curves Tank C xpenmaa,JIA/ur 20GA3 tuta Xinkl) K.ifxnmmt,I Afar 26396 tides . ]'ankl> Experiment. ff-Wu TuU C TankF. Ejcfw Fig. 48. O. K. II. 31 59. THIRD REPORT OF THE COMMITTEE APPOINTED TO INVES- TIGATE THE ACTION OF WAVES AND CURRENTS ON THE BEDS AND FORESHORES OF ESTUARIES BY MEANS OF WORKING MODELS. [From the "British Association Report," 1891.] THE Committee held a meeting in the rooms of Mr G. F. Deacon, 32 Victoria Street, Westminster (July 29, 1891), and considered the results obtained since the last report. Professor Reynolds reported that by the date of the meeting of the British Association the objects of the investigation would be accomplished, and suggested that it would not be necessary to continue the investigation beyond that date or to apply to the Association for reappointment. These suggestions were adopted, and it was resolved that the thanks of the Committee be communicated to the Council of the Owens College for the facilities afforded for conducting the experiments in the Whitworth Engineering Laboratory. Having considered the disposal of the apparatus, which has no pecuniary value, the Committee resolved to recommend the Association to place it at the disposal of the Owens College. At a second meeting held in the Committee room of Section G at Cardiff the report submitted by Professor Reynolds was adopted. I. INTRODUCTION TO REPORT III. 1. In accordance with the suggestions in the Second Report, read at the Leeds meeting of the British Association, the investigation has been continued with a view 59] ON THE ACTION OF WAVES AND CURRENTS. 4.S3 (1) To obtain further information as to the final condition of equilibrium with long tidal rivers entering the head of a V-shaped estuary. (2) To obtain a more complete verification of the value of the criterion of similarity. (3) To investigate the effect of tides in the generator diverging from simple harmonic tides. (4) To determine the comparative effect of tides varying from spring to neap. Opportunity has also been taken : (5) To investigate the effect of prolonging the walls of the river into the estuary through the bar, which was below low water, with prolongations reaching up to low water, and others reaching up to half-tide this being done in both models, so that the similarity of the effects might be seen ; and (6) To investigate the effect of rendering the estuaries unsymmetrical by means of large groins, and so to test the laws of similarity obtained in the symmetrical estuaries as applied to unsymmetrical estuaries. 2. The two models have been continuously occupied in these investi- gations, when not stopped for surveying or arranging fresh experiments. In this way each of the models has run 600,000 tides, corresponding to 840 years. These tides have been distributed over six experiments in the large tank E, and four in the small tank F, in number from 50,000 to 250,000. 3. The experiments have all been conducted on the same system as described in the previous reports. All the experiments but one have been made in tanks E and F, without further modification ; and in all these land water to the extent of (V5 per cent, of the tidal capacity per tide has been introduced at the top of the river. Initially, the sand has been laid to the level of half- tide from Section 13 up the river to Section 26 down the estuary. The vertical sand gauges distributed along the middle line of the estuary have been read and recorded each day. Tide curves have been taken at frequent intervals. Contour surveys have been made, generally after 16,000 tides, and again after 32,000 ; while in the longer experiments further surveys have been made. With the spring and neap tide, the rate of action being much the slower, intervals between the surveys have been longer. In all 26 complete surveys have been made, and 20 plans showing contours corresponding to 312 484 ON THE ACTION OF WAVES AND CURRENTS. [59 every 6 feet, reduced to a 30-foot tide, together with sections and tide curves (page 506), are given in this report. The general conditions of each experiment, together with the general results obtained, are shown in the table, while a description of each experi- ment is given in VI. The Committee have been fortunate in retaining the services of Mr Greenshields, who has carried out the experiments, observing and recording the results, besides executing such modifications as have been required, designing the compound harmonic gearing for the spring and neap tides, which has answered excellently. Mr Bamford has kindly continued his assistance in conducting the inves- tigations and reducing the results. II. GENERAL RESULTS AND CONCLUSIONS. 4. The conditions of equilibrium with a long tidal river entering at the top of a V-shaped estuary. The experiments in tanks C and E made last year led to the conclusion stated in Art. 11 of the Second Report: 'that the effect of a river 50 miles long, when reduced to a 30-foot tide, increasing gradually in width until it enters the top of a V-shaped estuary, is entirely to change the character of that estuary. The time occupied by the water in getting up the river and in returning causes this water to run down the estuary while the tide is low, and necessitates a certain depth at low water, which causes the channel to be much deeper at the head of the estuary. In its effects on the lower estuary the experiments with the tidal river are decisive, but as regards the action of silting up the river further investi- gation is required, both to establish similarity in the models, and to ascertain the ultimate condition of final equilibrium.' From this year's experiments, III., IV., V., VI., and VII., in tank E, and V. and VI. in tank F, it appears tJiat if the length of the tidal river, reduced to a 30 foot tide, is 50 miles ; or taking R for the length of the tidal river in miles and hfor the rise of tide at the mouth of the estuary in feet, if R = 8-5 VA the river will keep open so that the tide will rise to the top, the sand falling gradually from the top of the river to the level of about mean tide at the mouth. That the depth of water in the river and at the top of the estuary increases rapidly with the length of the river, and when 59] ON THE ACTION OF WAVES AND CURRENTS. 485 the level of the sand at the mouth of the river will be more than h feet below the level of low water, and tlie bottom will be below low water level for more than half the length of the river above its mouth. 5. The similarity of the results in the tanks E and F. The experiments in the tanks E and F this year confirm those of last year, in showing that during the early stages of forming the estuary from sand at the level of mean tide, the action in the river is different in the small tank F from what it is in the large tank E, although the value of the criterion of simi- larity h 3 e* may be but little below 0'09. It was not found practicable to get the value of the criterion any greater in tank F, but it was found on diminishing the rise of tide in the large tank E until the criterion had a value 0'09, that the results were still similar, although the rate of action and the increase in the size of the ripple in- dicated that the limit was being approached. That the dissimilarity in tank F was only the result of a phase in the formation of the estuary was also definitely shown by the effects of dredging out the sand, which was above the initial level in the river during the early stages of the Experi- ments V. and VI., after which the action in tank F resumed the same course as that in E, and led to the same final condition of equilibrium, showing by the rate of action and size of ripple that the limit of similarity was approached. It thus appears that, with such arrangements as these tanks represent, there are two possible conditions of final equilibrium. The one is that which has uniformly been presented by tank E, and in Experiment V. in tank F after dredging ; namely, the tide rising up to the top of the river and keeping the sand low in the estuary. The other, that which was presented in Experiments I., II., III., and IV., in tank F ; namely, the sand at the top of the estuary rising to high water level, as it would do if there were no river, choking the mouth of the river, except so far as necessary to allow the land water to pass, and so preventing any tidal action from the river. Which of these two conditions the river will assume during the process of forming the estuary appears to be a critical matter, decided by whether the tidal action of the river in lowering the sand at the head of the estuary pre- dominates over the tendency of the tide in the estuary to raise the sand at the mouth of the river. There is a possible condition of instability between the river and the estuary. The emphatic difference in the action of the long tidal river, and * h IR the actual rise in feet, e the vertical exaggeration as referred to a 30-foot tide. 486 ON THE ACTION OF WAVES AND CURRENTS. [59 mere tidal capacity at the head of the estuary, in keeping down the sand at the head of the estuary; and, further, the very great effect which an in- crease in the length of the river has on the depth of water in the estuary and in the river are clearly shown*. In Experiments III. and V. in tank E, an increase of from 50 to 70 miles in the length of the river in V. causing the depth of water to increase from by 40 to 30 feet all down the river and estuary, lowering the sand in the lower river and upper estuary from the level of half-tide to 28 feet below low water. In neither of these experi- ments was the condition of instability reached, but 50 miles was very near the limit. In such a state any diminution of the upper tidal waters of the river, by shortening the river or by land reclamation, might well have caused the critical stage to be passed and caused the river to silt up just as in the other way the increasing of the tidal capacity high up the river by dredging in Experiment V., tank F, caused the critical stage of silting up to be passed and the river to open out. The sand actually removed in this experiment by dredging was 8 per cent, of the tidal capacity, or 400 million cubic yards, removed at the rate of 7 million cubic yards a year. In most navigable rivers two processes have been going on dredging and land reclamation the first tending greatly to improve the rivers and estuaries, the second to deteriorate them so that any improvement has been a question of balance. Where the rivers have improved they will probably continue to improve so long as dredging goes on, but if the dredging should stop, for example in the Thames, there would in all probability be a gradual deterioration, possibly ending in the silting up of the tidal river. 6. The effect of Tides deviating from the simple Harmonic Law. One attempt was made to study this question, when it was found that it would require such modifications in the gearing as were not practicable in the time, and so it was abandoned. 7. The action of Tides varying from Spiking to Neap. The rates of action and conditions of final equilibrium in rectangular tanks, in V-shaped estuaries with a long tidal river, and in each estuary rendered unsymmetrical by large groins, have been investigated with tides varying harmonically from spring to neap, and again to spring in 29 tides. The ratio of these at spring and neap being 3 to 2 as compared with uniform tides, having the same rise as the spring tides, also for uniform tides having the same rise as the mean of spring and neap, the results showing definitely : * See page 50G in which the sections of the rivers and estuaries in tank C, Experiment II., and tank E, Experiments III. and IV. are plotted to the same vertical and horizontal scales. 59] ON THE ACTION OF WAVES AND CURRENTS. 487 (1) That the condition of Final Equilibrium in all cases with spring and neap tides ivas the same as that with uniform tides having the^same rise as springs, and much greater, essentially different, from that with a uniform tide //lining a rise equal to the mean rise of spring and neap tides. (2) That the Rate of Action with the varying tide is much smaller than that of a uniform tide having the rise of the spring tide. The ratios being definite, about 2'5 to 1. (3) That the limits of similarity obtained for all spring tides hold approximately for tides varying from spring to neap. 8. The effects of prolonging the rivers into the estuaries by walls below high water. Experiments V. in tanks E and F having arrived at similar final conditions of equilibrium (in which the depth of the rivers for some distance above their mouths was reduced to a 30-foot tide, nearly 30 feet at low water, while the sand in the estuaries gradually rose from the mouths of the rivers until it reached to within 12 feet of low water at a distance of 14 miles below the mouth and then fell again, all the sand being below this level, there being passes which formed a crooked deep water channel), opportunity was taken to prolong the banks of the river by walls at first up to low water and extending through the bar to a distance of 44 miles from the mouths of the rivers. Then raising these walls to half-tide, and finally carrying the walls forward slowly in tank E at a rate of half a mile a year (700 tides), and in tank F dredging from between the walls at a rate of seven million cubic yards a year (700 tides). This was done in the first place as a further test of the similarity of the action in the two tanks, and secondly as affording an interesting study as to the effect of vertical walls in the direction of the current in the bed of a tide- way. The effect of these walls at the level of low water and at half-tide were precisely similar in both tanks ; in neither case did they produce any sensible effect at all on the level of the sand between them. At the level of half-tide they caused in both tanks a slight silting up outside the walls and also a slight silting up in the river above its mouth, which effects were very much increased when the walls were raised to half-tide. On the walls being removed in tank E and then gradually carried forward, the silting up behind the wall and deterioration of the river increased, but there was no improve- ment in navigable depth between the walls. The dredging in tank F, so long as it was continued, added about 20 feet on a 30-foot tide or 10 feet on a 15-foot tide, to the navigable depth between the walls, but there was the same silting up behind the walls and the same deterioration in the river. It thus appears that the similarity of the results in both tanks supports 488 ON THE ACTION OF WAVES AND CURRENTS. [59 the conclusion that vertical walls having the horizontal direction of the current in a straight tideway and terminating well below high water, produce but little effect on the distribution of the sand between them, so long as the passage is freely open at both ends, but that if the passage be blocked at one end they form a bay in which the sand rises at the head. 9. The effects of the tide in estuaries not symmetrical. Having so far, in accordance with the original scheme of this investigation (First Report, 1889, p. 5), simplified the circumstances which influence the distribution of sand by maintaining the lateral boundaries perfectly symmetrical, and as nearly rectilinear as practicable, and having found definite laws connecting the distributions of sand in the beds of the model estuaries with the period and rise of the tide and the length of the estuary, besides the laws connecting the period of the tide with the horizontal and vertical scales under which the models give similar results, there remained two questions : (1) How far such discrepancies as appear between the general distri- butions of sand found in the models, and those observed in actual estuaries, are attributable to irregularities in the boundaries of the latter ? (2) How far the influence of these boundaries is subject to the same laws of similarity as those already obtained ? The original experiments of the author in models of the Mersey which led to the appointment of the Committee (see page 326) had to a great extent answered these questions, showing that similar irregularities in the lateral boundaries exercise similar and predominating influences on the lateral dis- tributions of the sand in the models and in the estuaries. It seemed, however, desirable, so far as time allowed, to confirm these results of the author's and make this investigation complete in itself, by carrying out experiments in both models similar to those already carried out, except that the boundaries should be boldly irregular. Such experiments also afforded opportunity for studying some general effects of great importance. The relation between the depths of water and the rise of tide had come out very definite in the symmetrical experiments, and it was desirable to see how far these relations would be disturbed by lateral irregularities. For instance: (1) Would bold irregularities in the boundaries of the estuary alter the depth of water in the river ? Bold irregularities in the boundaries, causing the water to take a sinuous course, would have the effect of virtually narrowing and increasing the length of the estuary, and by causing eddies would obstruct the passage of the water to some extent. Lengthening the estuary would tend to increase its depth 59] ON THE ACTION OF WAVES AND CURRENTS. 489 at corresponding points, and obstructing the water would tend to diminish the tidal action in the river ; at all events, until the estuary bad-increased in depth. (2) At the mouth of the estuary the flow of water had so far been straight up and down, and equal all across the estuary. By rendering the mouth unsymmetrical, circulation would be set up which would render the up-currents stronger at one part and the down-currents stronger at another, an effect which would correspond to some extent to that of tidal currents across the mouth of the estuary. (3) The large tidal sand ripples below low water in the model estuaries, with the flood and ebb taking the same course, constitute a feature which it is impossible to overlook, yet the existence of corresponding ripples had been entirely overlooked in actual estuaries, until, when they were looked for, they were found to exist, having been first seen in the models. The reason that they were overlooked before is, no doubt, explained by the fact that the bottom is not visible below low water in actual estuaries ; but this is not all. In the estuaries, these ripples, where found, have been confined to the bottoms and sides of the narrow channels between high sand-banks, and they do not occur on the level sands below low water towards the mouths of estuaries to anything like the same extent as in the models. By rendering the estuary unsymmetrical and so causing the ebb and flood to take different courses, this effect, as explaining the greater prevalence of ripples with sym- metrical estuaries, would be tested. These considerations led to the repetition of Experiment V. in tank F, at first with a single groin extending from the right bank into the middle of the estuary at the mouth, and subsequently to the introduction of three more groins from alternate sides of the estuary to the middle, up the estuary, and then to the introduction of similar groins into tank E, during Experi- ment VII., with spring and neap tides. The result of these experiments is to show conclusively : (1) That the laws of similarity found for symmetrical channels with uniform tides hold with sinuous channels for uniform or- varying tides. (2) That the greater uniformity of the depth of sand on cross section.* of models with symmetrical boundaries than with actual estuaries, does not exist when the banks are equally irregular. (3) That the circulation caused by the unequal flow of the tide in model estuaries tends greatly to take the sand out, and that the natural tendency in an cufuiiry to xntrp the boundaries so as to increase its sinuosities tends greatly to the deepening of the channels. 490 ON THE ACTION OF WAVES AND CURRENTS. [59 (4) That in the models with boldly irregular boundaries the tidal ripples are much less frequent than in the symmetrical models, being confined to places where there are no cross currents, as in actual estuaries. 10. Conclusion of the Investigation. It seems that the objects of this investigation have now been accomplished. The investigation of the action of tides on the beds of model estuaries has been found perfectly practicable. Two tanks have been kept running night and day from June 22, 1889, to August 1891, and have each accomplished upwards of 1,200,000 tides, representing the experience of 2,000 years. Such difficulties as protecting the sand from extraneous disturbance and keeping it free from fouling, regulating the levels of the water, the tidal periods, the rise of tide, forms of the tide curve and the supply of land water, observing and recording the results, have all been fairly overcome, though none of the precautions taken could have been safely dispensed with. The limits to the conditions under which the results will conform to the simple hydrokinetic law of similarity have been fairly established ; while above these limits the applicability of the simple hydrokinetic law to these experiments has been abundantly verified in models varying in scale from six inches to a mile to an inch and a half to the mile, and with vertical exaggerations, as compared with a 30-foot bide, ranging from 60 to 100. The laws of the distribution of the sand in a tideway under circumstances of progressing complexity have been determined, and have been verified, not only by repetitions of the same experiment, but also by producing similar distributions under different circumstances, which circumstances, however, conformed to the laws of hydrokinetic similarity. Thus the distri- butions of sand in simple rectangular estuaries, V-shaped estuaries, and V-shaped estuaries with a long tidal river, have all been investigated and found to be definite. Investigations have also been made, with definite results, of the separate effects of land water in moderate quantities, and of the length of the tidal river on the depth of water in the river and estuary, and of the effect of bold irregularities in the configuration of the lateral boundaries of the estuaries, also of training walls in deep water. And, lastly, the comparative rates and ultimate action of uniform tides, and tides varying from spring to neap, have been determined. It thus appears that this system of investigation has been tested over a great portion of the ground it is likely to cover, and that most of the difficulties that are likely to occur have been met, and the necessary pre- cautions found. 59] ON THE ACTION OF WAVES AND CURRENTS. 491 It would seem, therefore, by carefully observing these precautions, the method may now be applied with confidence to practical problems. - III. THE APPARATUS. 11. General Working of the Apparatus. All the apparatus has worked well, although certain repairs have been rendered necessary by wear; thus, the motor has required new pins, not much, considering it has made over 200 million revolutions. The knife edges, on which the generator of the large tank rests, which are of cast-iron, and 2 inches long, and each carry about 1,000 lb., were found to have, after one million oscillations, worn down ^ of an inch, until they had become so locked in the Vs as to stop the motor. 12. The modifications in the Tanks have this year been confined to the introduction of training walls and groins. These have been made of paper saturated with solid paraffin (which gradually became warped by the pressure), sheet zinc, and sheet lead or wood, as was most convenient. In the last experiment the large tank was modified by taking out the partition boards and stopping the opening at the end so as to reproduce the original rectangular tank A. iR. 1. 492 ON THE ACTION OF WAVES AND CURRENTS. [59 13. Gearing for the Spring and Neap Tides. This arrangement, de- signed by Mr Greenshields, accomplished the result very neatly and effect- ually with a minimum of new appliances. It admits of any degree of adjustment in the ratio of maximum and minimum tides, and works easily and well. On commencing the work with spring and neap tides it was found essential to have an indicator of the phase of the tide, which would be easily visible without having to examine the gearing. For this, a counter, having twenty-nine teeth in the escapement wheel, which carried a long finger over the face, was constructed by Mr Greenshields, and worked well, proving a great convenience. IV. DESCRIPTION OF THE EXPERIMENTS ON THE MOVEMENT OF SAND IN A TIDEWAY, FROM SEPTEMBER 4, 1890, TO AUGUST 1891. 14. Experiment III., Plan 1, Tanks E and F, Fig. 4, Page 507. These experiments were intended as a repetition of Experiments I. C and D. (Second Report, p. 429), which were only continued to 36,000 tides. The only difference in the conditions being that, while in Experiment I. the sand was initially laid up to the top of the river, Section 38, in Experiment III. the sand was only laid up the river to Section 13. These experiments were carried on during the vacation, Mr Foster kindly keeping the tanks running and reading the counters daily. In this way 47,000 tides were run in tank E, and 66,000 in F, when the surveys for Plan 1 were taken. These surveys show a rather more advanced state than is shown in Plan 2, Experiment I., but they present exactly the same characters. In tank E the sand in the estuary is slightly lower in the longer experiment than in the shorter, but shows the same distribution. In both experiments in tank E the level of the sand at the mouth of the river is that of mean tide, and in both experiments the level of the sand reaches the H.W.L. in the generator at Section 11, or 13 miles up from the mouth, and in both the tide continued to rise to the top of the river. In tank F, also, both experiments show the same general distribution of sand in the estuary and river. In the estuary the phenomenon, previously observed, with a low value for the criterion, namely, the large ripple, is more pronounced in the longer experiment ; but in both experiments the river has become barred at an early stage, showing that the conditions in F, during the formation of the estuary, have been below those essential for similarity. The rise of tide observed at the end of the Experiment III. in both E and F is below those observed at the earlier stages. In tank E the rise of 59] ON THE ACTION OF WAVES AND CURRENTS. 493 tide with the same rise in the generator has fallen to 0125 foot at 47,000 tides, though it was 0140 foot at 32,000; and in F it was 0*095 fooLat 66,000 against 0'096 foot at 32,000. This phenomenon, which becomes more pronounced in some of the later experiments, is accounted for by the im- proved tideway as the experiment gets older, allowing the estuary to empty itself more completely. It requires notice, since it renders estimates, such as the value of the criterion of similarity, based upon the rise of tide, difficult. The same quantity of water passes up and down the estuary, but does not effect the same rise of tide at the generator, which falls as the experiment gets older, while the rise of tide up the estuary increases at the same time. 15. Experiments on Increased Length of Tidal River. Experiments IV., E and F, with Land Water, Figs. 3, 5, 11, pp. 506, 508, 514, October 22 to November 17, 1890. The sand laid 0'333 foot in E, and 0187 in F from Section 13 up the river to Section 26 down the estuary. Mean rise of the tide, 0*310 in E, 0197 in f. Rise of the generators the same as before, periods 33'47 in E, 22'21 in F. The conditions were thus the same as in Experiment III., with the exception that the tidal periods were reduced in the ratio 1 to V2. As reduced to a 30-foot tide, this would have the effect of increasing the horizontal scales in the ratio \/2 to 1. Thus, while in Experiment III. the estuaries from generator to mouth of tidal river represented about 50 miles, and the rivers 54 miles ; in Experiment IV. the estuaries were 70, and the rivers 76. With the same tide at the mouth, the elongation of the estuary would cause the tide to rise higher at the mouth of the river, but as there was only the same quantity of water from the generator, the tides with the longer estuaries were smaller at the generators, which would again diminish the tides at the mouths of the rivers. The tides observed at the mouths of the rivers were somewhat higher than in Experiment III. And this fact must be allowed for in considering the results as representing the effect of increasing the lengths of the rivers on the distribution of sand. In tank E the effect was very remarkable. For the first 5,000 tides the sand rose up the river as far as it was laid, the head of the sand gradually going forward, and the sand falling at the top of the estuary and in the mouth of the river. Somewhat the same appearances appeared in tank F, though it soon became apparent that the advance of the head of the sand was much slower in F, and also the lowering of the sand at the top of the estuary. Sand was going up the river, but it accumulated in the lower reaches. 494 ON THE ACTION OF WAVES AND CURRENTS. [59 In E, at 9,000 tides, there was an almost sudden change ; the sand in the river was rapidly earned to the top, leaving the lower reaches empty. After 11,000 tides the bottom of the river was swept clean from the mouth to Section 15 (30 miles), and then a steady downward movement of the sand went on, all down the estuary, until there was deep water all the way down from 10 miles below the head of the river. The clearing of the bottom of the river of sand evidently increased the action of the river, increasing greatly the rise of tide. In tank F the result was very different ; instead of the sand shifting suddenly up the river, the sand reached Section 15, and then barred the river at Section 11, the river then gradually filling up. At 38,000 tides, when the second survey was made, the tide was still rising at the top of the river, and the head of the sand still proceeding forwards. The experiment was continued to 81,000 tides, and the head of the sand reached Section 19, the tide still rising at the head very slightly. This shows that the conditions of similarity were more nearly fulfilled in the river in tank F in this experi- ment than in III. The values of the criterion, however, given in the table, are lower in IV. than in III. This is because these values are calculated from the rise in the generators, which were in these experiments O'llO in tank E, and 0'081 in F, against 0'125 and 0'095 in Experiments III. With the same water going out of the generator there must have been higher tides at the mouths of the rivers in IV., and as the vertical exaggeration in Experi- ment IV. was \/2 times larger than in I. and III., assuming the rise of tide in tanks E and F, Experiments III. and IV., to be as in Experiments L, the values of the criterion in Experiments IV. would be at least 0'261 and 0103. This is in accordance with the observed results. It seems therefore that in order to apply the criterion to the conditions of similarity at the top of a long estuary with a tidal river, the actual rise of the tide at the mouth of the river should be taken in estimating the value of the criterion for similarity at these points. It appears, however, that in every case where the criterion, estimated from the tides in the generator, exceeded the value '09, the conditions of similarity have been fulfilled, while in no case has it fallen decidedly below this value without decided symptoms of dissimilarity having appeared, so that this value for the criterion seems to be established as a good working rule for the formation of an estuary from sand at the level of half-tide. If the bottom of the estuary is modelled the case is different, but the occurrence of large ripples, in experiments in tank F and in Experiment V. in tank E, when the value of the criterion fell as low as '08, shows that the similarity of the ripple depends on the same value of the criterion as the formation of the estuary. 59] ON THE ACTION OF WAVES AND CURRENTS. 495 16. Experiments with Limiting Value of Criterion. Experiment V. with Land Water, Tank E, Figs. 6, 7, and 12, pp. 509...,/?-ow November 20 to December, 24, 189(). : The conditions of this experiment were designed to bring the value of the criterion, estimated from the rise of tide in the generator in the final condition of equilibrium, to 0'09, keeping the horizontal scale as nearly as possible the same as in IV., and diminishing the rise of tide so as to increase the proportional depth of sand in the river, and thus prevent the bottom being swept clean when the final condition was reached. The length of the crank working the generator in IV. had been 4'437 inches; this was reduced to 377 inches in V., reducing the rise of the tide in the ratio 0'85. To keep the horizontal scale the same the period 33'3 seconds was increased to 36 seconds, leaving the product p ^h constant. This reduced the vertical exaggeration e in the ratio 0'85. Thus the value of h*e is reduced (0'85) 4 or 0'52. Now the value of the criterion in Experiment IV., just before the bottom was swept with sand, was greater than 0'18, which, multiplied by 0'52, gives 0-093. As carried out at the final condition shown in Plan 3, Page 510, the period was 35'6 seconds, the rise of tide 0107, and the value of the criterion 0'0912. This low value of the criterion showed itself in the rate of progress of the experiment. It was 13,000 tides before the sand in the river reached Section 19, against 4,000 in Experiment IV., and 25,000 against 9,000 in IV. before reaching the head of the river. In the early stage of the experiment it seemed doubtful whether the sand was going to bar the river as in Experiment IV., tank F. Except in rate of action, however, the motion of the sand followed the same course as in Experiment IV., taking a sudden shift at about 20,000 tides, and then rapidly lowering the sand at the head of the estuary. At the mouth of the river the bottom of the tank was reached after 50,000 tides, but only between the ripple bars, so that it was not swept clean. The ripples in this experiment were very much larger than anything before in tank E, showing that the criterion was approaching its critical value. The final condition of the estuary, as shown in Plan 3, after 36,000 tides, shows conclusively the effect of the upper tidal water in a long river on the bed of the lower estuary. Below Section 19, 32 miles from the top of the river, there is no sand above the level of low water in the estuary, and from this the sand falls uniformly to the mouth of the river, where there is a depth of water, at low tide, of 30 feet. In the head of the estuary there is 496 ON THE ACTION OF WAVES AND CURRENTS. [59 a bar the top of which is only 12 feet below low water ; this is at Section 9, or 18 miles below the mouth of the river ; below this point the sand gradually falls to the generator. Comparing this with the results in Experiments I. and III., where the reduced length of the river is only some 50 miles, but in which the rise of tide at the mouth of the river was somewhat greater, the effect of the extra 20 miles length in the river is seen to have improved the general and navigable depth of the river and estuary, from the top of the river to a distance of 40 miles down the estuary, by from 40 to 30 feet. 17. The effects of dredging in the river, Experiment V., in Tank F,from November 19 to December 23, 1890, Plan 3, Page 510. The initial conditions of this experiment were the same as those of Experiment IV. in tank F, except that the mean level of the tide was raised to O'OIG above the initial level of the sand, and the period was increased from 22 to 23'3 seconds. The experiment was undertaken with the intention of ascertaining (1) whether raising the mean level of the tide above the initial level of the sand, without altering the rise of tide, would prevent the river becoming barred ; and, supposing this did not succeed, (2) to ascertain whether, if the bar, which had hitherto formed in the river during the early stages of the experiments in tank F, were kept down by dredging out the sand as it rose above the initial level, the later stages would follow the same course as in tank E. The results were remarkable, and bring out the critical character of the conditions at the mouth of the river. The experiment was allowed to run 30,000 tides, during which the progress of the sand was much more rapid than in IV., reaching Section 19 in 6,000 tides, as against 36,000 in Experiment IV. and 13,000 in Experiment V, E., and reaching Section 23 in 16,000. At this point it stuck, and the sand accumulated at the head of the estuary and in the river, which became barred at Section 19, on reaching 30,000 tides. It thus appears that lowering the initial level of the sand produced an effect on the first action very nearly equal to increasing the rise of tide by double the amount, but that as the sand distributed itself this effect passed off. At 30,000 tides the bar in the river was dredged down to the initial level of the sand, and this level was maintained by daily dredging till 70,000 tides had been run, 0'08 cubic foot of sand in all being removed. At this stage the sand in the river suddenly shifted up to the top as in Experiments IV. and V., E. The sand at the mouth of the river and top of 59] ON THE ACTION OF WAVES AND CURRENTS. 497 the estuary falling until the bottom appeared, dredging was discontinued. At 95,000 tides the final condition had been reached, which _yvas almost identical over the whole estuary with that of Experiment V. E after 60,000 tides, as shown in Plan 3, Experiment V., E and F. The instability of the condition which may prevail at the mouth of a river is thus clearly shown, as well as the useful effect of improving the tideway by dredging in the upper reaches in the river. In three experi- ments in tank F, I., III., and IV., the river became completely barred, and the estuary became a bay with a stream of land water entering at its top; in Experiment V. the bar again formed, but on being kept down, by dredging, to the level of half-tide, till the sand had fallen at the head of the estuary, the river at length prevailed, and the sand was washed out till there was 30 feet of water at low tide. The time occupied and amount of sand removed, in producing this effect, were considerable. The tidal capacity of the river and estuary is 1 cubic foot; this reduced to a 30-foot tide is 21,700 million cubic yards, or on a 15-foot tide is 5,422 million. The amount of dredging, 0'08 cubic foot in all, represents 1,743 million cubic yards on a 30-foot tide, or 437 millions on a 15-foot tide. This was distributed over 40,000 tides, or sixty years, so that even with the 15-foot tide it would represent 7 million cubic yards a year. After the dredging the rise of tide fell from '081 to '073 foot, which would result from the lowering of the sand which was above low water. 18. Experiments with Training Walls. Experiment V. (continued} with Training Walls, Tanks E and F, from January 7 to February 20, 1891, Plan 4, Page 516. Having arrived at similar final conditions of equilibrium in tanks E and F, in which the sand was entirely below low water from Section 19 up the rivers (32 miles from the top of the river) to the generators, and iti which there were bars in the estuary below the mouths of the rivers, reducing the depth of water at low tide from 28 feet in the river to a minimum of 12 on the top of the bars, it seemed an opportunity not to be lost for testing the similarity of the effect in the two tanks of prolonging the rivers by training walls through the bars. With this view, walls of thick paper saturated with paraffin, pushed vertically into the sand and extending up to low water, were run out from the end of the river, preserving the same divergence as the walls of the river to Section 22, or 40 miles on a 30-foot tide, the tanks being stopped for the purpose. These walls produced no apparent effect whatever on the depth of sand between the walls, during 20,000 or 30,000 tides. They were then replaced o. R. ii. 32 498 ON THE ACTION OF WAVES AND CURRENTS. [59 at the upper end by walls of sheet zinc extending to half-tide, which did produce an apparent effect, inasmuch as the sand accumulated outside the walls, forming an apparent channel within ; also the sand rose in the river, doing away with the appearance of a bar. These effects were similar in both models after 40,000 tides had been run. The old walls were removed in both tanks and replaced by walls com- mencing at f tide at the mouths of the rivers, and falling during the first 4 or 5 miles to half-tide, at which they were continued to Section 22. In tank E the walls were advanced gradually from the mouth of the river at a rate of about half a mile in 700 tides (a year). The result of this is shown in Plan 4, Page 516, tank E. There is no improvement in the navig- able depth of the river. In tank F the walls were put in and then the tops of the ripple bars were daily dredged off between the walls. This was continued for 10() ; 000 tides, during which 5 per cent, of the tidal capacity was removed, or about 1,000 million cubic yards on a 30-foot tide, or 250 millions on a 15-foot tide, which represents 7 millions annually on the 30-foot tide, or 1*8 millions on a 15-foot tide. The effect, as shown in Plan 4, tank F, Page 516, is to add some 20 feet to the depth on a 80-foot tide, or 10 feet on a 15-foot tide. The silting up behind the walls is the same as in tank E, and the detriment to the navigable depth of the river is also similar. 19. Experiment V. (continued] with Tide deviating from the Simple Harmonic in Tank E, February 23 to March 12, 1891. This was meant as a preliminary experiment. The balance of the generator was altered to give a rise of tide in 17 seconds and a fall in 20. The experiment was run for about 40,000 tides, and a survey taken, which showed little or no effect. On carefully examining the tide curves it was found that they showed very little inequality in the rise and fall. On attempting to increase this by further altering the balance, it was found that this could not be done. To continue this part of the investigation it would have been necessary to introduce complex gearing. Time did not suffice for this, and the study was not carried further. 20. Experiments with Tides varying from Spring to Neap, Tank E, V., VL, VII., VIII., Tank A, XIII. Figs. 11, 12, 13, 15, pp. 514..., March 20 to August 1891. The gearing for tank E having been modified so as to cause a rise in the generator, varying to over an interval of 29 tides, the variation being harmonic and adjustable, so as to admit of any relation between the maximum and minimum rise. These were adjusted so that the mean rise was the same as the rise in 59] ON THE ACTION OF WAVES AND CURRENTS. 499 Experiment V., the spring and neap rises being in the ratio 3 to 2. A drain with an adjustable orifice was put in the bottom of the tank-to-drain off nearly all the fresh water, and the scummer adjusted so as to draw off the excess of land water at low spring tide level ; this being adjusted by trial until, when running, the mean tide level was the same as before. Experiment V. was then restarted, without the sand having been dis- turbed, to afford a preliminary trial of the apparatus, the period being that of Experiment V., 36 seconds. This was continued 18,000 tides, till the apparatus was completely in hand ; then the sand was relaid for Experiment VI., Fig. 11, Page 514, in which the conditions were the same as V., except the tide. The mean rise in the generator was the same in VI. as in V., and the ratio of the spring and neaps 3 to 2. This brought the rise in the generator at spring tides in VI. greater than that in Experiment IV., in the ratio of 11 to 1. The action on the sand was much more rapid than in Experiment V. with the uniform tide, being nearly as quick as in IV. The sand reaching the top of the river in 13,000 tides, as against 10,000 in IV. and 25,000 in V., and the bottom of the river being swept as clean in 17,000 tides in VI., as in 14,000 in IV. In other respects the action in VI. very closely resembled that in IV. The rate of action was a little slower, but the action itself seemed rather stronger, as corresponding to a higher tide. Surveys were taken at 20,000 and 34,000 tides. The experiment was then stopped, in order to make the conditions comparable with those of Experi- ment V.; it being quite clear that the action of spring and neap tides, having a mean rise equal to that of a uniform tide, was not only much more rapid, but led to a different state of final equilibrium. Experiment VII., Plan 1, Page 515. In this the tide was adjusted until the rise of the generator at spring tide was the same as that for the uniform tide in V., the other conditions being all the same. The character of the action now became identical with what it had been in Experiment V., but the rate was decidedly slower. Thus the sand moving up the river reaches : Section 19 after 13,000 in V. and 39,000 in VI. 27 after 20,000 51,000 in VI. The survey taken after 18,000 tides in Experiment V., Tank E, and 51,000 VII., are almost identical, the latter being a little the forwardest. It thus appears that spring and neap tides, having a ratio 3 to 2, produce the same result as two-fifths the same number of tides all springs. 322 500 ON THE ACTION OF WAVES AND CURRENTS. [59 So far neither of these estuaries had reached the condition of final equi- librium, but the similarity that the Plans 1, Experiments V. and VII. present, seemed sufficient assurance that this would be the same. It was intended to repeat Experiment V., tank A, as soon as the tank had been re-formed to its rectangular shape ; in the meantime groins were introduced in tank E similar to those which had been used in Experiment VI. F, and Experiment VII. E was continued, to ascertain how far similar effects would be produced by varying and uniform tides in estuaries with similar but boldly irregular outlines. Experiment VII. E, Plan 2, Page 517 was continued with groins to 123,000 tides. Similar groins had affected the condition of the sand in the estuary and river in Experiment VI., tank F, so that further comparison between Experiments VII. and V. cannot be made. Experiment XIII., Tank A, rectangular without land-water, spring and neap tides, Plan 3, Page 512, from July 10 to August 10, 1891. In this experiment the rates of spring and neap tides were 3 to 2, and the rise of tide at spring tides was 0*176, the same as in Experiment V., tank A. The tank was reduced to its original rectangular form (Report I.), namely, 4 feet broad, and 12 feet from the generators to the top. The sand was laid as in Experiment V., tank A, at a depth of 2 in. from Section 18 to the top of the tank, and the mean tide was adjusted in Experiment V., tank A. The period was 50 seconds, as in tank A. Thus the conditions of Experiment XIII. and V., tank A, were precisely the same, with the exception that while the tides in Experiment V. were all springs, those in Experiment XIII. varied from springs to neap; the object of Experiment XIII. being to compare the rate of action and final condition of equilibrium with varying tides, with the very definite results, as to the slopes of the sand, obtained in the rectangular tanks, and recorded in Report I., B. A. Report, 1889 (see page 380). These results are shown in the plans on page 510. The period in Experiment XIII., tank A, being shorter than in V. The actual slope is greater, but the slopes reduced to a 30-foot tide agree. 21. Experiments on Estuaries not Symmetrical. Experiment VI., in Tank F, with large groins, Plans 1 and 4, Pages 517, 518, from April 8 to June 16, 1891. This experiment was started under conditions in all respects similar to those in Experiment V., tank F, with the exception of a vertical groin extending from the right bank to the middle of the estuary, with an inclination of 45 towards the generator, and rising from the bottom of the tank above high water. This groin, which appears in the charts to represent an artificial structure, is, in fact, out of all proportion to anything 59] ON THE ACTION OF WAVES AND CURRENTS. 501 of that kind which has yet been attempted. As reduced to a 30-foot tide, it is 11 miles long, 100 feet high up to H.W.L., and half a mile broad. Thus it corresponds rather to such a natural feature as Spurn Head, at the mouth of the Humber, than to a breakwater such as that at Harwich. In starting the experiment, the end of the sand at Section 26 was 20 miles above the point of the groin at Section 36. The groin had deep water on both sides of it, so that its only effect was to deflect the flood on to the left bank of the estuary. This effect was very decided, the strength of the flood on the right carrying the sand up the estuary in spite of the effect of the ebb to bring it down. But this in itself was not so much ; it was the large eddy caused by the groin which produced the greatest effect. The water entering on the left of the estuary crossed over to the right, and returned along the right bank. In other words, during flood the right side of the estuary for 30 miles from the generator was in back water. This back water also gave the ebb a start down the right bank which rendered the ebb stronger on this side. The sand came down rapidly on the right side, and besides was carried H\< r from the left to the right, and formed a bank along the right middle of the estuary, reaching the generator after a very few tides. Round this bank the water circulated, carrying the sand with it up on the left and down on the right, the bank growing all the time. The ripple round this bank was very striking, arranged with the ripple heads all down on the right side and up on the left. After about 3,000 tides the sand began to pass from the point of this bar in a fine stream across the open channel, dividing this point from the point of the groin, and commenced the formation of a bank in the generator corresponding to that in the tank. This bank had to be removed from the generator, and after 6,000 tides 4 Ibs. of sand were so removed. In Experiment V. the first sand removed from the generator was after 120,000 tides had been run. The sand also went more rapidly up the river in Experiment VI. than in Experiment V. But this was accounted for by dredging in the river having begun much earlier, after 20,000 tides as against 30,000. In all 8 Ibs. of sand were removed from the river in Experiment VI., against 10 Ibs. in V., or about 0'004 of the tidal capacity in VI. against 08 in V. In both cases the dredging stopped when the sand began to shift up the river after 70,000 tides. At 100,000 tides a condition of final equilibrium had been arrived at. The sand in the river was just the same as in V., Plan 3, Experiments V. 502 ON THE ACTION OF WAVES AND CURRENTS. [59 and VI. in tank F. There is deep water in VI. up to Section 21, 30 miles from the generator, the levels of the sand being much the same from this point up as in V. A similar groin was then introduced at Section 16, extending from the left bank to the middle of the estuary. This groin was 4| miles long and 100 feet high to H.W.L., and 50,000 more tides were run, the river all the time slightly improving. Thus having brought deep water up to Section 14, or about 44 miles from the generator, a groin extending from the right bank to mid-channel at Section 8, about 2'5 miles long and 70 feet high, and another from the left bank to mid-channel at Section 5, 2 miles long and 70 feet high, were put in. The first effect of these groins was to raise the sand slightly in the mouth of the river; but this improved again, and after 50,000 more tides there was deep water extending from the mouth of the river to the generator, and the river was better than in Experiment V. with the training walls, though not quite so good as before these were put in. In the meantime the banks had risen in the estuary below the groins, extending down from nearly H.W.L. to the point of the next groin, where there was a pass with water nearly to the bottom of the tank. The sand carried down into the generator during the experiment amounted to 69 Ibs., or 57 per cent, of the tidal capacity. In Experiment V. 24 Ibs. were removed in like manner, or 20 per cent, of the tidal capacity. 37 per cent, of the tidal capacity on a 30-foot tide would represent a mean increase of depth over the entire estuary of 1 1 feet ; and as the increase was by no means over the whole estuary, the increase in the channels and lower estuary was much more than this, and although by this time the sand in the estuary had for the most part become quite yellow, sand was still being carried down into the generator. In the meantime, as already stated, groins similar to those in Experi- ment VI. in tank F, had been introduced into experiment VII. in tank E, after 64,000 tides had been run with spring and neap tides. 60,000 more tides, which would be equivalent to about 27,000 spring tides, were run, the effect being that, notwithstanding the difference in the initial condi- tions, the state of the lower estuary was closely approximating to the state of VI. in F after 36,000 tides (Plan 2, Experiment VII, tank E; VI., tank F). In the upper estuary in VII., tank E, the distribution of the sand is precisely similar to that in VI., tank F, but there is rather more of it, which is explained partly by the fact of the difference in the equivalent 59] ON THE ACTION OF WAVES AND CURRENTS. 503 tides run, 30,000 in E as against 50,000 in F, after the upper groins were put in, and partly by the much greater amount of sand still leftinlhe lower estuary in tank E. Had it been possible to run 250,000 more spring and neap tides in VII, tank E, there is every reason to suppose that the final condition would have been precisely similar to that obtained in Experiment VI. in tank F. TABLE I. GENERAL CONDITIONS Per- Horizontal scale Shape of the Estuary cent- nge of Land Water Experi ment Tank Plan Plan Oil Page Shortest period in seconds Vertica scale Inches 1 in. to a mile 50 _; 0-5 III E 1 507 46-16 14,901 4-25 240 miles 1 V n F 1 507 30-53 25,844 2-45 315 ' 11 IV E 1 514 33-47 20,550 3-01 240 11 11 F 1 22-20 38,256 1-65 365 n 11 E 2 508 33-20 22,090 2-78 273 11 11 F 2 508 22-03 38,788 1 -f>3 370 11 V E 1 515 35-6 19,558 3-24 246 11 F 1 509 23-68 36,310 1-74 375 11 E 2 509 35-6 19,972 3-172 256 "3 11 11 F 2 23-32 36,890 1-718 375 H 11 11 E 3 510 35-60 20,833 3-03 280 C o ^3 11 11 F 3 510 23-32 38,955 1-63 416 ** g Train- I 11 11 E 4 516 35-60 20,691 3-06 275 B ing | 1 o Walls ( 11 11 F 4 516 23-32 39,700 1-60 435 W 1 Quick ( rise 11 11 E 5 35-60 21,285 2-97 291 I f 11 11 11 6 35-60 19,095 3-318 234 op 11 VI 11 1 35-78 18,230 3-475 215 Spring and 11 5) 11 2 514 36-26 20,000 3-168 252 Neap Tides 11 VII i 1 515 35-10 19,756 3-207 244 1 " 11 2 517 35-10 20,890 3-033 273 , 1 11 11 4 518 Unsym- 11 VI F 1 517 23-40 39,564 1-605 434 metrical 11 11 2 23-40 38,465 1-647 411 11 11 n 3 23-40 39,854 1-589 411 1 \ 11 11 11 4 518 23-40 39,280 1-613 428 , fe ( Spring and = 1 Neap Tides o-o XIII A 3 512 48-00 12,473 5-08 182 p g> I Uniform I Tides 11 V " 1 511 50-00 11,758 5-45 170 AND RESULTS OF EXPERIMENTS. Else of tide in feet Vertical exaggera- tion on a 30-feet tide e Criterion of similarity Height of initial sand in feet Height of mean tide in feet Excess of mean tide over initial sand in feet d Number of tides from the start Remarks C'=h 3 e C' = (h + 2d)e 0-125 62-00 0-121 0-333 0-322 47,183 Normal. 0-095 81-84 0-070 0-070 0-187 0-187 66,369 River blocked. 0-125 85-63 0-167 0-333 0-310 18,530 Eiver cleaned. 0-082 104-57 0-057 0-187 0-182 0-005 21,135 River blocking. 0-110 80-98 0-108 0-333 0-308 37,755 River cleaned. 0-081 104-73 0-056 0-187 0-179 0-008 38,719 (River nearly \ blocked. 0-122 79-53 0-144 0-333 0-336 17,923 Slow. 0-080 96-82 0-049 0-187 0-203 0-016 19,416 Quicker. 0-117 77-88 0-124 0-333 0-321 37,359 River cleaned. 0-080 98-32 0-050 0-165 0-187 0-203 0-016 37,181 ^Blocking ( Dredged. 0-107 74-48 0-091 0-333 0-320 65,404 River clear. 0-072 93-49 0-035 0-187 0-207 0-020 95,558 River clear. 0-109 0-069 75-18 91 -32 0-097 0-030 : 0-333 0-187 0-306 0-204 0-017 167,186 255,200 [Similar. 0-103 73-08 0-080 0-333 0-335 208,264 Failure. 0-128 81-47 0-170 0-333 0-328 226,930 Preliminary. 0-139 84-46 0-2268 0-333 0-325 20,822 Quick. 0-119 79-33 0-1336 0-333 0-317 34,394 River clear. 0-123 81-00 0-1507 - 0-333 0-333 51,591 Normal. 0-110 76-60 0-1017 0-333 0-332 101,790 122,989 0-069 91-01 0-0299 0-187 0-192 0-005 18,972 0-073 93-60 0-0360 0-187 0-193 0-006 36,511 0-068 93-33 0-0284 0-187 0-193 0-006 99,558 0-070 91-66 0-0314 0-187 0-192 0-005 196,651 __ 0-165 0-176 IJS-.-i 1 69-16 0-3084 0-3769 0-250 0-250 51,240 16,282 f Similar. 506 s s 507 I !* , ' 508 : : r is 509 510 ii * I 511 MJy? /turyr yyiay jnoyutv 512 513 0. R. II. 33 514 -Ult il 515 > 332 516 i#R 517 =1 d < jfiSja If 518 s 60. ON TWO HARMONIC ANALYZERS. [From the Fourth Volume of the Fourth Series of " Memoirs and Proceedings of the Manchester Literary and Philosophical Society." Session 1890 91.] (deceived April 2nd, 1891.) THE object of these instruments is to afford a ready means of ascertaining the periods of free vibration of structures or members of structures. If any portion of a material structure (i.e., an elastic structure) is disturbed from its normal position of equilibrium and suddenly released, the structure is thrown into a complex state of vibration, which gradually subsides. While the vibration lasts each point in the structure goes through movements which may be very complex, but which are, nevertheless, compounded of simple periodic or harmonic movements, each simple movement taking place in a definite direction, as well as having a definite period. The art of measuring and recording the complex movements at a point of the earth during an earthquake has long been a study, and the seismometer of Professor Ewing has been applied to record the movements of points of various structures when subjected to disturbances. The principle of these seismometers consists in attaching a weight to the point of the structure to be examined, by attachments of such slight elasticity, that the disturbances communicated to the weight are insensibly small, and the weight remains sensibly steady amid the surrounding vibrations, and forms a steady observa- tory from which the vibrations may be measured. This measurement is effected by causing pencils vibrating with the structure to describe lines on cards attached to the steady weight, or vice versa, the cards being fixed, or having a time movement. In this way the complex motions of the points are beautifully recorded, as in Prof. Ewing's experiments on the Tay Bridge, and Prof. J. Milne's numerous experiments in railway carriages, &c. Such curves represent the complex movements of the point of the structure examined ; and any analysis of the motion into its simple periodic components remains to be accomplished by mathematical reduction or by 520 ON TWO HARMONIC ANALYZERS. [60 such instrumental synthesis as that which may be effected in Sir William Thomson's " Harmonic Analyzer." The Harmonic Analyzers about to be described differ essentially from the seismometer in that they do not measure or record the actual motions of the structure, while they single out and exaggerate any component periodic motion according to its period and direction, which are defined in the instru- ments. The principle of these Harmonic Analyzers is that of the accumu- lation of motion which takes place, when a weight is subject to a periodic disturbance which coincides in period and direction with that of free vibra- tion of which the weight is susceptible. If a small weight w be elastically attached to a much heavier weight so that it requires a definite force (El) to disturb the weight (w) through a distance I, the large weight remaining at rest ; then, if released after any disturbance, the small weight w will vibrate in the direction of disturbance, and with a constant period of: A / V ^r seconds, i.e. in the period of free vibration of the small weight. If the small weight be at rest and the large weight be subject to a periodic disturbance having a period 1/w; then, if this period is larger than the period of free vibration of the small weight, i.e., if - is smaller than 2-n- A / - n V gE' the small weight will follow essentially the movements of the larger weight as if rigidly attached, while if the period of motion of. the larger weight is smaller than that of the period of free vibration of the small weight, the small weight will remain virtually at rest. But when the period of motion of the large weight coincides with the period of free vibration of the small weight, the small weight will take and accumulate the disturbance, oscillat- ing with increasing amplitude until it reaches such an extent that the energy dissipated is equal to that received from the disturbance. If the elasticity of the connections be fairly perfect, the amplitude of the small weight will be very considerable, although the disturbing motion is otherwise insensible. If the small weight (w) has only one degree of freedom, i.e., if the elasticity of the connections is not equal in all directions, there will be three axes of elasticity, and if the elasticities along two of these directions are much greater than the third this is the direction of freedom ; then, when the period of free vibration along the third axis, i.e., in the direction of freedom, coincides with the period of disturbance, the small weight will only take up 60] ON TWO HARMONIC ANALYZERS. 521 the disturbance when this has a component in the direction of freedom ; that is, if the direction of the disturbance is at right angles to the direction of freedom, there will be no vibration. So that in this way the direction of the disturbance may be ascertained, or vice versa. Similar results follow if, instead of the disturbance coming through the elastic supports, the body be subject to a synchronous periodic force. If the period of the force were not synchronous with any of the three periods of free vibration corresponding respectively to the three axes of elasticity, the resulting vibration would, as before, merely correspond with the time effect of the force, but on coincidences with any one of these, unless the direction of the disturbance were at right angles to that of the axis of elasticity, the body would accumulate the disturbance. It thus appears that, if a structure is in a state of vibration, the periods of free vibration and their directions may be ascertained by an Harmonic Analyzer consisting of a small weight with elastic attachments, so adjustable that the period of free vibration of the weight can be varied to any required extent, and the direction of such free vibration turned through all requisite angles. This may be accomplished in many ways. That which I have so far adopted with satisfactory success has been very simple. 2 l 6i l 5"l l 6' Fig. 1. It consists, as shown in Fig. 1, essentially of a base formed of a bar of hard wood, one-and-a-half inches square, and two feet long, a cross notch being cut in one end to enable this end to be held against any point of the structure with less chance of slipping. About four inches from the notched end, right across the axis of this bar, is a hole, in which is fitted, with moderate tightness, a piece of straight steel wire, one-eighth of an inch in diameter, and 18 inches long. On one end of the wire is a ball of lead, 522 ON TWO HARMONIC ANALYZERS. [60 about 2 oz., through the centre of which is a small hole at right angles to the wire, in which is fixed a small graphite pencil. On the other end of the wire is a carrier, to afford handhold for the purpose of adjusting the wire in the hole. When the carrier is pushed right up to the wood, the ball, if disturbed, will vibrate in any direction perpendicular to the wire so as to make about 200 oscillations a minute, which is slower than any period it is required to measure. As the carrier is pulled back, and the wire between the base and the ball shortened, the rate of vibration increases, until, when the wire is only 1^ inches long, the ball, when disturbed, gives out an audible note of about 2,000 vibrations a minute. The instrument is used by holding in one hand the longer end of the wood, and pressing the notched end hard against the point of the structure of which the motion is to be analyzed, the carrier having previously been pushed up to the wood, then, with the free hand, the carrier is pulled steadily back, the ball being carefully watched. As by the shortening of the wire between the base and the ball the free period of vibration of the ball is diminished, and comes near to any period amongst the vibrations in the structure, the ball is seen to take up the vibration in beats with intervals of rest; and a very little more careful adjustment is sufficient to bring the period into coincidence, when the ball continues vibrating with the structure, having the appearance in Fig. 2 The period of the Analyzer having been thus adjusted to that of one of the periods of free vibration of the structure, the period is ascertained either by adjusting the Analyzer so that the pencil in the ball may oscillate in con- tact with the paper on a chronograph, or by measuring the distance of the ball from the wood on a scale, previously adjusted by aid of the chronograph to give the number of vibrations per minute. Extreme accuracy of determining the periods has not so far been an im- portant consideration. The readings on the chronograph were only taken to about 10/ . But that the Analyzer is susceptible of much greater accuracy is shown by the fact that several different adjustments to the same period in the structure brought the wire into exactly the same position. 60] ON TWO HARMONIC ANALYZERS. 523 Its power of analyzing complex vibrations is so far unqualified. It was invented for the purpose of determining the period of a particular vibration in a very stiff iron structure subject to the periodic disturbance of the belts from two engines running at high speed, and the centrifugal action of such want of balance as there might be in heavy pulleys, three feet in diameter, and running at 500 revolutions per minute. The vibration was very slight nothing more than a slight tremor could be felt with the hand. The periodic disturbances were about 500 per minute, and these came out clearly, but small, in the Analyzer when adjusted to these periods but the periods of free vibration of one of the members, 720 per minute, caused an am- plitude of half an inch in the ball, and that of another, 1,270, was easily identified. The instrument already described can clearly only be used on a structure while it is so disturbed as to set its members vibrating. Such disturbances can generally be set up by a shock of some sort, but when it is necessary to cause artificial disturbance, it is better to adopt a periodic disturbance of such varying period as will come gradually into coincidence with the periods of free vibration, bringing these vibrations out separately, when they will be readily identified with the Analyzer, if not otherwise perceptible. For this purpose, in 1887, I adopted the following method: A small cast-iron pulley, 6 inches in diameter, very much out of balance, was mounted on a small frame that could be clipped on to any part of the structure, and a cord passed over this pulley on to a larger wheel, which was turned by hand. In this way the unbalanced wheel was driven at a gradually increasing rate until steady vibrations in the structure were observed, then these coincided with the period of the unbalanced wheel, and this was ascertained to be about 1,200 by counting the revolutions of this hand-wheel. At this speed the disturbing force resulting from the un- balanced weight, 2 Ibs. on a radius of 2 inches, would be 40 Ibs. The structure thus under examination was an iron standard, very stiff. A theodolite was adjusted, with the cross wires on a mark on the top of the standard, which, when the period of the small unbalanced wheel coincided with that of free vibration, was seen to move as much as one-twentieth of an inch. Chains were then attached to the top of the standard, and by means of blocks, a horizontal force of a ton was thrown on to the top of the standard, when it did not yield more than two-hundredths of an inch. So that the deviation caused by the periodic force of 40 Ibs., in such coincidence with the period of free vibration as could be attained with the hand-wheel, was three times as great as that which resulted from a direct statical force of one ton. 61. STUDY OF FLUID MOTION BY MEANS OF COLOURED BANDS. [From the " Proceedings of the Royal Institution of Great Britain."] (Read June 2, 1893.) IN his charming story of The Purloined Letter, Edgar Allan Poe tells how all the efforts and artifices of the Paris police to obtain possession of a certain letter, known to be in a particular room, were completely baffled for months by the simple plan of leaving the letter in an unsealed envelope in a letter-rack, and so destroying all curiosity as to its contents ; and how the letter was at last found there by a young man who was not a professional member of the force. Closely analogous to this is the story I have to set before you to-night how certain mysteries of fluid motion, which have resisted all attempts to penetrate them are at last explained by the simplest means and in the most obvious manner. This indeed is no new story in science. The method adopted by the minister, D, to secrete his letter, appears to be the favourite of Nature in keeping her secrets, and the history of science teems with instances in which keys, after being long sought amongst the grander phenomena, have been found at last not hidden with care, but scattered about, almost openly, in the most commonplace incidents of every-day life which have excited no curiosity. This was the case in physical astronomy to which I shall return after having reminded you that the motion of matter in the universe naturally divides itself into three classes. 1. The motion of bodies as a whole as a grand illustration of which we have the heavenly bodies, or more humble, but not less effective, the motion of a pendulum or a falling body. 61] STUDY OF FLUID MOTION BY MEANS OF COLOURED BANDS. 525 2. The relative motion of the different parts of the same fluid or elastic body for the illustration of which we may go to the grand phenomena pre- sented by the tide, the whirlwind, or the transmission of sound, but which is equally well illustrated by the oscillatory motion of the wave, as shown by the motion of its surface, and by the motion of this jelly, which, although the most homely illustration, affords by far the best illustration of the pro- perties of an elastic solid. S. The inter-motions of a number of bodies amongst each other to which class belong the motions of the molecules of matter resulting from heat, as the motions of the molecules of a gas, in illustration of which I may mention the motions of individuals in a crowd, and illustrate by the motion of the grains in this bottle when it is shaken, during which the white grains at the top gradually mingle with the black ones at the bottom which interdiffusion takes an important part in the method of coloured bands. Now of these three classes of motion, that of the individual body is incomparably the simplest. Yet, as presented in the phenomena of the heavens, which have ever excited the greatest curiosity of mankind, it defied the attempts of all philosophers for thousands of years, until Galileo discdvered the laws of motion of mundane matter. It was not until he had done this, and applied these laws to the heavenly bodies, that their motions received a rational explanation. Then Newton, taking up Galileo's parable and completing it, found that its strict application to the heavenly bodies revealed the law of gravitation, and developed the theory of dynamics. Next to the motions of the heavenly bodies, the wave, the whirlwinds, and the motions of clouds, had excited the philosophical curiosity of man- kind from the earliest time. Both Galileo and Newton, as well as their followers, attempted to explain these by the laws of motion, but although the results so obtained have been of the utmost importance in the develop- ment of the theory of dynamics, it was not till this century that any considerable advance was made in the application of this theory to the explanation of fluid phenomena, and although during the last fifty years splendid work has been done, work which, in respect of the mental effort involved, or the scientific importance of the results, goes beyond that which resulted in the discovery of Neptune, yet the circumstances of fluid motion are so obscure and complex, that the theory has yet been interpreted only in the simplest cases. To illustrate the difference between the interpretation of the theory of the heavenly bodies and that of fluid motion, I would call your attention to the fact that solid bodies, on the behaviour of which the theory of the motion of the planets is founded, move as one piece, so that their motion 526 STUDY OF FLUID MOTION BY MEANS OF COLOURED BANDS. [61 is exactly represented by the motion of their surfaces ; that they are not subject to any internal disorder which may affect their general motion. So surely is this the case, that even those who have never heard of dynamics can predict with certainty how any ordinary body will behave under any ordinary circumstances, and so much so that any departure is a matter of surprise. Thus I have here a cube of wood, to one side of which a string is attached. Now hold it on one side, and holding the string you naturally suppose that when I let go it will turn down so as to hang with the string vertical ; it does not do so, that is a matter of surprise ; I place it on the other side and it still remains as I place it. If I swing it as a pendulum it does not behave like one. Would Galileo have discovered the laws of motion had his pendulum behaved like this ? Why is its motion peculiar ? There is internal motion. Of what sort ? Well, I think my illustration may carry more weight if I do not tell you ; you can all, I have no doubt, form a good idea. It is not fluid motion or I should feel bound to explain it. You have here an ordinary looking object which behaves in an extraordinary manner, which is yet very decided and clear, to judge by the motion of its surface, and from the manner of the motion I wish you to judge of the cause of the observed motion*. This is the problem presented by fluids, in which there may be internal motion which has to be taken into account before the motion of the surface can be explained. You can see no more of what the motion is within a homogeneous fluid, however opaque or clear, than you can see what is going on within the box. Thus, without colour bands the only visual clue to what is going on within the fluids is the motion of their bounding surfaces. Nor is this all ; in most cases the surfaces which bound the fluid are immovable. In the case of the wave on water the motion of the surface shows that there is motion, but because the surface shows no wave it does not do to infer that the fluid is at rest. The only surfaces of the air within this room are the surfaces of the floor, walls, and objects within it. By moving the objects we move the air, but how far the air is at rest you cannot tell unless it is something familiar to you. Now I will ask you to look at these balloons. They are familiar objects enough, and yet they are most sensitive anemometers, more sensitive than anything else in the room ; but even they do not show any motion ; each of them forms an internal bounding surface of the air. I send an aerial * In this experiment a cubical box of wood, apparently a solid block, contained a heavy spinning top. 61] STl'DY OF FLUID MOTION BY MEANS OF COLOURED BANDS. 5*27 messenger to them, and a small but energetic motion is seen by which it acknowledges the message, and the same message travels through- the rest, as if a ghost touched them. It is a wave that moves them. You do not feel it, and. but for the surfaces of the air formed by the balloons, would have no notion of its existence*. In this tank of beautifully clear distilled water, I project a heavy ball in from the end, and it shows the existence of the water by stopping almost dead within two feet. The fact that it is stopped by the water, being familiar, does not raise the question, Why does it stop? a question to which, even at the present day, a complete answer is not forthcoming. The question is, however, suggested, and forcibly suggested, when it appears that with no greater or other evidence of its existence, I can project a disturbance through the water which will drive this small disc the whole length of the tank. 1 have now shown instances of fluid motion of which the manner is in no way evident without colour bands, and were revealed by colour bands, as I showed in this room sixteen years ago. At that time I was occupied in setting before you the manners of motion revealed, and I could only inci- dentally notice the means by which this revelation was accomplished. Amongst the ordinary phenomena of motion there are many which render evident the internal motion of fluids. Small objects suspended in the fluid are important, and that their importance has long been recognised is shown by the proverb straws show which way the wind blows. Bubbles in water, smoke and clouds, afford the most striking phenomena, and it is doubtless these that have furnished philosophers with such clues as they have had. But the indications furnished by these phenomena are imperfect, and, what is more important, they only occur casually, and in general only under circumstances of such extreme complexity that any deduction as to the elementary motions involved is impossible. They afford indication of commotion, and perhaps of the general direction in which the commotion is tending, but this is about all. For example, the different types of clouds ; these have always been noticed and are all named. And it is certain that each type of clouds is an indication of a particular type of motion in the air; but no deductions as to what definite manner of motion is indicated by each type of cloud have ever been published. Before this can be done it is necessary to reverse the problem, and find to what particular type of cloud a particular manner of motion would give * By means of a large box, having a hinged door on one side, and a circular aperture on the side opposite, invisible vortex rings of air were projected towards the balloons. 528 STUDY OF FLUID MOTION BY MEANS OF COLOURED BANDS. [61 rise. Now a cloud, as we see it, does not directly indicate the internal motion of which it is the result. As we look at clouds, it is not in general their motion that we notice, but their figure. It is hard to see that this figure changes while we are watching a cloud, though such a change is continually going on, but is apparently very slow on account of the great distance of the cloud and its great size. However, types of clouds are determined by their figure, not by their motion. Now what their figure shows is not motion, but is the history or result of the motion of particular strata of the air in and through surrounding strata. Hence, to interpret the figures of the clouds we must study the changes in shape of fluid masses, surrounded by fluid, which result from particular motions. The ideal in the method of colour bands is to render streaks or lines in definite position in the fluid visible, without in any way otherwise interfering with these properties as part of the homogeneous fluid. If we could by a wish create coloured lines in the water, these would be ideal colour bands. We cannot do this, nor can we exactly paint lines in the air or water. I take this ladle full of highly coloured water, lower it slowly into the surface of the surrounding water till that within is level with that without ; then turn the ladle carefully round the coloured water ; the mass of coloured water will remain where placed. I distribute the colour slowly. It does not mix with the clear water, and although the lines are irregular they stand out very beautifully. Their edges are sharp here. But in this large sphere, which was coloured before the lecture, although the coloured lines have generally kept their places, they have, as it were, swollen out and become merged in the surrounding water in consequence of molecular motion. The sphere shows, however, one of the rarest phenomena in Nature the internal state in almost absolute internal rest. The forms resemble nothing so much as stratus clouds, as seen on a summer day, though the continuity of the colour bands is more marked. A mass of coloured water once introduced is never broken. The discontinuity of clouds is thus seen to be due to other causes than mere motion. Now, having called your attention to the rarity of water at rest, I will call your attention to what is apt to be a very striking phenomenon, namely, that when water is contained, like this, in a spherical vessel of which you cannot alter the shape, it is impossible by moving the vessel suddenly to set up relative motion in the interior of the water. I may swing this vessel about and turn it, but the colour band in the middle remains as it was, and when I stop shows the water to be at rest. This is not so if the water has a free surface, or if the fluid is of unequal density. Then a motion of the vessel sets up waves, and the colour band 61] STUDY OF FLUID MOTION BY MEANS OF COLOURED BANDS. 529 shows at once the beautifully lawful character of the internal motion. The colour bands move backwards and forwards, showing how the water is dis- torted like a jelly, and as the wave dies out the colour bands remain as they were to begin with. This illustrates one of the two classes of internal motion of water or fluid. Wherever fluid is not in contact with surfaces over which it has to glide, or surfaces which fold on themselves, the internal motions are of this purely wave character. The colour bands, however much they may be distorted, cannot be relatively displaced, twisted, or curled up, and in this case motion in water once set up continues almost without resistance. That wave motion, in water with a free surface, is one of the most difficult things to stop, is directly connected with the difficulty of setting still water in motion ; in either case the influence must come through the surfaces. Thus it is that waves once set up will traverse thousands of miles, establishing communica- tion between the shores of Europe and America. Wave motion in water is subject to enormously less resistance than any other form of material motion. In wave motion, if the colour bands are across the wave they show the motion of the water ; nevertheless, their chief indication is of the change of shape while the fluid is in motion. This is illustrated in this long bottle, with the coloured water less heavy than the clear water. If I lay it down in order to establish equilibrium, the blue water has to leave the upper end of the bottle and spread itself over the clear water, while the clear water runs under the coloured. This sets up wave motion, which continues after the bottle has come to rest. But as the colour bands are parallel with the direction of motion of the waves, the motion only becomes evident in thickening and bending of the colour bands. The waves are entirely between the two fluids, there being no motion in the outer surfaces of the bottle, which is everywhere glass. They are owing to the slight differences in the density of the fluids, as is indicated by the extreme slowness of the motion. Of such kind are the waves in the air, that cause the clouds which make the mackerel sky, the vapour in the tops of the waves being condensed and evaporated again as it descends, showing the results of the motion. The distortional motions, such as alone occur in simple wave motion, or where the surfaces of the fluid do not fold in on themselves, or wind in, are the same as occur in any homogeneous continuous material which com- pletely fills the space between the surfaces. If plastic material is homogeneous in colour, it shows nothing as to the internal motion; but if I take a lump built of plates, blue and white, say a square, then I can change the surfaces to any shape without folding or o. R. n. 34 530 STUDY OF FLUID MOTION BY MEANS OF COLOURED BANDS. [61 turning the lump, and the coloured bands which extend throughout the lump show the internal changes. Now the first point to illustrate is that, however I change its shape, if I bring it back to the original shape the colour bands will all come back to their original positions, and there is no limit to the extent of the change that may thus be effected. I may roll this out to any length, or draw it out, and the diminution in thickness of the colour bands shows the extent of the distortion. This is the first and simplest class of motion to which fluids are susceptible. By this motion alone elements of the fluid may be, and are, drawn out to an indefinitely fine line, or spread out in an indefinitely thin sheet, but they will remain of the same general figure. By reversing the process they change back again to the original form. No colour band can ever be broken, even if the outer surface be punched in till the punch head comes down on the table ; still all the colour bands are continuous under the punch, and there is no folding or lapping of the colour bands unless the external surface is folded. The general idea of mixture is so familiar to us that the vast generaliza- tion to which these ideas afford the key, remains unnoticed. That continued mixing results in uniformity, and that uniformity is only to be obtained by mixing, will be generally acknowledged, but how deeply and universally this enters into all the arts can but rarely have been apprehended. Does it ever occur to any one that the beautiful uniformity of our textile fabrics has only been obtained by the development of processes of mixing the fibres ? Or, again, the uniformity in our construction of metals ; has it ever occurred to any one that the inventions of Arkwright and Cort were but the application of the long-known processes by which mixing is effected in culinary opera- tions ? Arkwright applied the draw-rollers to uniformly extend the length of the cotton sliver at the expense of the thickness ; Cort applied the rolling- mill to extend the length of the iron bloom at the expense of its breadth ; but who invented the rolling-pin by which the pastry-cook extends the length at the expense of the thickness of the dough for the pie-crust ? In all these processes the object, too, is the same throughout to obtain some particular shape, but chiefly to obtain a uniform texture. To obtain this nicety of texture it is necessary to mix up the material, and to accom- plish this it is necessary to attenuate the material, so that the different parts may be brought together. The readiness with which the fluids are mixed and uniformity obtained is a by- word; but it is only when we come to see the colour bands that we realize that the process by which this is attained is essentially the same as that so laboriously discovered for the arts as depending first on the atten- uation of each element of the fluid as I have illustrated by distortion. 61] STUDY OF FLUID MOTION BY MEANS OF COLOURED BANDS. 531 In fluids, no less than in cooking, spinning, and rolling this attenuation is only the first step in the process of mixing all involve the second process, that of folding, piling, or wrapping, by which the attenuated layers are brought together. This does not occur in the pure wave motion of water, and constitutes the second of the two classes of motion. If a wave on water is driven beyond a certain height it leaps or breaks, folding in its surface. Or, if I but move a solid surface through the water it introduces tangential motion, which enables the fluid to wind its elements round an axis. In these ways, and only in these ways, we are released from the restriction of not turning or lapping. And in our illustration, we may fold up our dough, or lap it roll it out again and lap it again ; cut up our iron bar, pile it, and roll it out again, or bring as many as we please of the attenuated fibres of cotton together to be further drawn. It may be thought that this attenuation and wrapping will never make perfect admixture, for, however thin, each element will preserve its characteristic, the coloured layers will be there, however often I double and roll out the dough. This is true. But in the case of some fluids, and only in the case of some fluids, the physical process of diffusion completes the admixture. These colour bands have remained in this water, swelling but still distinct ; this shows the slowness of diffusion. Yet such is the facility with which the fluid will go through the process of attenuating its elements and enfolding them, that by simply stirring with a spoon these colour bands can be drawn and folded so fine that the diffusion will be instantaneous, and the fluid become uniformly tinted. All internal fluid motion other than simple distortion, as in wave motion, is a process of mixing, and it is thus from the arts that we get the clue to the elementary forms and processes of fluid motion. When I put the spoon in and mixed the fluid you could not see what went on it was too quick. To make this clear, it is necessary that the motion should be very slow. The motion should also be in planes at right angles to the direction in which you are looking. Such is the instability of fluid that to accomplish this at first appeared to be difficult. At last, however, as the result of much thought, I found a simple process which I will now show you, in what I think is a novel experiment, and you will see, what I think has never been seen before by any one but Mr Foster and myself, namely, the complete process of the formation of a cylindrical vortex sheet resulting from the motion of a solid surface. To make it visible to all I am obliged to limit the colour band to one section of the sheet, otherwise only those immediately in front would be able to see between the con- volutions of the spiral. But you will understand that what is seen is a section, a similar state of motion extending right across the tank. From the surface you see the plane vane extending half-way down right across the tank; this is attached to a float. 342 532 STUDY OF FLUID MOTION BY MEANS OF COLOURED BANDS. [61 Out of the tube I now institute a colour band on the right of the vane. There is no motion in the water, and the colour descends slowly from the tube. I now give a small impulse to the float to move it to the right, and at once the spiral form is seen from the tube. Similar spirals would be formed all across the tank if there were colours. The float has moved out of the way, leaving the revolving spiral with its centre stationary, showing that the horizontal axis of the spiral is half-way between the bottom and surface of the tank, in which the water is now simply revolving round this axis. This is the vortex in its simplest and rarest form (for a vortex cannot exist with its ends exposed). Like an army it must have its flanks protected ; hence a straight vortex can only exist where it has two surfaces to cover its flanks, and parallel vertical surfaces are not common in nature. The vortex can bend, and, as with a horse-shoe axis, can rest both its flanks on the same surface, as this piece of clay, or with a ring axis, which is its commonest form, as in the smoke ring. In both these cases the vortex will be in motion through the fluid, and less easy to observe. These vortices have no motion beyond the rotation because they are half-way down the tank. If the vane were shorter they would follow the vane ; if it were longer they would leave it. In the same way, if instead of one vortex there were two vortices, with their axis parallel, extending right across, the one above another, they would move together along the tank. I replace the float by another which has a vane suspended from it, so that the water can pass both above and below the vane extending right across the middle portion of the tank. In this case I institute two colour bands, one to pass over the top, the other underneath, the vane, which colour bands will render visible a section of each vortex just as in the last case. I now set the float in motion and the two vortices turn towards each other in opposite directions. They are formed by the water moving over the surface of the vane, downwards to get under it, upwards to get over it, so that the rotation in the upper vortex is opposite to that in the lower. All this is just the same as before, but that instead of these vortices standing still as before they follow at a definite distance from the vane, which con- tinues its motion along the tank without resistance. Now this experiment shows, in the simplest form, the modus operandi by which internal waves can exist in fluid without any motion in the external boundary. Not only is this plate moving flatwise through the water, but it is followed by all the water, coloured arid uncoloured, enclosed in these cylindrical vortices. Now, although there is no absolute surface visible, yet there is a definite surface which encloses these moving vortices, and separates them from the water which moves out of their way. This surface will be 61] STUDY OF FLUID MOTION BY MEANS OF COLOURED BANDS. 533 rendered visible in another experiment I shall show you. Thus the water which has only wave motion is bounded by a definite surface, the motion of which corresponds to the wave; but inside this closed surface there is also water, so that we cannot see the surface, and this water inside is moving round and round, but so that its motion at the bounding surface is every- where the same as that of the outside water. The two masses of water do not mix. That outside moves over the bounding surface, out of the way of and past the vortices, while the vortices move round arid round inside the surface in such a way that they are moving in exactly the same manner at the surface as the wave surface outside. This is the key to the internal motion of water. You cannot have a pure wave motion inside a mass of fluid with its boundaries at rest, but you have a compound motion, a wave motion outside, and a vortex within, which fulfils the condition that there shall be no sliding of the fluid over fluid at the boundary. A means, which I hope may make the essential conditions of this motion clearer, occurred to me while preparing this lecture, and to this I will now ;isk your attention. I have here a number of layers of cotton-wool (wadding). Now I can force any body along between these layers of wadding. They yield, as by a wave, and let it go through ; but the wadding must slide over the surface of the body so moving through it. And this it must not do if it illustrate the conditions of fluid motion. Now there is one way, and only one way, in which material can be got through between the sheets of wadding without slipping. It must roll through ; but this is not enough, because if it rolls on the under surface it will be slipping on the upper. But if we have two rollers, one on the top of the other, between the sheets, then the lower roller rolls on the bottom sheet, the upper roller rolls against the upper sheet, so that there is no slipping between the rollers or the wadding, and, equally important, there is no slipping between the rollers, as they roll on each other. I have only to place a sheet of canvas between the rollers and draw it through ; both the flannel rollers roll on the canvas and on the wadding, which they pass through without slipping, causing the wadding to move in a wave outside them, and affording a complete parallel of the vortex motion. I will now show by colour bands some of the more striking phenomena of internal motion, as presented by Nature's favourite form of vortex, the vortex ring, which may be described as two horse-shoe vortices with their ends founded on each other. To show the surface separating the water moving with the vortex, from 534 STUDY OF FLUID MOTION BY MEANS OF COLOURED BANDS. [61 that which gives way outside, I discharge from this orifice a mass of coloured water, which has a vortex ring in it formed by the surface as already described. You see the beautifully defined mass moving on slowly through the fluid, with the proper vortex ring motion, but very slow. It will not go far before a change takes place, owing to the diffusion of the vortex motion across the bounding surface ; then the coloured surface will be wound into the ring which will appear. The mass approaches the disc in front. It cannot pass, but will come up and carry the disc forward; but the disc, although it does not destroy the ring, disturbs the motion. If I send a more energetic ring it will explain the phenomenon I showed you at the beginning of this lecture ; it carries the disc forward as if struck with a hammer. This blow is not simply the weight of the coloured ring, but of the whole moving mass and the wave outside. The ring cannot pass the disc without destruction, with the attendant wave. Not only can a ring follow a disc, but as with the plane vane so with the disc, if we start a disc we must start a ring behind it. I will now fulfil my promise to reveal the silent messenger I sent to those balloons. The messenger appears in the form of a large smoke ring, which is a vortex ring in air rendered visible by smoke instead of colour. The origination of these rings has been carefully set so that the balloons are beyond the surface which separates the moving mass of water from the wave, so that they are subject to the wave motion only. If they are within this surface they will disturb the direction of the ring, if they do not break it up. These are, if I may say so, the phenomenal instances of internal motion of fluids. Phenomenal in their simplicity, they are of intense interest, like the pendulum, as furnishing the clue to the more complex. It is by the light we gather from their study that we can hope to interpret the parallel of the vortex wrapped up in the wave, as applied to the wind of heaven, and the grand phenomenon of the clouds, as well as those things which directly concern us, such as the resistance of ships. 62. ON THE DYNAMICAL THEORY OF INCOMPRESSIBLE VIS- COUS FLUIDS AND THE DETERMINATION OF THE CRITERION. [Fro7n the " Philosophical Transactions of the Royal Society," 1895.] (Read May 24, 1894.) SECTION I. Introduction. 1. THE equations of motion of viscous fluid (obtained by grafting on certain terms to the abstract equations of the Eulerian form, so as to adapt these equations to the case of fluids subject to stresses depending in some hypothetical manner on the rates of distortion, which equations Navier* seems to have first introduced in 1822, and which were much studied by Cauchyt and PoissonJ) were finally shown by St Venant and Sir Gabriel Stokes||, in 184=5, to involve no other assumption than that the stresses, other than that of pressure uniform in all directions, are linear functions of the rates of distortion, with a coefficient depending on the physical state of the fluid. By obtaining a singular solution of these equations as applied to the case of pendulums in steady periodic motion, Sir G. StokesH was able to compare the theoretical results with the numerous experiments that had * Mem. de I'Academie, vol. vi. p. 389. t Mem, des Sacmitx Klrnngers, vol. I. p. 40. J Mem. de VAcademie, vol. x. p. 345. B.A. Report, 1840. || Cambridge Phil. Tnin*., 1845. IT Ibid., vol. ix. 1857. 536 THE DYNAMICAL THEORY OF INCOMPRESSIBLE VISCOUS FLUIDS [62 been recorded, with the result that the theoretical calculations agreed so closely with the experimental determinations as seemingly to prove the truth of the assumption involved. This was also the result of comparing the flow of water through uniform tubes with the flow calculated from a singular solution of the equations, so long as the tubes were small and the velocities slow. On the other hand, these results, both theoretical and practical, were directly at variance with common experience as to the resistance encountered by larger bodies moving with higher velocities through water, or by water moving with greater velocities through larger tubes. This discrepancy Sir G. Stokes considered as probably resulting from eddies, which rendered the actual motion other than that to which the singular solution referred, and not as disproving the assumption. In 1850, after Joule's discovery of the Mechanical Equivalent of Heat, Stokes showed, by transforming the equations of motion with arbitrary stresses so as to obtain the equations of ("Vis-viva") energy, that this equation contained a definite function, which represented the difference between the work done on the fluid by the stresses and the rate of increase of the energy, per unit of volume, which function, he concluded, must, according to Joule, represent the Vis- viva converted into heat. This conclusion was obtained from the equations irrespective of any particular relation between the stresses and the rates of distortion. Sir G. Stokes, however, translated the function into an expression in terms of the rates of distortion, which expression has since been named by Lord Rayleigh the Dissipation- Function. 2. In 1883 I succeeded in proving, by means of experiments with colour bands the results of which were communicated to the Society* that when water is caused by pressure to flow through a uniform smooth pipe, the motion of the water is direct, i.e., parallel to the sides of the pipe, or sinuous, i.e., crossing and re-crossing the pipe, according as U m , the mean velocity of the water, as measured by dividing Q, the discharge, by A, the area of the section of the pipe, is below or above a certain value given by where D is the diameter of the pipe, p the density of the water, and K a numerical constant, the value of which according to my experiments, and, as I was able to show, to all the experiments by Poiseuille and Darcy, is for pipes of circular section between 1900 and 2000, * Phil. Trains., 1883, Part III. p. 935. (See this vol. p. 51.) 62] AND THE DETERMINATION OF THE CRITERION. 537 or, in other words, steady direct motion in round tubes is stable or unstable according as > 1900 or < 2000, the number K being thus a criterion of the possible maintenance of sinuous or eddying motion. 3. The experiments also showed that K was equally a criterion of the law of the resistance to be overcome which changes from a resistance proportional to the velocity, and in exact accordance with the theoretical results obtained from the singular solution of the equation, when direct motion changes to sinuous, i.e., when DU m p - = K. P 4. In the same paper I pointed out that the existence of this sudden change in the law of motion of fluids between solid surfaces when P proved the dependence of the manner of motion of the fluid on a relation between the product of the dimensions of the pipe multiplied by the velocity of the fluid, and the product of the molecular dimensions multiplied by the molecular velocities which determine the value of for the fluid, also that the equations of motion for viscous fluid contained evidence of this relation. These experimental results completely removed the discrepancy previously noticed, showing that, whatever may be the cause, in those cases in which the experimental results do not accord with those obtained by the singular solution of the equations, the actual motions of the water are different. But in this there is only a partial explanation, for there remains the mechanical or physical significance of the existence of the criterion to be explained. 5. [My object in this paper is to show that the theoretical existence of an inferior limit to the criterion follows from the equations of motion as a consequence : (1) Of a more rigorous examination and definition of the geometrical basis on which the analytical method of distinguishing between molar- motions and heat-motions in the kinetic theory of matter is founded ; and 538 THE DYNAMICAL THEORY OF INCOMPRESSIBLE VISCOUS FLUIDS [62 (2) Of the application of the same method of analysis, thus definitely founded, to distinguish between mean-molar-motions and relative-molar- motions, where, as in the case of steady-mean-flow along a pipe, the more rigorous definition of the geometrical basis shows the method to be strictly applicable, and in other cases where it is approximately applicable. The geometrical relation of the motions respectively indicated by the terms mean-molar-, or MEAN-MEAN-MOTION, and relative-molar-, or RELATIVE- MEAN- MOTION, being essentially the same as the relation of the respective motions indicated by the terms molar-, or MEAN-MOTION, and relative-, or HEAT-MOTION, as used in the theory of gases. I also show that the limit to the criterion obtained by this method of analysis, and by integrating the equations of motion in space, appears as a geometrical limit to the possible simultaneous distribution of certain quantities in space, and in no wise depends on the physical significance of these quan- tities. Yet the physical significance of these quantities, as defined in the equations, becomes so clearly exposed as to indicate that further study of the equations would elucidate the properties of matter and mechanical principles involved, and so be the means of explaining what has hitherto been obscure in the connection between thermodynamics and the principles of mechanics. The geometrical basis of the method of analysis used in the kinetic theory of gases has hitherto consisted : (1) Of the geometrical principle that the motion of any point of a mechanical system may, at any instant, be abstracted into the mean-motion of the whole system at that instant, and the motion of the point relative to the mean-motion ; and (2) Of the assumption that the component, in any particular direction, of the velocity of a molecule, may be abstracted into a mean-component- velocity (say u) which is the mean-component-velocity of all the molecules in the immediate neighbourhood, and a relative-velocity (say ), which is the difference between u and the component- velocity of the molecule*; u and being so related that, M being the mass of the molecule, the integrals of (M%), and (Mug), &c., over all the molecules in the immediate neighbourhood are zero, and 2 [M (u, + ) L> ] = 2 [M(u? + 2 )]t- The geometrical principle (1) has only been used to distinguish between the energy of the mean-motion of the molecule, and the energy of its internal motions taken relatively to its mean-motion ; and so to eliminate the internal motions from all further geometrical considerations which rest on the as- sumption (2). * "Dynamical Theory of Gases," Phil. Trans., 1866, p. 67. t Phil. Trans., 1866, p. 71. 62] AND THE DETERMINATION OF THE CRITERION. 539 That this assumption (2) is purely geometrical, becomes at once obvious, when it is noticed that, the argument relates solely to the distribution in space of certain quantities at a particular instant of time. And it appears that the questions as to whether the assumed distinctions are possible under any distributions, and, if so, under what distribution, are proper subjects for geometrical solution. On putting aside the apparent obviousness of the assumption (2), and considering definitely what it implies, the necessity for further definition at once appears. The mean-component-velocity (u) of all the molecules in the immediate neighbourhood of a point, say P, can only be the mean-component-velocity of all the molecules in some space (S) enclosing P. u is then the mean- component-velocity of the mechanical system enclosed in S, and, for this system, is the mean-velocity at every point within S, and, multiplied by the entire mass within S, is the whole component momentum of the system. But according to the assumption (2), u with its derivatives are to be con- tinuous functions of the position of P, which functions may vary from point to point even within S', so that u is not taken to represent the mean- component-velocity of the system within S, but the mean-velocity at the point P. Although there seems to have been no specific statement to that effect, it is presumable that the space S has been assumed to be so" taken that P is the centre of gravity of the system within S. The relative, positions of P and S being so defined, the shape and size of the space S requires to be further defined, so that u, &c., may vary continuously with the position of P, which is a condition that can always be satisfied if the size and shape of S may vary continuously with the position of P. Having thus defined the relation of P to $ and the shape and size of the latter, expressions may be obtained for the conditions of distribution of u, for which S (^/) taken over S will be zero, i.e., for which the condition of mean- momentum shall be satisfied. Taking S lt u 1} &c., as relating to a point P l and S, u, &c., as relating to P, another point, of which the component distances from P] are x, y, z; P l is the C.G. of Si, and by however much or little S may overlap Si, S has its centre of gravity at x, y, z, and is so chosen that u, &c., may be continuous functions of x,y,z\ u may, therefore, differ from Ui even if P is within $,. Let u be taken for every molecule of the system Si. Then according to assumption (2), 2 (Mu) over Si must represent the component of momentum of the system within S l} that is, in order to satisfy the condition of mean- momentum, the mean-value of the variable quantity u over the system S t must be equal to HI the mean-component-velocity of the system Si, and this is a condition which, in consequence of the geometrical definition already 540 THE DYNAMICAL THEORY OF INCOMPRESSIBLE VISCOUS FLUIDS [62 mentioned, can only be satisfied under certain distributions of u. For since u is a continuous function of x, y, z, M(ii-u^) may be expressed as a function of the derivatives of u at P l multiplied by corresponding powers and products of #, y, z, and again by M ; and by equating the integral of this function over the space Si to zero, a definite expression is obtained, in terms of the limits imposed on x, y, z, by the already-defined space Si for the geometrical condition as to the distribution of u under which the condition of mean-momentum can be satisfied. From this definite expression it appears, as has been obvious all through the argument, that the condition is satisfied if u is constant. It also appears that there are certain other well-defined systems of distribution for which the condition is strictly satisfied, and that for all other distributions of u the condition of mean-momentum can only be approximately satisfied to a degree for which definite expressions appear. Having obtained the expression for the condition of distribution of u, so as to satisfy the condition of mean-momentum, by means of the expression for M(u-u'), &c., expressions are obtained for the conditions as to the distribution of , &c., in order that the integrals over the space S L of the products Jf(ttff), &c. may be zero when 2 [M (u. Wi)] = 0, and the con- ditions of mean-energy satisfied as well as those of mean-momentum. It then appears that in some particular cases of distribution of u, under which the condition of mean-momentum is strictly satisfied, certain conditions as to the distribution of , &c., must be satisfied in order that the energies of mean- and relative-motion may be distinct. These conditions as to the distribution of , &c., are, however, obviously satisfied in the case of heat- motion, and do not present themselves otherwise in this paper. From the definite geometrical basis thus obtained, and the definite expressions which follow for the condition of distribution of u, &c., under which the method of analysis is strictly applicable, it appears that this method may be rendered generally applicable to any system of motion by a slight adaptation of the meaning of the symbols, and that it does not necessitate the elimination of the internal motion of the molecules, as has been the custom in the theory of gases. Taking u, v, w to represent the motions (continuous or discontinuous) of the matter passing a point, and p to represent the density at the point, and putting u, &c., for the mean-motion (instead of u as above), and u', &c., for the relative-motion (instead of as before), the geometrical conditions as to the distribution of u, &c., to satisfy the conditions of mean-momentum and mean-energy are, substituting p for M, of precisely the same form as before, and as thus expressed, the theorem is applicable to any mechanical system however abstract. 62] AND THE DETERMINATION OF THE CRITERION. 541 (1) In order to obtain the conditions of distribution of molar-motion, under which the condition of mean-momentum will be satis_fied, so that the energy of molar-motion may be separated from that of the heat- motion, u, &c., and p are taken as referring to the actual motion and density at a point in a molecule, and S l is taken of such dimensions as may corre- spond to the scale, or periods in space, of the molecular distances, then the conditions of distribution of u, under which the condition of mean-momentum is satisfied, become the conditions as to the distribution of molar-motion, under which it is possible to distinguish between the energies of rnolar- motions and heat-motions. (2) And, when the conditions in (1) are satisfied to a sufficient degree of approximation by taking u to represent the molar-motion (u in (1)), and the dimensions of the space S to correspond with the period in space or scale of any possible periodic or eddying motion, the conditions as to the distribution of u, &c. (the components of mean-mean-motion), which satisfy the condition of mean-momentum, show the conditions of mean-molar-motion, under which it is possible to separate the energy of mean-molar-motion from the energy of relative-molar- (or relative-mean-) motion. Having thus placed the analytical method used in the kinetic theory on a definite geometrical basis, and adapted so as to render it applicable to all systems of motion, by applying it k> the dynamical theory of viscous fluid, I have been able to show : Feb. 18, 1895.] (a) That the adoption of the conclusion arrived at by Sir Gabriel Stokes, that the dissipation function Represents the rate at which heat is pro- duced, adds a definition to the meaning of u, v, w the components of mean or fluid velocity which was previously wanting. (6) That as the result of this definition the equations are true, and are only true, as applied to fluid in which the mean-motions of the matter, excluding the heat-motions, are steady. (c) That the evidence of the possible existence of such steady mean- motions, while at the same time the conversion of the energy of these mean- motions into heat is going on, proves the existence of some discriminative cause, by which the periods in space and time of the mean- motion are prevented from approximating in magnitude to the corresponding periods of the heat-motions, and also proves the existence of some general action by which the energy of mean-motion is continually transformed into the energy of heat-motion, without passing through any intermediate stage. (d) That as applied to fluid in unsteady mean-motion (excluding the heat-motions), however steady the mean integral flow may be, the equations 542 THE DYNAMICAL THEORY OF INCOMPRESSIBLE VISCOUS FLUIDS [62 are approximately true in a degree which increases with the ratios of the magnitudes of the periods, in time and space, of the mean-motion, to the magnitude of the corresponding periods of the heat-motions. (e) That if the discriminative cause and the action of transformation are the result of general properties of matter, and not of properties which affect only the ultimate motions, there must exist evidence of similar actions as between the mean-mean-motion, in directions of mean-flow, and the periodic mean-motions taken relative to the mean-mean-motion but excluding heat- motions. And that such evidence must be of a general and important kind, such as the unexplained laws of the resistance of fluid motions, the law of the universal dissipation of energy, and the second law of thermo- dynamics. (/) That the generality of the effects of the properties on which the action of transformation depends, is proved by the fact that resistance, other than proportional to the velocity, is caused by the relative (eddying) mean- motion. (g) That the existence of the discriminative cause is directly proved by the existence of the criterion, the dependence of which on circumstances which limit the magnitudes of the periods of relative-mean-motion, as com- pared with the heat-motion, also proves the generality of the effects of the properties on which it depends. (k) That the proof of the generality of the effects of the properties on which the discriminative cause, and the action of transformation depend, shows that if in the equations of motion the mean-mean-motion is dis- tinguished from the relative-mean-motion in the same way as the mean- motion is distinguished from the heat-motions (1) the equations must contain expressions for the transformation of the energy of mean-mean- motion to energy of relative-mean-motion ; and (2) that the equations, when integrated over a complete system, must show that the possibility of relative- mean-motion depends on the ratio of the possible magnitudes of the periods of relative-mean-motion, as compared with the corresponding magnitude of the periods of the heat-motions. (i) That when the equations are transformed so as to distinguish between the mean-mean-motions, of infinite periods, and the relative-mean- motions of finite periods, there result two distinct systems of equations, one system for mean-mean-motion, as affected by relative-mean-motion and heat- motion, the other system for relative-mean-motion as affected by mean-mean- motion and heat-motions. (j) That the equation of energy of mean-mean-motion, as obtained from the first system, shows that the rate of increase of energy is diminished by 62] AND THE DETERMINATION OF THE CRITERION. 543 conversion into heat, and by transformation of energy of mean-mean-motion in consequence of the relative-mean-motion, which transformation is ex- pressed by a function identical in form with that which expresses the conversion into heat ; and that the equation of energy of relative-mean- motion, obtained from the second system, shows that this energy is in- creased only by transformation of energy from mean-mean-motion expressed by the same function, and diminished only by the conversion of energy of relative-mean-motion into heat. (k) That the difference of the two rates (1) transformation of energy of mean -mean-motion into energy of relative-rneari-motion as expressed by the transformation function, (2) the conversion of energy of relative-mean-motion into heat, as expressed by the function expressing dissipation of the energy of relative-mean-motion, affords a discriminating equation as to the conditions under which relative-mean-motion can be maintained. (I) That this discriminating equation is independent of the energy of relative-mean-motion, and expresses a relation between variations of mean- rnean-motion of the first order, the space periods of relative-mean-motion, and fji/p, such that any circumstances which determine the maximum periods of the relative-mean-motion, determine the conditions of mean-mean-motion under which relative-mean-motion will be maintained, that is, determine the criterion. (m) That as applied to water in steady mean-flow between parallel plane surfaces, the boundary conditions, and the equation of continuity, impose limits to the maximum space periods of relative-mean-motion, such that the discriminating equation affords definite proof that when an in- definitely small sinuous or relative disturbance exists, it must fade away if is less than a certain number, which depends on the shape of the section of the boundaries, and is constant as long as there is geometrical similarity. While for greater values of this function, in so far as the discriminating equation shows, the energy of sinuous motion may increase until it reaches to a definite limit, and rules the resistance. (??) That besides thus affording a mechanical explanation of the existence of the criterion K, the discriminating equation shows the purely geometrical circumstances on which the value of K depends, and although these circum- stances must satisfy geometrical conditions required for steady mean-motion other than those imposed by the conservations of mean-energy and momentum, the theory admits of the determination of an inferior limit to the value of K under any definite boundary conditions, which, as determined for the par- ticular case, is 517. 544 THE DYNAMICAL THEORY OF INCOMPRESSIBLE VISCOUS FLUIDS [62 This is below the experimental value for round pipes, and is about half what might be expected to be the experimental value for a flat pipe, which leaves a margin to meet the other kinematical conditions for steady mean-mean- motion. (o) That the discriminating equation also affords a definite expression for the resistance, which proves that, with smooth fixed boundaries, the con- ditions of dynamical similarity under any geometrical similar circumstances depend only on the value of where b is one of the lateral dimensions of the pipe ; and that the expression for this resistance is complex, but shows that above the critical velocity the relative-mean-motion is limited, and that the resistances increase as a power of the velocity higher than the first. SECTION II. The Mean-motion and Heat-motions as distinguished by Periods. Mean- mean-motion and Relative-mean-motion. Discriminative Cause and Action of Transformation. Two Systems of Equations. A Discrimi- nating Equation. 6. Taking the general equations of motion for incompressible fluid, subject to no external forces to be expressed by du f d . d ~di = ~\fa (PXX + puu) + dy dv ( d , d dw _ ( d d d ~dt~ ~ \dx (PXZ + pwu > + dy (JV + P wv "> + fa (P ...... ax with the equation of continuity 0=du/dx+dv/dy + dw/dz ....................... '....(2), where p m , &c., are arbitrary expressions for the component forces per unit of area, resulting from the stresses, acting on the negative faces of planes perpendicular to the direction indicated by the first suffix, in the direction indicated by the second suffix. 62] AND THE DETERMINATION OF THE CRITERION. 545 Then multiplying these equations respectively by u, v, w, integrating by parts, adding and putting 2E for p(u*+v* + w 2 ) and transposing, the rate of increase of kinetic energy per unit of volume is given by d , d . d T~ ('Upxx) + -j- ( u Pvx) + j~ ' ax ay dz d_ t \ v fJXIl) I ~J (.vPw) ' dx dy d , d , d (d d d d\ T d U + * 'Si + ' dy + '" Tz) E = ~ 1 + dec du du du 1 dv dv dv ~7 "~ Pw J r Pzv ~j~ dx yy dy y dz dw dw . dw .(3). The left member of this equation expresses the rate of increase in the kinetic energy of the fluid per unit of volume at a point moving with the fluid. The first term on the right expresses the rate at which work is being done by the surrounding fluid per unit of volume at a point. The second term on the right therefore, by the law of conservation of energy, expresses the difference between the rate of increase of kinetic energy and the rate at which work is being done by the stresses. This difference has, so far as I am aware, in the absence of other forces, or any changes of potential energy, been equated to the rate at which heat is being converted into energy of motion, Sir Gabriel Stokes having first indicated this* as resulting from the law of conservation of energy then just established by Joule. 7. This conclusion, that the second term on the right of (3) expresses the rate at which heat is being converted, as it is" usually accepted, may be correct enough, but there is a consequence of adopting this conclusion which enters largely into the method of reasoning in this paper, but which, so far as I know, has not previously received any definite notice. * Cambridge Phil. Trans., vol. ix. p. 57. o. B. n. 35 546 THE DYNAMICAL THEORY OF INCOMPRESSIBLE VISCOUS FLUIDS [62 The Component Velocities in the Equations of Viscous Fluids. In no case, that I am aware of, has any very strict definition of u, v, w, as they occur in the equations of motion, been attempted. They are usually defined as the velocities of a particle at a point (x, y, z) of the fluid, which may mean that they are the actual component-velocities of the point in the matter passing at the instant, or that they are the mean- velocities of all the matter in some space enclosing the point, or which passes the point in an interval of time. If the first view is taken, then the right-hand member of the equation represents the rate of increase of kinetic energy, per unit of volume, in the matter at the point ; and the integral of this expression over any finite space S, moving with the fluid, represents the total rate of increase of kinetic energy, including heat-motion, within that space ; hence the differ- ence between the rate at which work is done on the surface of S, and the rate at which kinetic energy is increasing can, by the law of conservation of energy, only represent the rate at which that part of the heat which does not consist in kinetic energy of matter is being produced, whence it follows : (a) That the adoption of the conclusion that the second term in equation (3) expresses the rate at which heat is being converted, defines u, v, w, as not representing the component-velocities of points in the passing matter. Further, if it is understood that u, v, w, represent the mean velocities of the matter in some space, enclosing x, y, z, the point considered, or the mean-velocities at a point taken over a certain interval of time, so that 2 (pu), 2 (pv), 2 (pw) may express the components of momentum, and 22 (pv) ;?/2 (pw), &c., &c., may express the components of moments of momentum, of the matter over which the mean is taken ; there still remains the question as to what spaces and what intervals of time. (b) Hence the conclusion that the second term expresses the rate of conver- sion of heat, defines the spaces and intervals of time over which the mean- component-velocities must be taken, so that E may include all the energy of mean-motion, and exclude that of heat-motions. Equations Approximate only except in Three Particular Gases. 8. According to the reasoning of the last article, if the second term on the right of equation (3) expresses the rate at which heat is being converted into energy of mean-motion, either pu, pv, pw express the mean components of momentum of the matter, taken at any instant over a space S enclosing the point x, y, z, to which u, v, w refer, so that this point is the centre of gravity of the matter within S and such that p represents the mean density of the matter within this space; or pu, pv, pw represent the mean components of momentum taken at x, y, z over an interval of time T, such that p is 62] AND THE DETERMINATION OF THE CRITERION. 547 the mean density over the time r, and if t marks the instant to which n, v, w refer, and t' any other instant, 2 [(t t') p], in which p is the actual density, taken over the interval T is zero. The equations, however, require, that so obtained, p, u, v, w shall be continuous functions of space and time, and it can be shown that this involves certain conditions between the distribution of the mean-motion and the dimensions of S and T. Mean- and Relative-motions of Matter. Whatever the motions of matter within a fixed space S may be at any instant, if the component- velocities at a point are expressed by u, v, w, the mean-component-velocities taken over S will be expressed by (4). If then u, v, w are taken at each instant as the velocities of x, y, z, the instantaneous centre of gravity of the matter within S, the component momentum at the centre of gravity may be put pu = pu + pu ................................. (5 ), where u' is the motion of the matter, relative to axes moving with the mean velocity, at the centre of gravity of the matter within S. Since a space S of definite size and shape may be taken about any point x, y, z in an indefinitely larger space, so that x, y, z is the centre of gravity of the matter within S, the motion in the larger space may be divided into two distinct systems of motion, of which u, v, w represent a mean-motion at each point and u', v', w' a motion at the same point relative to the mean-motion at the point. If, however, u, v, w are to represent the real mean-motion, it is necessary that 2 (pu'}, 2 (pv), 2 (pw r ) summed over the space 8, taken about any point, shall be severally zero ; and in order that this may be so, certain conditions must be fulfilled. For taking x, y, z, for 0, the centre of gravity of the matter within S, and x ', i/, z' for any other point within S, and putting a, b, c for the dimensions of S in directions x, y, z, measured from the point x, y, z; since u, v, w are continuous functions of x, y, z by shifting S so that the centre of gravity of the matter within it is at x ', y', z', the value of u for this point is given by + &c ................... (6), where all the differential coefficients on the left refer to the point x, y, z; and in the same way for v and w. Subtracting the value of il thus obtained for the point x' , y', z', from that 352 548 THE DYNAMICAL THEORY OF INCOMPRESSIBLE VISCOUS FLUIDS [62 of u at the same point, the difference is. the value of u at this point, whence summing these differences over the space 8 about G at x, y, z, since by definition when summed over the space 8 about G 2[p(tt-u e )] = and 2[p(V-#)] = (7), > (8 A). s ( P ,,-) . - iS o (. - .')'] + That is it \ G I \M U \ n 1+5- + &c. rV/ dt is everywhere constant, the conditions are perfectly satisfied. The conditions are also satisfied if the mean-motion is that of uniform expansion or contraction, or is that of a rigid body. These three cases, in which it may be noticed that variations of mean- motion are everywhere uniform in the direction of motion, and subject to steady variations in respect of time, are the only cases in which the condi- tions (8 A), (8fi), can be perfectly satisfied. The conditions will, however, be approximately satisfied, when the variations of n, v, w of the first order are approximately constant over the space S. In such case the right-hand members of equations (8 A), (8fi), are neglected, and it appears that the closeness of the approximations will be measured by the relative magnitude of such terms as d?u d*u .,, du du a dtf> &C " T d? M com P ared Wlth dx> dt' &C ' Since frequent reference must be made to these relative values, and, as in periodic motion, the relative values of such terms are measured by the period (in space or time) as compared with a, b, c and r, which are, in a sense, the periods of u', v', w', I shall use the term period in this sense, taking note of the fact that when the mean-motion is constant in the direction of motion, or varies uniformly in respect of time, it is not periodic, i.e., its periods are infinite. 9. It is thus seen that the closeness of the approximation with which the motion of any system can be expressed as a varying mean-motion together with a relative-motion, which, when integrated over a space of which the dimensions are a, b, c, has no momentum, increases as the magni- tude of the periods of w, v, w in comparison with the periods of u', v', w', and is measured by the ratio of the relative orders of magnitudes to which these periods belong. Heat- 1 notions in Matter are Approximately Relative to tlie Mean- motions. The general experience that heat in no way affects the momentum of matter, shows that the heat-motions are relative to the mean-motions of matter taken over spaces of sensible size. But, as heat is by no means the only state of relative-motion of matter, if the heat-motions are relative to all mean-motions of matter, whatsoever their periods may be, it follows that there must be some discriminative cause which prevents the existence of relative-motions of matte]' other than heat, except mean-motions with 550 THE DYNAMICAL THEORY OF INCOMPRESSIBLE VISCOUS FLUIDS [62 periods in time and space of greatly higher orders of magnitude than the corresponding periods of the heat-motions otherwise, by equations (8A), (8B), heat-motions could not be to a high degree of approximation relative to all other motions, and we could not have to a high degree of approximation, du du du \ dv dv dv d~ x *PyyJy+ p *yTz dw dw dw where the expression on the right stands for the rate at which heat is con- verted into energy of mean-motion. Transformation of Energy of Relative-mean-motion to Energy of Heat- motion. 10. The recognition of the existence of a discriminative cause, which prevents the existence of relative-mean-motions with periods of the same order of magnitude as heat-motions, proves the existence of another general action by which the energy of relative-mean-motion, of which the periods are of another and higher order of magnitude than those of the heat- motions, is transformed to energy of heat- motion. For if relative-mean-motions cannot exist with periods approximating to those of heat, the conversion of energy of mean-motion into energy of heat, proved by Joule, cannot proceed by the gradual degradation of the periods of mean-motion until these periods coincide with those of heat, but must, in its final stages, at all events, be the result of some action which causes the energy of relative-mean-motion to be transformed into the energy of heat- motions, without intermediate existence in states of relative-motion, with intermediate and gradually diminishing periods. That such change of energy of mean-motion to energy of heat may be properly called transformation, becomes apparent when it is remembered that neither mean-motion nor relative-motion has any separate existence, but are only abstract quantities, determined by the particular process of abstraction, and so changes in the actual-motion may, by the process of abstraction, cause transformation of the abstract energy of the one abstract- motion, to abstract energy of the other abstract-motion. All such transformation must depend on the changes in the actual-motions, and so must depend on mechanical principles and the properties of matter, and hence the direct passage of energy of relative-mean-motion to energy of 62] AND THE DETERMINATION OF THE CRITERION. 551 heat-motions is evidence of a general cause of the condition of actual- motion which results in transformation which may be called ~ihe cause of transformation. The Discriminative Cause, and the Cause of Transformation. 1 1 . The only known characteristic of heat-motions, besides that of being relative to the mean-motion, already mentioned, is that the motions of matter which result from heat are an ultimate form of motion which does not alter so long as the mean-motion is uniform over the space, and so long as no change of state occurs in the matter. In respect of this characteristic, heat-motions are, so far as we know, unique, and it would appear that heat- motions are distinguished from the mean-motions by some ultimate properties of matter. It does not, however, follow that the cause of transformation, or even the discriminative cause, are determined by these properties. Whether this is so or not can only be ascertained by experience. If either or both these causes depend solely on properties of matter which only affect the heat- motions, then no similar effect would result as between the variations of mean-mean-motion and relative-mean-motion, whatever might be the difference in magnitude of their respective periods. Whereas, if these causes depend on properties of matter which affect all modes of motion, distinctions in periods must exist between mean-mean-motion and relative- mean-motion, and transformation of energy take place from one to the other, as between the mean-motion and the heat-motions. The mean-mean-motion cannot, however, under any circumstances stand to the relative-mean-motion in bhe same relation as the mean-motion stands to the heat-motions, because the heat-motions cannot be absent, and in addition to any transformation from mean-mean-motion to relative-mean- motion, there are transformations both from mean- and relative-mean- motion to heat-motions, which transformation may have important effects on both the transformation of energy from mean- to relative- mean-motion, and on the discriminative cause of distinction in their periods. In spite of the confusing effect of the ever present heat-motions, it would, however, seem that evidence as to the character of the properties on which the cause of transformation and the discriminative cause depend, should be forthcoming as the result of observing the mean- and relative-mean-motions of matter. 12. To prove by experimental evidence that the effects of these properties of matter are confined to the heat- motions, would be to prove a negative ; but if these properties are in any degree common to all modes of matter, then at first sight it must seem in the highest degree improbable that the effects of these causes on the mean- and relative-mean-motions 552 THE DYNAMICAL THEORY OF INCOMPRESSIBLE VISCOUS FLUIDS [62 would be obscure, and only to be observed by delicate tests. For properties which can cause distinctions between the mean- and heat-motions of matter so fundamental and general, that from the time these motions were first recognized the distinction has been accepted as part of the order of nature, and has been so familiar to us that its cause has excited no curiosity, must, if they have any effect at all, cause effects which are general and important on the mean-motions of matter. It would thus seem that evidence of the general effects of such properties should be sought in those laws and phenomena known to us as the result of experience, but of which no rational explanation has hitherto been found ; such as the law that the resistance of fluids moving between solid surfaces and of solids moving through fluids, in such a manner that the general-motion is not periodic, is as the square of the velocities, the evidence covered by the law of the universal tendency of all energy to dissipation, and the second law of thermodynamics. 13. In considering the first of the instances mentioned, it will be seen that the evidence it affords as to the general effect of the- properties, on which depends transformation of energy from mean- to relative-motion, is very direct. For, since my experiments with colour bands have shown that when the resistance of fluids, in steady mean flow, varies with a power of the velocity higher than the first, the fluid is always in a state of sinuous motion, it appears that the prevalence of such resistance is evidence of the existence of a general action, by which energy of mean-mean-motion, with infinite periods, is directly transformed to the energy of relative-mean- motion, with finite periods, represented by the eddying motion, which renders the general mean-motion sinuous, by which transformation the state of eddying- motion is maintained, notwithstanding the continual transforma- tion of its energy into heat-motions. We have thus direct evidence that properties of matter which determine the cause of transformation, produce general and important effects which are not confined to the heat-motions. In the same way, the experimental demonstration I was able to obtain, that relative-mean-motion in the form of eddies of finite periods, both as shown by colour bands and as shown by the law of resistances, cannot be maintained except under circumstances depending on the conditions which determine the superior limits to the velocity of the mean-mean-motion, of infinite periods, and the periods of the relative-mean-motion, as defined in the criterion DU m /f* = K (10), is not only a direct experimental proof of the existence of a discriminative cause which prevents the maintenance of periodic mean-motion except with periods greatly in excess of the periods of the heat-motions, but also indicates that the discriminative cause depends on properties of matter which affect the mean-motions as well as the heat-motions. 62] AND THE DETERMINATION OF THE CRITERION. 553 Expressions for the Rate of Transformation and the Discriminative Cause. 14. It has already been shown (Art. 8) that the equations of motion approximate to a true expression of the relations between the mean-motions and stresses, when the ratio of the periods of mean-motions to the periods of the heat-motions approximates to infinity. Hence it follows that these equations must of necessity include whatever mechanical or kinematical principles are involved in the transformation of energy of mean-mean- motion to energy of relative-mean-motion. It has also been shown that the properties of matter, on which depends the transformation of energy of varying mean-motion to relative-motion, are common to the relative-mean- motion as well as to the heat-motion. Hence, if the equations of motion are applied to a condition in which the mean-motion consists of two components, the one component being a mean-mean-motion, as obtained by integrating the mean-motion over spaces Si taken about the point x, y, z, as centre of gravity, and the other component being a relative-mean-motion, of which the mean components of momentum taken over the space ^ everywhere vanish, it follows : (1) That the resulting equations of motion must contain an expression for the rate of transformation from energy of mean -mean-motion to energy of relative-mean-motion, as well as the expressions for the transformation of the respective energies of mean- and relative-mean-motion to energy of heat- motion. (2) That, when integrated over a complete system these equations must shod,' that the possibility of the maintenance of the energy of relative-mean- motion depends, 'whatsoever may be the conditions, on the possible order of magnitudes of the periods of the relative-mean-motion, as compared with the periods of the heat-motions. The Equations of Mean- and Relative-mean-motion. 15. These last conclusions, besides bringing the general results of the previous argument to the test point, suggest the manner of adaptation of the equations of motion, by which the test may be applied. Put u = u+u', v = v where u = ~ , ^,&c., &c .......................... (12), *(P) the summation extending over the space >j of which the centre of gravity is at the point a, y, z. Then since u, v, w are continuous functions of x, y, z, 554 THE DYNAMICAL THEORY OF INCOMPRESSIBLE VISCOUS FLUIDS [62 therefore u, v, w, and u, v, w', are continuous functions of x, y, z. And as p is assumed constant, the equations of continuity for the two systems of motion are : .(13); du dv dw _ _. du dv' dw' _ ~T~ T ~r~ H j " and -= | j- -f- = = (} dx dy dz dx dy dz also both systems of motions must satisfy the boundary conditions, whatever they may be. Further putting p xx , &c., for the mean values of the stresses taken over the space S l and P'XX=PXX~PXX (14), and defining S l to be such that the space variations of u, v, w are approximately constant over this space, we have, putting u'u', &c., for the mean values of the squares and products of the components of relative-mean-motion, for the equations of mean-mean-motion, du ( d . _ P ~jl = ~ l T ( Pxx r at [dx ^ pu'v') , d < - -7-v1 + ~T ( Pzx + PUW + OUW)\ dz ' ) &c. = &c. &c. = &c. .(15), which equations are approximately true at every point in the same sense as that in which the equations (1) of mean-motion are true. Subtracting these equations of mean-mean-motion from the equations of mean-motion, we have du! d -T- [p'xx f P (UU + U'll) + p (it'll! - u'u')} Ty { p ' yx + P ( UV> + u '^ + P ( u ' v ' ~ ^)1 (P'zx + P (UW' + U'w) + p (tl'w' - vfw' >&c., which are the equations of momentum of rclative-mean-motion at each point. Again, multiplying the equations of mean-mean-motion by u, v, w respectively, adding and putting 'IE =p(u~+ v- + vjfi), we obtain 62] AND THE DETERMINATION OF THE CRITERION. 555 dt *L dx d - ^ , dy dz pxx + pu'u')] -J- t M (P'J X + P U ' V ')] + j- L M (.Pzs + ptt'0] , ,^ + d_r-,- dz du _ du Pyx Ty dv dv . div dw ^ du dv dz . dw r-, du r , du -7 / du u u j- + u v -,- 4- u w -=- dx dy dz , dv dv , dv + vu -j- + v v -j- + vw -j- ax ay dz T-, dw r-, dw f,dw +w u -r- +w v -j- +w w T- dx dy dz ...(17), which is the approximate equation of energy of mean-mean-motion in the same sense as the equation (3) of energy of. mean-motion is approximate. In a similar manner multiplying the equations (16) for the momentum of relative-mean-motion respectively by u, v', w', and - adding, the result would be the equation for energy of relative-mean-motion at a point, but this would include terms of which the mean values taken over the space $, are zero, and, since all corresponding terms in the energy of heat are excluded, by sum- mation over the space S in the expression for the rate at which mean-motion is transformed into heat, there is no reason to include them for the space S t ; so that, omitting all such terms and putting 2#' = / 3(^ + 7 2 +w 7 .............................. (18), we obtain ( - , - \-J* + U J-+ V -J- + W ^- \dt dx dy dz - d\ ~ + JL K( d , j \P \P vx ax ^[^(P + , du' , du' p **fa +p *^7 d , , + j- C w dy 1 c^ dyt v d + d. d *'(?'. + pv'v')] + -j- [v' (p' zy + pv'w')] + [w' (p' yz + pw'v')] + [w' (p' u + pw'w')] dy , dv dx +Py dj, dv' du'\ dv> dx dy dw' f> u ' u 'fx +pU ' V ' cfy -7. dv -r-, dv + pvu d;K +p V v ^ r-, dw f-, dw . u, u 4- pw u -j- + pw v -j + pww -j- i dx ay dz puw j j dv , da- 556 THE DYNAMICAL THEORY OF INCOMPRESSIBLE VISCOUS FLUIDS [62 where only the mean values, over the space S 1} of the expressions in the right member are taken into account. This is the equation for the mean rate, over the space 8 lt of change in the energy of relative-mean-motion per unit of volume. It may be noticed that the rate of change in the energy of mean-mean- motion, together with the mean rate of change in the energy of relative- mean-motion, must be the total mean-rate of change in the energy of mean-motion, and that by adding the equations (17) and (19) the result is the -same as is obtained from the equation (3) of energy of mean-motion by omitting all terms which have no mean value as summed over the space $j. The Expressions for Transformation of Energy from Mean- mean- motion to Relative-mean-motion. 16. When equations (17) and (19) are added together, the only expres- sions that do not appear in the equation of mean-energy of mean-motion are the last terms on the right of each of the equations, which are identical in form and opposite in sign. These terms, which thus represent no change in the total energy of mean-motion, can only represent a transformation from energy of mean- mean- motion to energy of relative-mean-motion. And as they are the only expressions which do not form part of the general expression for the rate of change of the mean energy of mean- motion, they represent the total exchange of energy between the mean-mean-motion and the relative-mean- motion. It is also seen that the action, of which these terms express the effect, is purely kinematical, depending simply on the instantaneous characters of the mean- and relative-mean-motion, whatever may be the properties of the matter involved, or the mechanical actions which have taken part in determining these characters. The terms, therefore, express the entire result of transformation from energy of mean-mean-motion to energy of relative-mean-motion, and of nothing but the transformation. Their exist- ence thus completely verifies the first of the general conclusions in Art. 14. The term last but one in the right member of the equation (17) for energy of mean-mean-motiori, expresses the rate of transformation of energy of heat-motions to that of energy of mean-mean-motion, and is entirely independent of the relative-mean-motion. In the same way, the term, last but one on the right of the equation (19) for energy of relative- mean-motion, expresses the rate of transformation from energy of heat-motions to energy of relative-mean-motion, and is quite in- dependent of the niean-niean-motion. 62] AND THE DETERMINATION OF THE CRITERION. 557 17. In both equations (17) and (19) the first terms on the right express the rates at which the respective energies of mean- and relative-mean-motion are increasing on account of work done by the stresses on the mean- and relative-motions respectively, and by the additions of momentum caused by convections of relative-mean-motion by relative-mean-motion to the mean- and relative-mean-motions respectively. It may also be noticed that while the first term on the right, in the equation (19) of energy of relative-mean-motion, is independent of mean- mean-motion, the corresponding term in equation (17) for mean-mean-motion is not independent of relative-mean-motion. A Discriminating Equation. 18. In integrating the equations over a space moving with the mean- mean-motion of the fluid, the first terms on the right may be expressed as surface integrals, which integrals respectively express the rates at which work is being done on, and energy is being received across the surface, by the mean-mean-motion, and by the relative-mean-motion. If the space over which the integration extends includes the whole system, or such part that the total energy conveyed across the surface by the relative-mean-motion is zero, then the rate of change in the total energy of relative-mean-motion within the space, is the difference of the integral, over the space, of the rate of increase of this energy by trans- formation from energy of mean-mean-motion, less the integral rate at which energy of relative-mean-motion is being converted into heat, or, integrating equation (19), - d d _ d\ =,,, , j + j- + -j- 4 WT- 1 E dxdydz dx dy dz) -I ' ' au _T7T> du , ^77^7, du dz , dv -, dv -7-7 dv + pv'u' -.- + pvv + pvw dx dy dz a 77 dw dw T + PW V -j ix ay , du' , du' , du' dv , dv' , dv' , dw' , dw' , dw' ,dw dz I dxdydz (20). 558 THE DYNAMICAL THEORY OF INCOMPRESSIBLE VISCOUS FLUIDS [62 This equation expresses the fundamental relations : (1) That the only integral effect of the mean-mean-motion on the relative- mean-motion is the integral of the rate of transformation from energy of mean-mean-motion to energy of relative-mean-motion. (2) That, unless relative energy is altered by actions across the surface within which the integration extends, the integral energy of relative-mean- motion will be increasing, or diminishing, according as the integral rate of transformation from mean-mean-motion to relative-mean-motion is greater, or less than, the rate of conversion of the energy of relative-mean-motion into heat. 19. For p' xx , &c., are substituted their values as determined according to the theory of viscosity, the approximate truth of which has been verified, as already explained. Putting du' dv dw'\ du g + + / , &'., &c. we have, substituting in the last term of equation (20), as the expression for the rate of conversion of energy of relative-mean-motion into heat, [f[ d / IT\J j i /Y/T /<&*' M du/\ - If I -j- (pH) dxdydz = ||U_+_ + _) JJJ at JJJ [_ \dac dy dz ) du' dv dw'\ 2 ,M *V /*/ AA. ,M Af-^-l ...... \dy dz] \dz dx ) \dx dy J jj in which JJL is a function of temperature only ; or since p is here considered as constant, du'\' whence substituting for the last term in equation (20) we have, if the energy of relative-mean-motion is maintained, neither increasing nor diminishing, 7-7 du f-, du -j: du \UU -j- + U V j- + UW' - 1 - dx dy dz -P 4- v'u - - + v V 1- v'w' -,-\ dxdydz dx dy dz j r-jdw , ,dw r -,dw\ + wu , +wv -j- + ww -j- dx dy dz ! 62] AND THE DETERMINATION OF THE CRITERION. 559 dxdydz = Q ...(24), dw dv'\ 2 du d/uf\ z dx dy 1 which is a discriminating equation as to the conditions under which relative- mean-motion can be sustained. 20. Since this equation is homogeneous in respect to the component velocities of the relative-mean-motion, it at once appears that it is independent of the energy of relative-mean-motion divided by the p. So that if fijp is constant, the condition it expresses depends only on the relation between variations of the mean-mean-motion and the directional, or angular, distri- bution of the relative-mean-motion, and on the squares and products of the space periods of the relative-mean-motion. And since the second term expressing the rate of conversion of heat into energy of relative-mean-motion is always negative, it is seen at once that, whatsoever may be the distribution and angular distribution of the relative-mean-motion and the variations of the mean-mean-motion, this equation must give an inferior limit for the rates of variation of the components of mean-mean-motion, in terms of the limits to the periods of relative-mean-motion, and p/p, within which the maintenance of relative- mean-motion is impossible. And that, so long as the limits to the periods of relative-mean-motion are not infinite, this inferior limit to the rates of variation of the mean- mean -motion will be greater than zero. Thus the second conclusion of Art. 14, and the whole of the previous argument is verified, and the properties of matter which prevent the main- tenance of mean-motion, with periods of the same order of magnitude as those of the heat-motion, are shown to be amongst those properties of matter which are included in the equations of motion of which the truth has been verified by experience. The Cause of Transformation. 21. The transformation function, which appears in the equations of mean-energy of mean- and relative-mean-motion, does not indicate the cause y of transformation, but only expresses a kinematical principle as to the effect of the variations of mean-mean-motion, and the distribution of relative- mean-motion. In order to determine the properties of matter and the mechanical principles on which the effect of the variations of the mean- mean-motion on the distribution and angular distribution of relative-mean- motion depends, it is necessary to go back to the equations (16) of relative- 5GO THE DYNAMICAL THEORY OF INCOMPRESSIBLE VISCOUS FLUIDS [62 momentum at a point; and even then the cause is only to be found by considering the effects of the actions which these equations express in detail. The determination of this cause, though it in no way affects the proofs of the existence of the criterion as deduced from the equations, may be the means of explaining what has been hitherto obscure in the connection between thermodynamics and the principles of mechanics. That such may be the case, is suggested by the recognition of the separate equations of mean- and relative-mean-motion of matter. The Equation of Energy of Relative-mean-motion and the Equation of Thermodynamics. 22. On consideration, it will at once be seen that there is more than an accidental correspondence between the equations of energy of mean- and relative-mean-motion respectively, and the respective equations of energy of mean-motion and of heat in thermodynamics. If instead of including only the effects of the heat-motion on the mean- momentum, as expressed by p xx , &c., the effects of relative-mean-motion are also included by putting p xx for p xx + pu'u' , &c., and p yz for p yz + pw'v , &c., in equations (15) and (17), the equations (15) of mean-mean-motion become identical in form with the equations (1) of mean-motion, and the equation (17) of energy of mean-mean-motion becomes identical in form with the equation (3) of energy of mean-motion. These equations, obtained from (15) and (17), being equally true with equations (1) and (3), the mean-mean-motion in the former being taken over the space $1 instead of 8 as in the latter, then, instead of equation (9), we should have for the value of the last term du d(pH) -7 , rfw ft-SE + *" = - ti + >" lu +&c (2o) ' in which the right member expresses the rate at which heat is converted into energy of mean-mean-motion, together with the rate at which energy of relative-mean-motion is transformed into energy of mean-mean-motion ; while equation (19) shows whence the transformed energy is derived. The similarity of the parts taken by the transformation of mean-mean- motion into relative-mean-motion, and the conversion of mean-motion into heat, indicates that these parts are identical in form ; or that the conversion of mean-motion into heat is the result of transformation, and is expressible by a transformation function similar in form to that for relative-mean-motion, but in which the components of relative-motion are the components of the heat-motions, and the density is the actual density at each point. Whence it would appear that the general equations, of which equations (19) and (16) are respectively the adaptations to the special condition of uniform density, 62] AND THE DETERMINATION OF THE CRITERION. 561 must, by indicating the properties of matter involved, afford mechanical explanations of the law of universal dissipation of energy and of the second law of thermodynamics. The proof of the existence of a criterion, as obtained from the equations, is quite independent of the properties and mechanical principles on which the effect of the variations of mean-mean-motion on the distribution of relative-mean-motion depends. And as the study of these properties and principles requires the inclusion of conditions which are not included in the equations of mean-motion of incompressible fluid, it does not come within the purpose of this paper. It is therefore reserved for separate investigation by a more general method. The Criterion of Steady Mean-motion. 23. As already pointed out, it appears from the discriminating equation that the possibility of the maintenance of a state of relative-mean-motion depends on p/p, the variation of mean-mean-motion, and the periods of the relative-mean-motion. Thus, if the mean-mean-motion is in direction # only, and varies in direction y only, if u', v', w' are periodic in directions x, y, z, a being the largest period in space, so that their integrals over a distance a in direction x are zero, and if the co-efficients of all the periodic factors are a, then putting and taking the integrals, over the space a 3 , of the 18 squares and products in the last term on the left of the discriminating equation (24) to be /O 2 -18M ) a'a 3 , \ a / the integral of the first term over the same space cannot be greater than Then, by the discriminating equation, if the mean-energy of relative-mean- motion is to be maintained, pC? is greater than 700 . - % , CL P -V(S' = 700 . ...(26) is a condition under which relative-mean-motion cannot be maintained in a o. K. ii. 36 562 THE DYNAMICAL THEORY OF INCOMPRESSIBLE VISCOUS FLUIDS [62 fluid, of which the mean-mean-motion is constant in the direction of mean- mean-motion, and subject to a uniform variation at right angles to the direction of mean-mean-motion. It is not the actual limit, to obtain which it would be necessary to determine the actual forms of the periodic function for u, v', w', which would satisfy the equations of motion (15), (16), as well as the equation of continuity (13), and to do this the functions would be of the form ^ r A \ ( 2?r ' 2i A. cos \r \nt-\ x 1 V a where r has the values 1,2, 3, &c. It may be shown, however, that the retention of the terms in the periodic series in which r is greater than unity would increase the numerical value of the limit. 24. It thus appears that the existence of the condition (26) within which no relative-mean-motion, completely periodic in the distance a, can be maintained, is a proof of the existence, for the same variation of mean-mean- motion, of an actual limit of which the numerical value is between 700 and infinity. In viscous fluids, experience shows that the further kinematical con- ditions imposed by the equations of motion do not prevent such relative- mean-motion. Hence for such fluids equation (26) proves that the actual limit, which discriminates between the possibility and impossibility of relative-mean-motion completely periodic in a space a, is greater than 700. Putting equation (26) in the form /(pV-TOoA. V \dy/ pa? it at once appears that this condition does not furnish a criterion as to the possibility of the maintenance of relative-mean-motion, irrespective of its periods, for a certain condition of variation of mean-mean-motiori. For by taking a 2 large enough, such relative-mean-motion would be rendered possible whatever might be the variation of the mean-mean-motion. The existence of a criterion is thus seen to depend on the existence of certain restrictions to the value of the periods of relative-mean-motion on the existence of conditions which impose superior limits on the values of a. Such limits to the maximum values of a may arise from various causes. If dujdy is periodic, the period would impose such a limit, but the only restrictions which it is my purpose to consider in this paper, are those which arise from the solid surfaces between which the fluid flows. These restric- tions are of two kinds restrictions to the motions normal to the surfaces, 62] AND THE DETERMINATION OF THE CRITERION. 563 and restrictions tangential to the surfaces the former are easily defined, the latter depend for their definition on the evidence to be obtained -from experi- ments such as those of Poiseuille, and I shall proceed to show that these restrictions impose a limit to the value of a, which is proportional to D, the dimension between the surfaces. In which case, if /Y*?Y - E V\dy) '' = D' equation (26) affords a proof of the existence of a criterion of the conditions of mean- mean-motion under which relative or sinuous- motion can continuously exist in the case of a viscous fluid between two continuous surfaces perpendicular to the direction y, one of which is main- tained at rest, and the other in uniform tangential-motion in the direction x with velocity U. SECTION III. The Criterion of the Conditions under which Relative-mean-motion cannot be maintained in the case of Incompressible Fluid in Uniform Symmetrical Mean-flow between Parallel Solid Surfaces. Expression for the Resist- ance. 25. The only conditions, under which definite experimental evidence as to the value of the criterion has as yet been obtained, are those of steady flow through a straight round tube of uniform bore ; and for this reason it would seem desirable to choose for theoretical application the case of a round tube. But inasmuch as the application of the theory is only carried to the point of affording a proof of the existence of an inferior limit to the value of the criterion, which shall be greater than a certain quantity deter- mined by the density and viscosity of the fluid and the conditions of flow, and as the necessary expressions for the round tube are much more complex than those for parallel plane surfaces, the conditions here considered are those defined by such surfaces. Case I. Conditions. 26. The fluid is of constant density p and viscosity p, and is caused to flow, by a uniform variation of pressure dp/dx, in direction x between parallel surfaces, given by y=b , y = b (28), the surfaces being of indefinite extent in directions z and x. 362 564 THE DYNAMICAL THEORY OF INCOMPRESSIBLE VISCOUS FLUIDS [62 The Boundary Conditions. (1) There can be no motion normal to the solid surfaces, therefore v = when y=b ............................. (29). (2) That there shall be no tangential motion at the surface, therefore u = w=Q when y = b ......................... (30); whence by equation (21), putting u for u', p yx = fsdu/dy. By the equation of continuity du/da; + dv/dy + dw/dz = 0, therefore at the boundaries we have the further conditions, that when y = b , du/dx = dv/dy = dwfdz = ........................ (31). Singular Solution. 27. If the mean-motion is everywhere in direction x, then, by the equation of continuity, it is constant in this direction, and as shown (Art. 8) the periods of mean-motion are infinite, and the equations (1), (3), and (9) are strictly true. Hence if v = w = u' = v' = w' = ......................... (32), we have conditions under which a singular solution of the equations, applied to this case, is possible whatsoever may be the value of b , dpjdx, p and /*. Substituting for p xx , p yz , &c., in equations (1) from equations (21), and substituting u for u', &c., these become This equation does not admit of solution from a state of rest*; but assuming a condition of steady motion such that du/dt is everywhere zero, and dp/dx constant, the solution of * In a paper on the "Equations of Motion and the Boundary Conditions of Viscous Fluid," read before Section A at the meeting of the B. A., 1883, I pointed out the significance of this disability to be integrated, as indicating the necessity of the retention of terms of higher orders ;o complete the equations, and advanced certain confirmatory evidence as deduced from the theory of gases. The paper was not published, as I hoped to be able to obtain evidence of a more definite character, such as that which is now adduced in Articles 7 and 8 of this paper, which shows that the equations are incomplete, except for steady motion, and that to render then. mtegrable from rest the terms of higher orders must be retained, and thus confirms the argument I advanced, and completely explains the anomaly. (See Paper 46, page 132.) 62] AND THE DETERMINATION OF THE CRITERION. 565 (34). fj, /d*u d*u\ _ 1 dp _ p \dy~ dz*J p dx if u = du/dz = when y = b , _ 1 dp y 2 - 6 2 IS U -j x IJL dx 2 This is a possible condition of steady motion, in which the periods of u, according to Art. 8, are infinite ; so that the equations for mean-motion as affected by heat-motion, by Art. 8, are exact, whatever may be the values of u, b , p, p,, and dpjdx. The last of equations (34) is thus seen to be a singular solution of the equations (15) for steady mean-flow, or steady mean-mean-motion, when u, v', w', p, &c., have severally the values zero, and so the equations (16) of relative-mean-motion are identically satisfied. In order to distinguish the singular values of w, I put rb u = U, I udy = Zb U m ; j- (35). dp ZH TT TT 3 TT & 2 -; whence j = ~lh m ' ^ = n^m IT According to the equations, such a singular solution is always possible where the conditions can be realized, but the manner in which this solution of the equation (1) of mean-motion is obtained affords no indication as to whether or not it is the only solution as to whether or not the conditions can be realized. This can only be ascertained either by comparing the results as given by such solutions with the results obtained by experiment, or by observing the manner of motion of the fluid, as in my experiments with colour bands. The fact that these conditions are realized, under certain circumstances, has afforded the only means of verifying the truth of the assumptions as to the boundary conditions, that there shall be no slipping, and as to /* being independent of the variations of mean-motion. Verification of the Assumptions in the Equation of Viscous Fluid. 28. As applied to the conditions of Poiseuille's experiments and similar experiments made since, the results obtained from the theory are found to agree throughout the entire range so long as u', v, w' are zero, showing that if there were any slipping it must have been less than the thousandth part of the mean-flow, although the tangential force at the boundary was 0'2 gr. 566 THE DYNAMICAL THEORY OF INCOMPRESSIBLE VISCOUS FLUIDS [62 per square centimetre, or over 6 Ibs. per square foot, the mean flow 376 millims. (1*23 feet) per second, and du/dr = 215,000, the diameter of this tube being 0'014 millim., the length T25 millims., and the head 30 inches of mercury. Considering that the skin resistance of a steamer going at 25 knots is not 6 Ibs. per square foot, it appears that the assumptions, as to the boundary conditions and the constancy of p, have been verified under more exigent circumstances, both as regards tangential resistance and rate of variation of tangential stress, than occur in anything but exceptional cases. Evidence that other Solutions are possible. 29. The fact that steady mean-motion is almost confined to capillary tubes, and that in larger tubes, except when the motion is almost insensibly slow, the mean-motion is sinuous and full of eddies, is abundant evidence of the possibility, under certain conditions, of solutions other than the singular solutions. In such solutions u', v, w have values, which are maintained, not as a system of steady periodic motion, but such as has a steady effect on the mean- flow through the tube ; and equations (1) are only approximately true. The Application of the Equations of the Mean- and Relative- mean-motion. 30. Since the components of mean-mean-motion in directions y and z are zero, and the mean flow is steady, v = 0, w = Q, duldt=Q t du/dx=0 ............. (36), and as the mean values of functions of u', v, w' are constant in the direction of flow, ...... dx dx dx By equations (21) and (37) the equations (15) of mean-motion become du dp /d*u du\ (d ,-r-,. d ,-r-,.}\ -- - dw dp ~ 62] AND THE DETERMINATION OF THE CRITERION. The equation of energy of mean-mean-motion (17) becomes d (E) dp (d fdu\ d I dii\] \ = - u J L + p 4 j- [u -j- ) + -j- (u -j- )} dt ax [dy \ dy] dz \ dz)} d r -r* . d ,- -T-A) (/du\* . /du^ 507 ....(39). du -7-7 du + p luv -j- + uw -j- { dy dz Similarly the equation of mean-energy of relative-mean-motion (19) becomes dE' d dt ~ dy -[ (iz [ ' v} + W< -^ + pUw) + V ( p'zy + pv'lv Ty dydz = * j j 2 [rf ^ -^ - & ^J ^ J dy ag+i sin (2s+l)p],\ ' (68) ' where p = 7n//2& ) To satisfy the boundary conditions s = 2r, when s is even, s = 2r + 1, when s is odd. t = 2r + 1, when t is odd, = 2 (r + 1), when t is even. Since a = 0, when p = ^TT, ioo / \ A .(69). and since dfi/dy = 0, when p = %TT, 2 " {- (4r + 2) 6 4r+2 + 4 (r + 1) 6 4r+4 } = j From the form of K l it is clear that every term in the series for a and /3 increases the value of K 1 and to an extent depending on the value of r. K 1 will therefore be a minimum, when a = ! sin p + a s sin ft = b 2 sin 2p + 6 4 sin which satisfy the boundary conditions if (70), .(71). Therefore we have, as the values of a and /8, which render K l a minimum for any value of I a.1 a-i = sin p + sin 3p, /3/6 2 = sin 2p + ^ si And a ~r = cos + 3 cos %>, -^ -y- = 2 cos 2w + 2 cos 4w Trttj ay 7rb 2 ay 26 /arf/8 /3da\ 1 f -T 2 - ~ ^j )= - {- 3 sm - 3 sin 3p + sm 5 + sm lp\ \ a dt/ / 4 l ...(72) and integrating twice j - b a J -b a \ dy dy / TT T> .Li.' f T C 7 rutting j- L for I, the denominator of ^ KI, equation (67), becomes - l-325Xo 1 6 2 . /TOX (73). 62] AND THE DETERMINATION OF THE CRITERION. 575 In a similar manner the numerator is found to be - Y {L< (2a, 2 + 1-256 2 2 ) + 2# (lO^ 2 + 8& 2 2 ) + 82^ + 806 2 2 }, \ZOo/ and as the coefficients of a^ and 6 S are nearly equal in the numerator, no sensible error will be introduced by putting 2 = - o,, 3 Z< + 2 x 5'53 2 + 50 2> 0-408Z which is a minimum if = 1-62 ................................. (75) and ^ = 517 ................................. (76). Hence, for a flat tube of unlimited breadth, the criterion p2b U m /fji is greater than 517 ..................... (77). 37. This value must be less than that of the criterion for similar circumstances. How much less it is impossible to determine theoretically without effecting a general solution of the equations ; and, as far as I am aware, no experiments have been made in a flat tube. Nor can the experi- mental value 1900, which I obtained for the round tube, be taken as indicative of the value for a flat tube, except that, both theoretically and practically, the critical value of U m is found to vary inversely as the hydraulic mean depth, which would indicate that, as the hydraulic mean depth in a flat tube is double that for a round tube, the criterion would be half the value, in which case the limit found for K l would be about This is sufficient to show that the absolute theoretical limit found /< = ~^r ~ is of the same order of magnitude as the experimental value ; so that the latter verifies the theory, which, in its turn, affords an explanation of the observed facts. The State of Steady Mean-motion above the Critical Value. 38. In order to arrive at the limit for the criterion it has been necessary to consider the smallest values of u' t v, w', and the terms in the discriminating equation of the fourth degree have been neglected. This, however, is only necessary for the limit, and, preserving these higher terms, the discriminating equation affords an expression for the resistance in the case of steady mean- mean-motion. 576 THE DYNAMICAL THEORY OF INCOMPRESSIBLE VISCOUS FLUIDS [62 The complete value of the function of transformation as given in equation (64) is Whence putting U + u U, for u in the left member of equation (77), and integrating by parts, remembering the conditions, this member becomes ^npf dy r pu'v'dy + -f (u'vjdy (78), PO J -6 J -&o /* ' -&o in which the first term corresponds with the first term in the right member of equation (64), which was all that was retained for the criterion, and the second term corresponds with the second term in equation (64), which was neglected. Since by equation (35) we have, substituting in the discriminating equation (43), either ( [dldt(pH')dy ? [* (Mv 2 bjdp 2V U ^ + ^l- bn (UV)d V.6 3 ' A p _, ay uv ay J-b y J-b a d z u dp 3f-- < 80 >- Therefore, as long as - p -f- 3 r p* dx is of constant value, there is dynamical similarity under geometrically similar circumstances. The equation (79) shows that, when ~p~-f- is greater than K, 3 r p 2 dx uv' must be finite, and such that the last term in the numerator limits the rate of transformation, and thus prevents further increase of u'v'. 62] AND THE DETERMINATION OF THE CRITERION. 577 The last term in the numerator of equation (79) is of the order and degree p 2 Z 4 a 4 //i- as compared with Z 4 a 2 , 1 t? the order and degree of - -r (pH'} the first term in the numerator. It is thus easy to see how the limit comes in. It is also seen from equation (79) that, above the critical value, the law of resistance is very complex and difficult of interpretation, except in so far as showing that the resistance varies as a power of the velocity higher than the first. o. R. u. 37 63. EXPERIMENTS SHOWING THE BOILING OF WATER IN AN OPEN TUBE AT ORDINARY TEMPERATURES. (Exhibited before Section A, Brit. Assoc., 1894, at Oxford.) AMONG the many phenomena, the secrets of which have been preserved by the deadening influence of familiarity on curiosity, there is perhaps none more remarkable than that of the ' singing of the kettle on the hob,' which has many times been the subject of sentiment and verse but not, it would seem, hitherto a subject of physical study which like the study of the rain- bow might afford evidence as to the conditions under which we exist. That the cheering evidence of the readiness of the social gathering is not the only evidence to be obtained from the song of the kettle will in the first place be demonstrated in these experiments. Thus, having analyzed by experiment the physical causes of this sound and its variations, the purpose of the experiments is to demonstrate the relation which exists between sounds in the kettle and sounds produced by the motion of water, or any liquid, under certain common conditions. And, in the third place, to demonstrate the general fact that liquids flowing between fixed boundaries emit no sound as long as they continuously occupy the space between the boundaries, and thence to demonstrate that when such sound occurs it is evidence of the boiling of the water. If we place a kettle on the top of a fire, the first evidence of action is that of a somewhat feeble and intermittent hissing sound which at first increases and becomes continuous and then again subsides as the temperature in- creases. This is followed by a much more definite and harsher sound which comes on suddenly, somewhat increases in volume, then suddenly softens 63] EXPERIMENTS SHOWING THE BOILING OF WATER, ETC. 579 and is immediately followed by the exit of steam showing that the water is boiling. If a glass flask is substituted for the opaque kettle the causes of the sound and its variations become apparent. The water in the flask is under the pressure of the atmosphere at its upper surface, which pressure is increased at points below the surface by the water above ; so that the boiling-point at the bottom of the kettle is somewhat above that higher up. The water receives its heat from the fire below by conduction through the metal, or glass, and the water between the bottom and the point con- sidered. The conduction through water is very slow ; so that the water in imme- diate contact with the hot surface at the bottom becomes much hotter than the water immediately above. Water expands with heat. Hence this hot layer on the bottom is in unstable equilibrium, and vertical convection currents are set up which carry the hot water from the bottom into the colder water above. Owing however to the eddying motion which is a consequence of the resistance offered to the ascending currents by the water above, these currents do not follow a straight course but, somewhat rapidly, interweave, as thin sheets, with the surrounding water ; so that the heat is soon diffused through the flask, leaving very little variation of temperature except close to the bottom of the flask or kettle. These convection currents are most vigorous soon after the kettle is put on the fire, when there is the greatest difference of temperature between the water on the bottom and the water above. In this condition however there is no sound, since the vigour of the currents, owing to the greater density of the water above, carry away the water, heated on the bottom, before it has reached a sufficient temperature. Then as the water above acquires heat through the agency of these currents, these currents diminish in vigour but still continue. When a certain temperature, about 174 F., at the upper surface is reached (which depends on the amount of air occluded in the water) bubbles begin to collect on the surface at the bottom of the flask and then to rise in increasing numbers. These bubbles do not vanish but rise to the surface, increasing in size as they ascend. They are a consequence of the tension of the occluded air added to that of the vapour. When a bubble first appears there is a sharp but slight click and these clicks, as they become numerous, constitute the preliminary hiss, which nearly subsides before the temperature reaches 200 F. At about 10 below the boiling-point the harsh hiss comes on suddenly 372 580 EXPERIMENTS SHOWING THE BOILING OF WATER [63 and, in the glass flask, it may be observed that, simultaneous with this sound, there appear again bubbles on the bottom of the flask, which bubbles grow on the bottom gradually until they leave the surface, and start to rise, when unlike the previous bubbles of air they suddenly collapse with a sharp click, which being rapidly repeated causes the harsh hiss. The reason of the collapse of these bubbles is that they are bubbles of steam at the tem- perature of the boiling-point at the bottom of the flask, formed between the surface of the glass, or metal, and held down by capillary action until they are large enough to break away and ascend, when their ascension brings the steam into contact with the colder water above, when, being free from air, their collapse is sudden and sonorous. As the temperature still further increases and the difference of tem- perature between the water at the bottom and that which is above diminishes, the bubbles rise higher and higher before condensing but still collapse suddenly, until the bubbles rise to the surface, when the water boils and the sharp sound subsides as suddenly as it came on. This analysis of the sound phenomenon of the kettle, which owing to our familiarity with it, has hitherto attracted but little notice, throws very definite light on a fact of the greatest importance to physics, which it would appear has met with partial recognition only. The question as to whether the motion of continuous liquid between solid boundaries with which it is everywhere in contact can produce sound, as a consequence of the motion, has not I believe hitherto received any definite answer. The general association of sound with running water has doubtless obscured the subject, although for the most part where it occurs the source of such sound may be easily traced to the variation of the positions of the surface of the water, and particularly where the surface is discontinuous or intermittent. But, apart from such sources of sound, it is a matter of familiar obser- vation that the flow of water through pipes under great pressure, as when, in the water supply of a town, the water is brought from below the surface of a reservoir on a continuous slope into houses or mains several hundred feet below the reservoir, and is generally attended with a hissing noise ; and of this I believe no explanation has hitherto been given. Nor have I ever heard anyone suggest that there is any connection between the singing of the kettle and the hiss which almost invariably attends the opening of a tap in a pipe under considerable pressure as in a town's service. Yet when observed the hiss of the pipe closely resembles the harshest sound of the kettle. 63] IN AN OPEN TUBE AT ORDINARY TEMPERATURES. 581 It is now some years since I was led, as the result of hydrodynamical analysis applied to a fluid having the physical properties of water, to the conclusion that both these sources of sound have the same origin. In hydrodynamics it is customary to consider the physical properties of the fluid as consisting of incompressibility and perfect fluidity only, no account being taken of internal cohesion or of adhesion to solid surfaces, as between water and glass, and still less of any vapour tension in spaces not occupied by the water. With these limited properties the hydrodynamical problem only admits of solution when the circumstances are such that the pressure is every- where positive, so that there could be no possibility of disruption of the fluid. The case however is entirely changed when we recognise that the water has cohesion, depending on its freedom from occluded air as well as viscosity, and that where the water is discontinuous the spaces are filled with vapour at a tension corresponding to the temperature. It has long been known, as shown by Bernoulli, that when water flows along a contracting channel which it completely occupies, the pressure falls approximately according to the law that the sum of the intensity of pressure p and the product of the density of mass multiplied by the half of the vis viva is constant, or p + pv* = a constant. Thus if water flows from below the surface of a reservoir, of unlimited dimensions, through a conical tube, the small end of which is in connection with the receiver of an air-pump from which all air has been removed, the small end of the pipe being at the level of the surface of the reservoir, supposing that there is no vapour tension, and the pressure of the atmosphere 1470 Ibs. on the square inch, the water enters the receiver with a velocity of 46 "5 feet per second. If however the temperature of the water is 59 F. the vapour tension is 0'241 Ibs. per sq. inch ; so that on entering the receiver the water would boil, and if the water and vapour were continually removed the experiment might be continued indefinitely the water enters in the receiver at 59 F. and boiling, so as to maintain the vapour tension something less than 0'241 Ibs. per square inch. In this case we have a continuous stream of water boiling at the ordinary temperature 59 F. But this cannot be said to be boiling in an open tube. And it is important to notice that although the water at the neck entering the receiver would be boiling, the temperature of the receiver would be 582 EXPERIMENTS SHOWING THE BOILING OF WATER [63 maintained at, or about, 59, so that there would be no condensation of bubbles in the stream of water, and hence the only hissing sound would be that resulting from the disruption of the water as in the preliminary hiss in the kettle when the air bubbles are coming off. If instead of withdrawing the water and vapour by means of the air-pump we can by taking off the receiver and connecting the small end of the conical tube, so far contracting in the direction of flow, with a similar conical tube the other way about, i.e., expanding in the direction of flow and discharging into a reservoir at a lower level than that of the supply, make such arrangements that the momentum of the water entering the diverging pipe at the minimum section would be sufficient to sweep out the water and bubbles of vapour and air which had been formed in the contracting tube, and secure in the expanding pipe a law of pressure and velocity somewhat similar to that of the contraction : p + pip = a constant. Then as the bubbles of air and vapour in the stream would be carried with great velocity from the low pressure at the neck, where they formed, into the higher pressure in the wider portion of the expanding tube ; so that the pressure being greater than the vapour tension, condensation would ensue and the bubbles would collapse, producing the hiss of the kettle before boiling, and in this case we should have water boiling in an open tube. Although certain conditions are necessary a simple experiment shows that these may be realized. Take a glass tube, say, half-an-inch internal diameter and six inches long, and draw it down in the middle so as to form a restriction with easy gradual curves so that the inside diameter in the middle is something less than the tenth of an inch, leaving the parallel ends of the tube something like 2 inches each. And then connect one of these parallel ends by flexible hose to a water main which is controlled by a tap. Then, on first opening the tap, the water entering from the main at A will fill the tube as far as the restriction, and pass through the restriction, but it will not, in the first instance, of necessity fill the tube on the far side of the restriction. If the water is turned on very slowly and the open end of the tube is inclined upwards, then the water will accumulate and fill the tube, displacing the air. But if the water is turned on sharply so that when it reaches the neck it has a velocity of 40 or 50 feet a second, the water after passing the minimum section will preserve its velocity and shoot out as a jet from a squirt, not touching the sides of the glass, while if the open end of the tube be held downwards the water, whatever the velocity, will, after passing the restriction, run out of the tube without filling it. 63] IN AN OPEN TUBE AT ORDINARY TEMPERATURES. 583 In neither of these cases is there any hiss or sound except such as is caused by the free jet passing through the air. But on holding the open end of the tube upwards and quietly filling both limbs of the tube by opening the tap very quietly, as in case (1), and then turning on more water, the water will not shoot out in a jet but will come out like any other stream as it might do if there were no restriction. At first, while the velocity through the neck is below 50 feet per second, there is no sound, but as soon as a velocity of 54 feet per second is attained, or a little more, a distinct sharp hiss is heard exactly resembling that of the kettle or the hiss of the water through a tap. So far however this is no proof that the hiss is the result of the boiling or disruption of the water. But the hiss is not the only evidence afforded by the experiment. If the glass tube, through which the water is flowing at velocities below that at which the sound comes on, be carefully examined against a black ground to see whether there are any imperfections in the glass in the region of the neck, such as minute bubbles, and the positions of any such carefully located ; and if then, after increasing the flow so that the 584 EXPERIMENTS SHOWING THE BOILING OF WATER [63 hiss just begins, it be carefully examined again, a small white speck will be observable somewhere in the region of the minimum section a little before where the water enters the neck. This is always observable unless obscured by imperfection in the glass. And a crucial test is afforded by varying the tap so as to bring the hiss on and off; when it will be seen that the appear- ance and disappearance of the spot and the starting and stopping of the hiss are simultaneous. The white spot against a dark ground indicates reflection of light by a frost-like surface such as would be afforded by bubbles coming on and going off rapidly, and is thus a crucial proof of disruption in the water or between the water and the glass. The sound, when it first comes on, is generally loud enough to be heard distinctly over a lecture room, and any increase in the flow augments the sound as well as the size of the spot. See Fig. 4, page 583. During the experiment the water is quietly flowing out of the tube running with a full bore but with rather an uneven surface, which indicates some internal disturbance. If however the parallel parts of the tube leading to the open end be examined both when the hiss is on and off another phenomenon will be observed, which again furnishes evidence of the effects of boiling. When the hiss is on, the water in the tube will be somewhat opaque rather foggy which fog disappears after the hiss is stopped. This fog is caused by the separation of the air occluded in the water, and corresponds exactly to the separation of the air, as when the tem- perature of the water in the kettle is above 174 F. In the case of the tube the bubbles of air, which separate out, are very much smaller than those in the kettle on account of the greater violence of the action. If however instead of holding the tube with the mouth inclined upwards the tube be immersed in a beaker of water, the water coming out of the tube into the beaker will present the appearance of clouds of white smoke from a chimney. On close examination it is seen that the whiteness is due to minute bubbles, while the cloud-like appearance in the beakers is owing to the fact that the air in these bubbles is being somewhat rapidly again occluded by the water in the beaker, while the motion of the water in the beaker, disturbed by the flow from the tube, wafts the foggy water as it leaves the tube in directions which are continually changing until the reocclusion terminates their existence, leaving those parts of the water farthest from the mouth of the tube practically clear. In showing this experiment it is not my object to enter into the hydro- dynamical and physical considerations, on which the explanation of increase 63] IN AN OPEN TUBE AT ORDINARY TEMPERATURES. 585 of pressure as the water flows along the diverging tube depends. And I will conclude by pointing out that these considerations are entirely distinct from those on which the fall of pressure in the water proceeding along the converging channel depends. In the latter eddies or tumultuous motion the water has no function other than that of diminishing the rate at which the pressure falls, while in the former the rate of increase in the pressure depends entirely on this sinuous eddying or tumultuous motion. It has been proved definitely that water moving between solid boundaries has two manners of motion depending on whether the value of the quantity expressed by is greater or less than a certain numerical constant K. In a parallel pipe water in tumultuous motion entering with a velocity V such that - is greater than 1400, will, as it flows in the pipe at a steady rate, convert all the eddying motion into heat. While if water enters a pipe without tumultuous motion, such motion will be generated if - r\ir is greater than 1900. These limits have been for some time established as the limits of the criterion K for straight smooth pipes, and having thus found the limiting values of the purely numerical physical constant, it still remains to find the form of the function corresponding to - under other boundary conditions 586 EXPERIMENTS SHOWING THE BOILING OF WATER [63 such as parallel pipes with sections other than round and smooth and for converging or diverging boundaries. Such determinations present analytical difficulties which have not been altogether overcome. But it has been possible to obtain from analysis evidences that in converging pipes the critical velocity increases very rapidly with the rate of convergence and, on the other hand, that the critical velocity in diverging pipes diminishes very rapidly as the divergence increases. When the mean velocity of the water taken over the section of the pipe, whether parallel, converging or diverging, is greater than the critical velocity, there is a steady fall of pressure all along the channel and no rise of pressure in any part, as long as the flow is horizontal. But as soon as the rate of mean flow exceeds the critical velocity the motion becomes tumultuous the water moving in all directions across the channel as well as along the channel ; so that the continual mixing up of the water which has high forward velocity with that which has less, effected by the lateral motion, ensures a nearly uniform velocity of mean flow across the channel between the boundaries, except at the actual boundaries. The eddies or tumultuous motion represent an irreversible loss of head or vertical energy in the outflowing stream, but this loss is definitely controlled by the laws of momentum, and were it not for the resistance at the boundaries this law would admit of analytical expression. Thus, taking u for the velocity of flow, /(tan 6} as expressing the diverg- ence of the boundaries, # as the direction of flow, A for the area of the section, A c the area at the neck, PC + pu e * - /(tan 0) pu* . jl - ^J j = gpH. Such law is only approximately fulfilled on account of our want of definite knowledge of the resistance at the boundaries. But it is com- paratively easy to experimentally determine the value of the function /(tan 0} for some particular arrangement, and it is found that the same law holds for all geometrical similar arrangements however different the dimensions may be, provided that the velocity is inversely proportional to the linear dimensions. It also appears that if the divergence, as expressed by tan 0, is small, owing to the greater length the water has to traverse in the diverging channel to attain equal total divergences, the loss of head owing to the resistance at the boundaries exceeds the resistance where tan 6 is greater ; so that there is a particular value of tan 6 for which the loss of head is a minimum. And it is found by experiment that when tan 6 is such that the loss is a minimum the loss of head is about 0'4. Taking this to be the total loss of head in the whole arrangement, it follows as a direct consequence 63] IN AN OPEN TUBE AT ORDINARY TEMPERATURES. 587 that with the pressure of the atmosphere 1470 Ibs. per square inch, and the temperature 59 F. giving a vapour tension 0'241 per square -inch, the minimum pressure necessary to reduce the pressure at the neck to the vapour tension would be 1470 -0-241 0-6 + 0-241 = 24-34, or subtracting the pressure of the atmosphere the excess of pressure in the reservoir over and above that of the atmosphere is 9'64 Ibs. per square inch. In this case, supposing there were very little air occluded in the water there would be no boiling or rupture in the water, but with the usual amount of air the ruptures would occur under a somewhat less difference of head, such rupture corresponding to the preliminary discharge of air in the kettle. If the head is increased the point of rupture takes place earlier, that is, at a point before the neck is reached, and the supply of air being strictly limited the pressure will fall until the water boils, sending forth the hissing, or it may be screaming, sound resulting from the sudden condensation of the vapour entering the higher pressure after passing the neck and producing the further evidence of disruption already pointed out. And thus demon- strating that the only sound due to the flow of water between solid boundaries results from the boiling or disruption of the water, whether the actual source of the sound is the disruption or the subsequent condensa- tion of the vapour in the vacuum produced. 64. ON THE BEHAVIOUR OF THE SURFACE OF SEPARATION OF TWO LIQUIDS OF DIFFERENT DENSITIES. [From the Ninth Vol. of the Fourth Series of the " Memoirs and Proceed- ings of the Manchester Literary and Philosophical Society." Session 189495.] (Read March 19, 1895.) THE paradox first noticed by Benjamin Franklin which was brought before the Society by Dr Schuster at the last meeting, namely, that when a glass vessel containing water and oil, so that the oil floats on the top of the water, forming two surfaces, one the upper surface of the water and lower surface of the oil, the other the surface between the oil and the air, is moved with a periodic motion, the surface separating the two fluids is much more sensitive and much more disturbed than the upper surface, is very striking, even when the motion of the vessel is somewhat casual such as may be imparted by the hand. And the paradox becomes even more pronounced when the vessel is, by suspension or otherwise, subject to regular harmonic motion in one plane, and compared with a vessel similar in all respects and similarly situated, except that it contains one fluid only. For while the upper surface of the oil appears to follow the motion of the vessel, remaining very nearly perpendicular to the line of suspension, as it would if the whole mass were a solid, the free surface of the water in the vessel without oil has a decidedly greater amplitude than that of the line of suspension, though the oscillations are exactly in the same phase and the amplitude is still small. On the other hand the surface separating the oil and water has an oscillatory motion about the line of suspension, much greater in magnitude than that of the surface of the water in the vessel without oil, and in exactly the same or the opposite phase. Another very striking fact is that all the surfaces appear to remain plane surfaces when the motion is within certain 64] ON THE BEHAVIOUR OF THE SURFACE OF SEPARATION, ETC. 589 considerable limits. These motions, however, do depend on the relations between the length of the pendulum, the size of the vessel, and the depths of the fluids, the phase of the separating surface changing from the same to the phase opposite to that of the line of suspension if the pendulum is shortened. The solution of the problem presented by this paradox, although not altogether confirmed or fully worked out, appears to be in- dicated by the fact that the oscillations of all the surfaces are steady, and in the same or opposite phase with the line of suspension, that is, in the same or opposite phase with the disturbance, together with the fact that the free surfaces of the fluids remain nearly plane. For if any material system is, when disturbed, capable of oscillating in a particular period (its natural period), and such oscillation is subject to a viscous resistance, then if subject to a very gradually increasing disturbance, having a period longer than the natural period, the system will oscillate in the period of disturbance always in the same phase as the disturbing force. But if the disturbance has a period shorter than the natural period, the system will oscillate in the same period as the force, but in the opposite phase. Now, in the vessel with oil and water three systems of oscillation, or wave motions, are possible. If the vessel were completely full, so that there were no free surface, and if there were no oil, no oscillation would be possible except (1) the pendulous motion. If half full of oil and filled up with water, then, if disturbed and left, a wave motion (2) in its natural period would be set up in the surface between the oil and water. In the same way (3) if the vessel were half full of water without oil. But in the latter case (3) the natural period would be two or three times less than (2) between the oil and water. Now, when the vessel contains oil and water, disturbances (2) and (3) will both be set up, and might continue, till destroyed by viscosity, in their natural periods if these were the same, but the periods being different, the oscilla- tions in the period (3) would cause periodic disturbance in (2), and the natural period of (3) being much shorter than that of (2), the oscillation so maintained in (2) would be in opposite phase to (3), but, owing to viscosity, such maintenance would be of short duration. If, however, the natural period of the pendulous motion (1) of the vessel were in magnitude between the periods (3) and (2), smaller than (2) and greater than (3), then it would maintain an oscillation in the same period as the pendulous motion in (3) and also in (2), that in (3) having the same phase as the pendulum, that in (2) having the opposite phase. So far this explanation is only partial, as it is assumed that there will be a disturbance in (2) in the same phase as in (3). That this must be the case, however, becomes evident when it is considered that the motion of the water cannot be that of a solid, but must be irrotational, and that the disturbance arises from the non-spherical form of the surfaces of the fluids. If the surface of the vessel were flexible, the motion of the fluids would be essentially that of a 590 ON THE BEHAVIOUR OF THE SURFACE OF SEPARATION, ETC. [64 particular portion of the water in a long wave adjacent to the surface as shown in the figure. In this, the plain lines indicate lines in the water at rest, which take the position of the dotted lines when the wave surface has the position of the thick dotted line. The black circle indicates the surface of the spherical vessel ; and the dotted curve shows the shape this surface would become if it were subject to the same distortion as the water. In fact, the vessel is rigid, and the surface of the water must conform to it, which requires further internal distortional motion of the water. It is seen there is an excess of water at the top on the higher side and a deficiency on the lower, to supply which the upper surface must be still further tipped, while there is a deficiency on the higher side below and an excess on the lower side, to remedy which the lower surface must tip in the opposite direction. This is exactly what is seen with the oil and water, and is there though it cannot be seen in the water, although not to so great an extent because there is no possibility of an internal wave as between the oil and water. 65. ON METHODS OF DETERMINING THE DRYNESS OF SATURATED STEAM AND THE CONDITION OF STEAM GAS. [From Volume 41, Part I. of " Memoirs and Proceedings of the Manchester Literary and Philosophical Society." Session 1896-97.] (Read November 3, 1896.) WHEN, after all air has been expelled from a vessel partially filled with water and kept at rest at a constant temperature, equilibrium is established, the vapour is said to be dry saturated steam. It is easy to show that under these circumstances the pressure of the steam is a definite function of the temperature. But it has been found very difficult to show, by direct means, that the density of the steam is also an invariable function of the temperature, although many experiments, from the time of Watt, have indicated that this is the case; those of Fairbairn and Tate being the least open to criticism. That the density of dry saturated steam is a constant function of the temperature has, however, been completely established indirectly by the experiments of M. Regnault on the total heat of evaporation, although these experiments do not directly furnish a measure of the density. These experiments consisted in maintaining a vessel containing a definite quantity of water in steady constant condition as to temperature and pressure and quantity of water, by the steady admission of water at any constant tem- perature, and the withdrawal of the vapour in an upward direction, with a slow motion so as to preclude the convection of water out of the vessel by the steam, the steam so withdrawn being condensed in a calorimeter back again 592 ON METHODS OF DETERMINING THE DRYNESS OF [65 to water at any constant temperature. The results proving that the total amount of heat given up by the steam for each temperature in the boiler is consistently proportional to the weight of steam condensed. It thus appears that the density of saturated steam at constant tem- perature must be constant, and that gravity alone is sufficient to free the saturated steam from any water that may have been entangled with it by the action of boiling, provided the rate of flow over the surfaces is not sufficient to carry along with the steam any water there may be on the surfaces. It was only after the utmost care in securing these conditions that Regnault succeeded in obtaining consistent results which results have since been confirmed by many researches, including that of Messrs Harker and Hartog read before the Society last year. It is to be noticed that the whole theory of the properties of steam, as at present accepted, and all the steam tables, are founded on these experiments of Regnault's on the total heat of evaporation, so that if any other definition is given of dry saturated steam, than that of the vapour of water which results from boiling the water under constant pressure after it is drained of entangled water by gravitation, these properties and tables will not apply. Wet Steam. For the most part the precautions taken by Regnault are precisely those under which steam is produced in practice. That is to say, in practice the conditions in the boiler are maintained, as far as practicable, steady, and the steam is withdrawn in a vertical direction from the steam space over the water, where it is drained by gravitation. Owing, however, to exigencies as to space and weight, a great deal more steam is often generated in proportion to the space than was the case in the experiments. Also the velocity of the steam after entering the steam pipes is, in practice, often so great that, even where these are ascending, any water that may have been drawn in with the steam, or produced by condensation owing to the radiation of heat from the pipes, is swept along with the steam ; and where, as in cases like the locomotive, the engine is under the boiler, so that the pipes are descending, this must be so. Under such conditions the steam as it enters the engine will be accompanied by some water, and is then variously called "wet steam," "nearly dry steam," or "super-saturated" steam, though the last name is apparently intended to imply that, notwithstanding Regnault's experiments, the density of steam after drainage is not necessarily a definite function of the temperature or pressure. Whatever may be the cause of the water entering the engine with the steam, its presence in unknown quantity prevents Regnault's formula for the 65] SATURATED STEAM AND THE CONDITION OF STEAM GAS. 593 total heat of evaporation from being used to form a correct estimate of the quantity of heat received by the engine. For the only measures qthe steam supplied to the engine are obtained from the measures of the teed- water supplied to the boiler, or the water discharged from a surface condenser, so that, if an unknown quantity of water enters with the steam, estimates so formed must be in excess. This is a matter of very serious consideration in all attempts to obtain a comparison of the actual performance of an engine in work done, as com- pared with the theoretical performance under ideal conditions. And, as the modern practice of steam engineering is largely guided by the results of such attempts, methods of assuring dry steam or, failing that, of in some way measuring the percentage of water passing with the steam into the engine, have attracted a great deal of attention. For purely experimental purposes, it is always possible to supply the engine with dry steam, even where the boiler is at a distance, by passing the steam through a sufficiently large vessel close to the engine, so that the water may be disentangled by gravitation before the steam enters the engine. These are called water-separators. In some cases such separators form part of the engine, but, although their employment is becoming more common, it is only in comparatively few cases that this is practicable. In other cases, that is, in the great majority of cases, the desire to obtain some experimental evidence of the percentage of water in the steam as it enters the engine, has led to the use of methods of testing samples of the steam drawn continuously from the steam pipe close to the engine. Sampling the Steam. In such methods, the question of getting a fair sample of the steam as it enters the engine is quite distinct from that of testing the sample so obtained The water in the pipe, although moving in the direction of the steam, will not be uniformly distributed throughout the steam, and will, to a great extent, merely drift along the surface of the pipe and mostly on the lower surface, so that unless a sample taken from the lowest part of the pipe is found to be dry, in which case the steam is dry, such methods afford but little evidence as to the percentage of water entering the engine with the steam. Testing the Samples. For absolute dryness such samples may, where the pressure in the steam pipe is steady, be tested by allowing the sample to flow quietly through a separator, so as to drain out the water, the weight of which is then o. K. ii. 38 594 ON METHODS OF DETERMINING THE DRYNESS OF [65 observed. But any attempt to estimate the percentage of water in the sample involves the subsequent condensation and weighing of the steam in the sample, as well as the drained water, which are difficult and com- plicated operations. Besides this, the pressure in the steam pipe near the engine is generally subject to considerable periodic alterations, owing to the intermittent and periodic demand for steam in the engine, which introduces complications of unknown extent. Wire-drawing Calorimeters. With a view to obtaining a test for the samples of steam, which should be independent of the separator, the so-called Wire-drawing Calorimeter has been introduced. In this, the sample of steam, whether it has been first drained or not, is received quietly in a vessel at the same pressure as the steam pipe, where it is at steady known pressure ; from this it is allowed to escape continuously through a small orifice into a second larger vessel, main- tained at greatly lower pressure than the first. In this its temperature and pressure are measured, the steam then passing on into a condenser or into the atmosphere. The quantity of water present is then estimated from the observed pressures in the two vessels, and the difference between the observed temperature in the second vessel and the temperature of saturation at that pressure, as taken from Regnault's tables. Such calculations are at once seen to be based on Regnault's deter- mination of the relations between the pressure and temperature of saturated steam, together with the heat relations, whatever they may be, between saturated steam and superheated steam. And, as the second relation does not appear to be known except as a very rough approximation, the results so obtained must be doubtful. Results. The results obtained with these calorimeters have apparently revealed the presence of anything up to 5 per cent, more water in the samples than revealed by the simple separator, and this even when the steam has been drained in the separator before passing into the calorimeter. This apparent experimental evidence of previously unsuspected water carried by steam has necessarily excited great interest, and is naturally welcomed, as it apparently brings the engines by so much nearer per- fection. On second thoughts, however, a very serious consideration will present 65] SATURATED STEAM AND THE CONDITION OF STEAM GAS. 595 itself, namely, that if the drained steam from a separator contains latent water, the drained steam from the separator on which Regnauit "made his experiments must also have contained similar latent water, and hence the theoretical volumes of steam, which are based solely on these experiments, must be subject to identically the same corrections as the observed results, so that the discovery, if true, would thus leave the percentage of theoretical performance unchanged, while it would upset the truth of Regnault's results as to the properties of steam and, moreover, upset all other deductions from these properties, including the deductions involved in these estima- tions. That such is the case cannot be admitted until after the fullest con- sideration and verification of the experiments, and of the method of reduction by which the novel results have been obtained. These experiments are many, and the methods of reducing the results have not been very fully, although widely, published, but in all that I have seen the results have been deduced by means of the properties of steam as determined by Regnault's experiments, by a formula which is based on a misunderstanding of the meaning of " the specific heat, at constant pressure, for steam when in the gaseous state," as determined by Regnauit. And that this must have been the case with the other results would seem to follow from the fact that this formula, when based on the correct meaning, affords no definite result at all under the circumstances of the experiments. It has thus seemed to me important not only to call attention to the error in reduction by which certain of these results have been obtained, but also to indicate, and if possible to verify, a method by which experiments could be made, so that Regnault's determination of the specific heat of steam gas could be correctly used to ascertain whether or not such latent water does exist in drained steam that is, to ascertain whether Regnault's experiments on the specific heat of steam gas are consistent with his experiments on the latent heat of steam. In the present paper the purpose is limited to pointing out the theory of the reductions, and to giving indications of the method of experimenting, the general character of the apparatus, and the precautions necessary. The Theory of the Reductions. By the law of conservation of energy, when a steady stream of matter flows through a chamber with fixed walls, so that the condition within the chamber is steady, the energy of the matter which enters (potential and actual) is equal to the energy which leaves in the same time, and hence is equal to the energy of the matter which leaves, together with such energy as 382 596 ON METHODS OF DETERMINING THE DRYNESS OF [65 may escape into the walls of the chamber. Thus, if a stream of fluid flows in a horizontal direction through a fixed passage and if at A, Pi = pressure, TI = temperature, Vi= volume per Ib. of fluid, HI P]' Vi = mechanical equivalent of heat per Ib. of fluid, %! = velocity of fluid P 2 ' = pressure, \ TJ = temperature, F 2 ' = volume per Ib. of fluid, \ a t B, HZ P/F 2 ' = mechanical equivalent of heat per Ib. of fluid, u 2 = velocity of fluid and Hj = ihe mechanical equivalent of heat received through the surface per Ib. of fluid passing through ; then Also, if the fluid at A consists, per Ib., of 51 Ib. of steam and (1 Si) Ib. of water, and at B consists of 5 2 Ib. of steam and (1 $ 2 ) Ib. of water, and if ^ and h 2 are put for the mechanical equivalents of heat per Ib. of water respectively at the temperatures, T^ and T 2 , of saturated steam at pressures of PI and P 2 ' respectively, then T-f = T lt where P/ and T l are pressure and temperature corresponding to the initial state of saturated steam at A, and T 2 may be taken to correspond to the temperature of saturated steam at pressure P a '. And if, further, H^ equals the equivalent of the total heat of evaporation at pressure P/ per Ib., then H^S^H.-h^ + h, (2). And if, similarly, H z and A 2 correspond to the temperature of saturated steam at pressure P 2 ', then T 9 ) (3), 65] SATURATED STEAM AND THE CONDITION OF STEAM GAS. 59*7 where K is the mean specific heat of steam at constant pressure between the temperatures TJ the actual temperature at B, and T. 2 the temperature of saturated steam at the actual pressure (P 2 ') at B. It being noticed that, if 1 - S 2 is greater than nothing, T. 2 ' = T 2 , so that the last term in (3) vanishes. While, if (1 $ 2 ) is zero, this last term expresses the heat, whatever it may be, requisite to raise steam, at constant pressure PI, from the temperature of saturation T 2 to the observed temperature T 2 '. Substituting from equations (2) and (3) in equation (1), this becomes If then w 1( u 2 , and Hj are small enough to be neglected, since the values of H lt h lt HZ, h 2 , T. 2 are obtainable from Regnault's tables, when P/, P 2 ' or f\ are observed, all the remaining quantities may be known except S 1} S 2 , and K. And either, if S. 2 is not equal to unity, (TV T 2 ) = 0, and S^-AO + AI = &(#-*.) + * .................. (5), or, if (1-S 2 ) = 0, TJ .................. (6). Equation (5) gives S l in terms of S. 2 when T 2 ' T 2 , but, since S 2 is un- known, this is of no use ; while, if T 2 is greater than T 2 , equation (6) gives Si in terms of K which is a function of T 2 and TV, which has not been determined. If it were possible to determine the exact value of TV at which S z 1 = 0, then But, here again, this is practically impossible, since the only indication that S. 2 1 = is that T 2 is greater than T 2 as given by Regnault's tables for steam at P.,', and, for any such excess as can be observed, the value of K(T 2 T. 2 ) is considerable, since, at the point of saturation, K is apparently infinite, so that neither of these determinations is practical. With a view to getting over these difficulties, the course that has apparently been adopted is to obtain a condition such that the temperature (TV) after wire-drawing is from 10 to 20 F. higher than the saturation temperature (T 2 ), and then to assume that K is equivalent to the specific heat at constant pressure of steam gas as determined by Regnault, or that K = 772x048, an assumption which constitutes the error in reduction to which I have referred. 598 ON METHODS OF DETERMINING THE DRYNESS OF [65 The possibility of obtaining an accurate estimate. This depends on obtaining a certain condition in the experiment, and reducing by a formula proved by Rankine (Trans. Roy. Soc. Edinb., 1849, 1855). Rankine's formula is that the total heat to convert water from a liquid state at any particular temperature, say 32, to steam gas at any temperature (T 2 '), the operation being completed under constant pressure, is expressed by TT I @i + a ^Y ~ 32, G l being a quantity depending only on the initial state, and a being the specific heat at constant pressure of the steam gas, determined by Regnault to be 0-48. Taking the initial state to be at 32, Rankine obtained, as the most probable value, G! = 1092. It is to be noticed, however, that although this value 0'48, as obtained by Regnault, has been universally accepted, the experiments by which he obtained it were independent of the method by which he determined the total heat of evaporation of saturated steam, and that, as Regnault observes*, the smallness of the scale, as compared with that by which the total heats were determined, rendered it necessarily less accurate, as regards the measure- ment of the total quantities of heat observed, although the extreme care with which the numerous experiments in the four cases were made, seems to assure their relative accuracy. The experiments consisted in determining the total heat necessary to raise water from 32 F. or C. to temperatures of about 120C. and 220 C. under the pressure of the atmosphere, then taking the differences as being the heat necessary to raise water from 120 C. to 220 C. It thus involves the assumption that steam at 20 C. (or 36 F.) above the boiling point, is in the condition of steam gas. This is probably very near the truth. Had, however, the experiments been as absolutely accurate as those for the total heat of saturated steam, they would have afforded the means of comparing the two methods of Regnault by Rankine's thermodynamical formulae. As it is, such a comparison can be made. Thus, substituting the total heats as obtained in the experiments for specific heat iu Rankine's formula, the constant d is found to be not 1092, as given by Rankine, but between 1076'4 and 1053'7, with a mean of about 1055. * Him. Acad. Sci., Vol. xxvi. pp. 170, 909. 65] SATURATED STEAM AND THE CONDITION OF STEAM GAS. 599 Taking this value, the heat necessary to raise water from 32 to 248 F. at constant pressure of 14'7 Ibs. per square inch is 1055 + 0-48 (216) =1158-68. To raise water from 32 to saturated steam at 212 requires by Regnault's formula for total heat of saturated steam 10917 + '305 (180) = 1146-6. Hence, to raise saturated steam from 212 to 248 at constant pressure would require 12*08 T.U., which, divided by the difference of temperature, gives for the mean specific heat of steam from saturation at 212 to 248 F. at constant pressure 12 ' 08 --335 "36" 3i> ' which shows that the specific heat, at constant pressure, of steam rises with the temperature. And this, although in accordance with the results obtained by Regnault for other vapours, presents great thermodynamical difficulties ; since many experiments have shown that the steam, on being heated from saturation to 36 F. above, expands three or four times as much as it would if it were gas. It is to be noticed that an error of 3 per cent, in estimating the total quantity of steam, which in these experiments would only mean an error of 0-0004 in the actual weighings, would account for the differences in the values of (7, as determined by Rankine, and as estimated from Regnault's experiment on specific heat, while such an error on the determination of the specific heat would fall within the limits of experimental accuracy. It thus seems probable that Rankine's determinations of the constants in his formula are approxi- mately right. In order to make use of this formula in the reduction of the experiments under consideration, all that is necessary is to bring about, by means of wire- < Ira wing, the condition that T.,' shall be sufficiently larger than T. 2 to insure that the final condition approximates to that of steam gas. That this differ- ence must be more than 20 F. has been shown, but it would appear that with this difference the error is not great. To use the formula, (1092 + 0-48 (ZV-2 1 ,)} 772 is substituted for the right member of equation (6). ?/ ^ // " Hj . - being small, therefore } 2<7 2^7 &(#!-/!) + /*, = 772 {1092 + 0-48 (Tj - T}} (7), which only requires the experimental determination of T l and T 2 ' to give the value of S lt provided that the final condition is that of steam gas. 600 ON METHODS OF DETERMINING THE DRYNESS, ETC. [65 The means of assuring the condition of Steam Gas. Perhaps the most important fact to which attention is herein directed is that, although, as already stated, the limiting relations of temperature and pressure of steam gas are not known with any degree of precision, the wire- drawing experiments are capable of affording simple and direct evidence of the existence of such a final state. As the pressure of steam is reduced by wire-drawing, which is gradually increased, at first, owing to the great expansion, the temperature falls considerably, but, as the wire-drawing in- creases, by the diminution of pressure in the receiving vessel the fall of temperature gradually diminishes, until the gaseous state is produced, when the temperature T 2 ' will be unaffected by still greater wire-drawing. So that to insure a gaseous state, all that is necessary is to gradually diminish the pressure in the receiving vessel, maintaining that in the first vessel, until the temperature T in the receiving vessel becomes constant. The only doubt is whether this point can be practically reached, and this can only be determined by experiments. . The remarkable circumstance in the flow of gases, of which I published the explanation in a paper read before this Society in 1885, that when steam or gas flows through a restricted channel from one vessel into another, in which the pressure is less than half that of the first, the quantity which passes is independent of the pressure on the receiving side, must have an important place in simplifying the apparatus required for such experiment. Thus, with boiler pressure on one side of an orifice, opening into a vessel from which its escape is allowed by an adjustable valve, the whole experiment can be regulated by this valve, the quantity flowing through remaining con- stant for all pressures after the half is reached. The only precautions necessary for accuracy, are those to secure approxi- mately small velocities at the points where the temperature is measured, and those to render small the loss of temperature in the steam by radiation. And, although these must complicate the appliances, they appear to be practical. I may also notice that, should such experiments be accomplished, they will afford the means of verifying or correcting Rankine's value for C l} which he has only given as a probable approximate value. I hope these experiments may shortly be made, as Mr J. H. Grindley, B. Sc., Fellow of Victoria University, has undertaken the research in the Whitworth Engineering Laboratory, Owens College. 66. BAKERIAN LECTURE. ON THE MECHANICAL EQUIVALENT OF HEAT. [From the "Philosophical Transactions of the Royal Society of London," 1897.] (Read May 20, 1897.) PART I. By Professor OSBORNE REYNOLDS, F.R.S., and W. H. MOORBY, M.Sc., late Fellow of Victoria University and 1851 Exhibition Scholar. ON THE METHOD, APPLIANCES AND LIMITS OF ERROR IN THE DIRECT DETERMINATION OF THE WORK EXPENDED IN RAISING THE TEM- PERATURE OF ICE-COLD WATER TO THAT OF WATER BOILING UNDER A PRESSURE OF 29'899 INCHES OF ICE-COLD MERCURY IN MANCHESTER. BY OSBORNE REYNOLDS. The Standard of Temperature for the Mechanical Equivalent. 1. THE determination by Joule, in 1849, of the expenditure of mechanical effect (772 - 69 Ibs. falling 1 foot) necessary to raise the temperature of 1 Ib. of water, weighed in vacuo, 1 Falir. between the temperatures of 50 and 60 Fahr. (at Manchester), together with the second, in 1878, 772'55 ft.-lbs., to raise the temperature of 1 Ib. (weighed in vacuo) from 60 to 61 Fahr., at the latitude of Greenwich, established once for all the existence of a physically constant ratio between the work expended in producing heat and the heat produced; while the extreme simplicity of his methods, his marvellous skill as an experimenter, and the complete system of checks he adopted, have led to the universal acceptance of the numbers he obtained 602 ON THE MECHANICAL EQUIVALENT OF HEAT. [66 as being within the limits he himself assigned (I foot), of the true ratio of work expended in his experiments in producing heat and the heat produced as measured on the scale of the thermometer on which he spent so much time and care. The acceptance of J = 772, as the mechanical equivalent of heat, amounts to the acceptance of the scale between 50 and 60 on Joule's thermometer b as the standard of temperature over this range. Joule's thermometers are now in the custody of the Manchester Literary and Philosophical Society (having been confided to its care by Mr A. Joule); so that this material standard is available. But the standard of temperature actually established by Joule is universally available wherever the British standard of length is available, together with pure water and the necessary means and skill of expending a definite quantity of work in raising the temperature of water between 50 and 60 Fahr., since in this way the scale on any thermometer may be compared with that on Joule's. The difficulty of access to Joule's thermometer, and the inherent difficulty of making an accurate determination of the equivalent, have limited the number of such comparisons. The most serious attempts have been made with the very desirable object of determining the mechanical equivalent of a thermal unit, measured on the scale of pressures of gas at constant volumes, first recognised by Joule as the nearest approximation to absolute temperature. The results of these comparisons have been various, all having apparently shown that Joule's standard degree of temperature is less than the one- hundred-and-eightieth part between freezing and boiling points on the scale of pressure of gas at constant volume, the differences being from 01 to I'O per cent. Joule himself contemplated comparing his thermometer with the scale of air pressures, but did not do so. So that only indirect comparisons have been possible. Him, who was the first to follow Joule, in one of his researches introduced a method of measuring the work done which afforded much greater facility for applying the work to the water than the falling weights used by Joule in his first determination, and this was adopted by Joule in his second determination. But notwithstanding the greater facilities enjoyed by sub- sequent observers, owing to the progress of physical appliances, the inherent difficulties remained. The losses from radiation and conduction could only be minimised by restricting the range of temperature, and this insured thermometric difficulties, particularly with the air thermometer, which, it seems, does not admit of very close reading. This, together with certain criticisms, of which some of the methods employed admit, appear to have 66] ON THE MECHANICAL EQUIVALENT OF HEAT. 603 left it still an open question what exact rise in the temperature in the scale of air pressures corresponds to the 772 ft.-lbs. 2. The research, to the method and appliances for which this paper relates, has been the result of the occurrence of circumstances which offered an opportunity, such as might not again occur, of obtaining the measure, in mechanical units, of the heat in water between the two physically fixed points of temperature to which all thermometrical measurements are referred, and of thus placing the heat as defined in mechanical units, on the same footing as the unit of heat as defined by temperature, without the inter- vention of scales, the intervals of which depend on the relative expansions of different materials such as mercury and glass. It has been, so far as I am concerned, undertaken with considerable hesitation, on account of the responsibility even in attempting such a deter- mination, and the harm to science that might follow from further confusion owing to error in what, in spite of opportunities, must be the extremely difficult task of making such complex determinations within less than the thousandth part. These considerations, together with my inability to find the large amount of time necessary for making the observations, prevented any attempt until July, 1894. At that time Mr W. H. Moorby offered to devote his time to the research, and so relieve me of all responsibility except that which attached to the method and the appliances ; and having, from experience, the highest opinion of Mr Moorby 's qualifications for carrying out the very arduous research, there seemed to be no further excuse for delay, particularly as after seeing the appliances in the laboratory both Lord Kelvin and Dr Schuster expressed strongly their opinion as to the value of the research. The Opportunity for the Research. 3. This consisted in the inclusion in the original equipment of the laboratory, in 1888, of the following appliances : (1) A set of special vertical triple-expansion steam-engines, with separate boiler, closed stoke-hole, and forced blast ; these engines being specially arranged to give ready access to the shafts, 3 feet above the floor, and being capable of running at any speeds up to 400 revolutions per minute, and working up to 100H.-P. (Plate 1). (2) Three special hydraulic brake dynamometers, on separate shafts, between and in line with the engine shafts, with faced couplings, so that one brake shaft could be coupled with the shaft of each engine to work its own shaft ; or the brakes on the high-pressure and intermediate engines could be removed, and their shafts coupled by means of intermediate shafts, 604 ON THE MECHANICAL EQUIVALENT OF HEAT. [66 so that all three engines worked on the brake connected with the low- pressure engine. These brakes, which are shown (Plate 1), are separately capable of absorbing any power up to a maximum of 30 horse-power at 100 revolutions, and increasing as the cube of the speed ; so that a single brake is capable of absorbing the whole power of the engine at any speed above 100 revolutions a minute. The whole of the work is absorbed by the agitation of the water contained in the brake, while the heat so generated is discharged by a stream of water through the brake, with no other functions than of affording the means of regulating, independently, the temperature of the brake and the quantity of water in the brake. The moment of resistance of the brake at any speed is a definite function of the quantity of water in the brake. And as, except for this moment, the unloaded brake is balanced on the shaft, the load being suspended from a lever on the brake at 4 feet from the axis of the shaft, if the moment of resistance of the brake exceeds the moment of the load, the lever rises, and vice versa. By making the lever actuate the valve which regulates the discharge from the brake, and thus regulate the effluent stream, the quantity of water in the brake is continually regulated to that which is just sufficient to suspend the load with the lever horizontal, and a constant moment of resistance is maintained whatever may be the speed of the engines. (3) Manchester town's water, of a purity expressed by not more than 3 grams of salts in a gallon, brought into the laboratory in a 4-inch main at town's pressure (50 to 100 feet head), and distributed either direct from the main or at constant pressure from a service tank 10 feet above the floor of the laboratory. (4) Two tanks, each capable of holding 60 tons of water, one in the tower, 116 feet above the floor, the other 15 feet below the floor, connected by 4-inch rising and falling mains, each 500 feet long, passing in a chase under the floor. The rising main is in communication with a special quadruple centrifugal pump, 2 feet above the floor, capable of raising a ton a minute from the lower to the upper tank. (Shown in Plate 5.) Also a set of mercury balances, showing continually the levels of water in the two tanks, and the pressures in the rising, falling, and town's mains. (Shown in Plate 2.) (5) A special quadruple vortex turbine, supplied from the falling main and discharging into the lower tank, capable of exerting 1 H.-P., and available for steady speeds at all parts of the laboratory. (Shown in Plate 5.) (6) A supply of power to the laboratory by an engine and boiler, quite distinct from the experimental engine, and distributed by convenient shafting which is always running. (Shown in Plate 1.) 66] ON THE MECHANICAL EQUIVALENT OF HEAT. 605 The Measurement of the Work. 4. Of the appliances mentioned, the brake on the low-pressure engine is the centre of interest, as it was by this that the work was measured, as well as converted into heat. The existence of the appliances was largely due to the interest in educa- tional work taken by Mr William Mather, who, together with the other members of the firm of Mather arid Platt, not only placed at my disposal the facilities of their works, but, inspired the enthusiasm which alone rendered the execution of such novel and special work possible. The development of the brake dynamometer, from its introduction by Prony, has an interesting and important history, but into this it is not necessary to enter. The purpose of these dynamometers is to afford con- tinuous frictional resistance, adapted to the power exerted by the prime mover in causing a shaft to revolve, and of a kind that is definitely measure- able. To fulfil the first of these conditions, the mean moment of resistance of the brake must just balance the mean moment of effort of the engine, and the means of escape of heat from the brake must be sufficient to allow all the heat generated to depart, without accumulating to an extent which may interfere with the action of the appliances. In the first brakes the resistance was obtained by the friction of blocks or straps pressed against a cylindrical wheel on the shaft, and, small powers being used, radiation and air-currents round the brake were found sufficient to carry off the heat, but, when larger powers were used, these sources of escape failed to keep the temperatures down to practical limits, which necessitated the application of currents of water to carry off the heat. The measurement of the work was invariably accomplished by attaching the brake blocks, or straps, to a lever, or arm, so that the whole brake would be free to revolve with the brake-wheel, except for the moment of the weight of the parts which, adjusted to the power of the engine, was kept in balance by the adjustment of the pressure of the blocks on the wheel. Then, since the work done is equal to the product of the mean moment of resistance, over the angle turned through, multiplied by the angle, if the resistance is constant over time, the moment of the brake, multiplied by the whole angle, measured the work done. It is however to be noticed that the assumption, that the time-mean of the moment on the brake is the same as would be the angle-mean of this moment, might involve an error of any extent, provided the resistance and the angular velocity varied in conjunction. And as steam engines invariably exert an effort, varying within the period of the revolution, while the friction and the pressure causing it are apt to respond to any variations of speed, it 606 ON THE MECHANICAL EQUIVALENT OF HEAT. [66 is probable that there has been some error from this cause in all such measurements, although not previously noticed. Hirn appears to have been the first to recognise that in a steady condition the resistance of fluid between the brake-wheel and the brake would answer instead of the solid friction, so that the mean time moment of effort exerted in turning a paddle in a case, containing water, with bafflers, would be strictly measured by the mean time moment of the case. And although subject to the same error from periodic motion as the friction brake, the facility this fluid brake offered for cooling arid regulating led to its simul- taneous adoption and development by several inventors, for measuring power the late William Froude, for the purpose of measuring the work of large engines, inventing that arrangement of paddle vanes and bafflers which gives the highest resistance, regulating the resistance by thin sluices between the vanes and bafflers, and always working with the case full of water. The brake under consideration differs from that of Mr Froude in only one fundamental particular the provision by which a constant pressure in the interior of the brake is secured by the admission of the atmosphere to that part of the brake where the dynamical effect of the water is to cause the lowest pressure this admits of working the brakes with any quantity of water from nothing to full, and thus allows of the regulation of the re- sistance, by regulating the quantity of water in the brakes, without sluices. The description of this brake has already been published, together with that of the engines*, but it will be convenient to give a short description. This brake consists primarily of (1) a brake wheel, 18 inches in diameter, fixed on the 4-inch brake shaft by set pins, so that it revolves with the shaft (Figs. 2 and 3), and (2) a brake (or brake case) which encloses the wheel, the shaft passing through bushed openings in the case which it fits closely, so as to prevent undue leakage of water while leaving shaft and brake-wheel free to turn in the case, except for the slight friction of the shaft (Figs. 1, 2 and 3). The outline of the axial section of the brake- wheel is that of a right cylinder, 4 inches thick. The cylinder is hollow in fact, made of two discs which fit together, forming an internal boss for attachment to the shaft, and also meet together at the periphery, forming a closed annular box, except for apertures to be further described (Fig. 3). In each of the outer disc faces of the wheel are 24 pockets, carefully formed, 4 inches radial, and 1 inches deep measured axially, but so inclined that the narrow partitions or vanes (^ inch) are nearly semicircular discs inclined at 45 to the axis ; the vane on one face being perpendicular to the vane on the opposite face (Fig. 2). * " Triple Expansion Engines," by Professor Osborne Reynolds, Minutes of Proceedings, Inst. C. E., vol. 99, 1889, p. 18. (See Paper 56, page 336.) 66] ON THE MECHANICAL EQUIVALENT OF HEAT. 607 The internal disc faces of the brake case, as far as the pockets are con- cerned, are the exact counterparts of the disc faces of the wheel,- except that there are 25 pockets, so that the partitions in the case are in the same planes as the partitions meeting them in the wheel, there being ^ inch clearance between the two faces. Fig. 1. The pairs of opposite pockets, when they come together, form nearly closed chambers, with their sections, parallel to the vanes, circular. In such spaces vortices in a plane inclined at 45 to the axis of the shaft may exist, in which case the centrifugal pressure on the outside of each vortex will urge the case and the wheel in opposite directions inclined at 45 to the direction of motion of the wheel, which will give a tangential stress over the disc faces of the wheel of 1/V2 of the sum of these vortex pressures. The existence and maintenance of these vortices is insured by the radial centrifugal force of the water in the pockets in the wheels owing to its motion. This is the late Mr W. Froude's arrangement. But an essential feature 608 ON THE MECHANICAL EQUIVALENT OF HEAT. [66 of the brake is the provision which insures the pressure of the atmosphere at the centre of the vortices, even when the pockets are only partially filled. Fig. 2. The vortex pressure is greatest at the outsides of the vortices, which occurs all over the annular surfaces of the pockets, but the actual pressure on these surfaces is not determined solely by the vortex motion unless the state of pressure at the centre of the vortices is fixed, for the vortex motion only determines the difference between these pressures. To insure the constant pressure, and at the same time to allow of the pockets being only partially full that is, to allow of hollow vortices with air cores at atmospheric pressure, it is necessary that there should be free access of air to the centres of the vortices, and as this access cannot be obtained through the water, which completely surrounds these centres, it is obtained by passages (^ inch diameter) within the metal of the guides, which lead to a common passage opening to the air on the top of the case (Figs. 2 and 3). To supply the brake with water there are similar passages in the vanes of the wheel leading from the box cavity, which again receives water through ports which open opposite an annular recess in one of the disc faces of the 66] ON THE MECHANICAL EQUIVALENT OF HEAT. 609 case into which the supply of water is led, by means of a flexible indiarubber pipe, from the supply regulating valve. Fig. 3. Fig. 4. The water on which work has been done leaves the vortex pockets by the clearance between the disc surfaces of the wheel and case, and enters the annular chamber between the outer periphery of the wheel and the cylindrical portion of the case, which is always full of water when the wheel is running, whence its escape is controlled by a valve in the bottom of the case, from which it passes to waste. By means of linkage connected with a fixed support and the brake case, an automatic adjustment of the inlet and outlet valves, according to the position of the lever, is secured without affecting the mean moment on the brake case. And this also affords means of adjusting the position of the lever. To admit of adjustment for wear, the shaft is coned over that portion which passes through the bushes, the bushes being similarly coned, and screwed into short sleeves on the casing, so that by unscrewing them the wear can be followed up and leakage prevented. The brake levers for carrying the load and balance weight, are such as to allow the load to be suspended from a groove parallel to the shaft, at 4 feet from the shaft, by a carrier with a knife edge, the carrier and the weights each being adjusted to 25 Ibs. (shown figs. 1 and 4). In addition to this load, a weight is suspended from a knife edge on the lever nearer the shaft, this weight being the piston of a dash-pot in which it hangs freely, except 39 O. B. II. 610 ON THE MECHANICAL EQUIVALENT OF HEAT. [66 for the viscous resistance of the oil. This weight being adjusted to exert a moment of 100 ft.-lbs., and again a travelling weight of 48 Ibs., is carried on the lever and worked by a screw with inch pitch, so that one turn changes the moment by 2 ft.-lbs., while a scale on the lever shows the position. A shorter lever on the opposite side of the case carries a weight of 74'6 Ibs., which is adjusted to balance the lever and sliding weight when the load is removed. The Accuracy of the Brake. 5. The principle of these hydraulic dynamometers is that when moment of momentum is introduced into a fixed space without altering the moment of momentum within that space, the rate at which moment of momentum leaves the space must equal the rate at which it enters. The brake- wheel imparts moment of momentum to the water within the case, and the friction of the shaft imparts moment of momentum to the case. The water in the case, when its moment of momentum is steady, imparts moment of momentum to the case as fast as it receives it, and the time mean of the moment of the load is equal to the time mean of the moment of the effort of the shaft. This is not affected by water entering and leaving the case at equal rates, provided it enters and leaves radially. The condition of steadiness is, however, essential, in order that the moment of effort shall be at each instant equal to the moment of resistance on the case ; any change in the moment of momentum of the water in the case being the result of the difference of the moment of effort on the shaft and that of resistance on the case. The Time-Mean of the Moment of Effort. 6. When, however, the shaft is run over an interval of time, the mean moment of resistance on the case, less the difference of the moments of momentum of the water, at the end and beginning of the interval, divided by the time, is the time-mean moment of effort on the shaft. The possible limit of this error may be estimated when the maximum moment of momentum of the water is known as well as the minimum moment of resistance, and the minimum interval of time. Thus taking the limits to be 30 Ibs. of water, with radius of gyration 0'66 foot, at 300 revolutions a minute (< 14), the interval of running 3600 seconds, the moment of the load 400 ft.-lbs., the limit of the time-mean of change of moment of momentum of the water is 14/3600, and this divided by the mean moment of resistance gives as the limits of relative error, 66] ON THE MECHANICAL EQUIVALENT OF HEAT. 611 + O'OOOOl. This is supposing the whole of the water to be absent at the beginning or end of the trial, while the actual difference never amounts to more than 2 or 3 Ibs., so that the limits do not exceed O'OOOOOl, which is neglected. The Angle-Mean of the Moment of Effort on the Shaft. 7. As already pointed out in Art. 4, when both the angular velocity of, and the moment of effort on, the shaft are subject to fluctuations of speed, the time-mean of the moment of effort may differ from the angle-mean. This applies to all brakes, but in hydraulic brakes, in which the resistance is proportional to the square of the speed, although lagging by an unknown interval, it becomes possible to estimate the possible limits of this error when the limits of fluctuation of speed are known. Taking the time-mean of the angular velocity, 2a 2 &> the extreme differences of speed, and assuming the variation to be harmonic, = a> 2 l ( - T 2 )} ...(3), where M is the time-mean of the moment of resistance. Also the rate at which work is done with uniform velocity, is J/o> , of which the mean is M (i) , and is the rate of work as measured by the mean moment on the case, multiplied by the mean-angular velocity. To a second approximation the rate of work with varying speed is Mo> = M a> { 1+ 2a 2 cos n (t - r l\ - T a ) + a 4 cos 2n (t - T, - r l\)} (H-OaCOSW^-^)} ...... (4), and from this it appears that the mean rate of work is 6> JU Q (1 + a* cos nl'a), which shows that the relative error in taking this as Jt/ <>o is +tt 4 coswjT 2 . Thus the error arising from fluctuations in speed of 2a 2 o) is within the limits a 4 , when the resistance varies as the square of the speed, as in the hydraulic brakes. Where, as in the brake under consideration, there is an automatic adjust- ment, by which the quantity of water in the brakes is adjusted to the speed, 392 612 ON THE MECHANICAL EQUIVALENT OF HEAT. [66 so as to maintain the resistance constant, there will be no error caused by such gradual variations of speed as result from changes in the boiler pressure, since the automatic adjustment can keep pace with them. But it takes time for the water to get in and out, and any variations, so rapid that, owing to the inertia of the brake case with its load, their effect has been reversed before the case has moved sufficiently to affect the water in the brake, will produce errors. Such cyclic variations of speed attend all motions derived from recipro- cating engines, and it is only these, and not the secular variations, that produce errors. The Variations in the Speed of Rotation of the Steam- Engine. 8. The cyclic variations all go through one or two complete periods in the time of revolution of the engine, and are approximately simple harmonic functions of the time. They arise from three distinct causes : (1) The varying energy of motion of the reciprocating parts ; (2) The varying moment of the effort of the steam pressures on the cranks ; (3) The effect of gravitation on the unbalanced parts in the engine. In the case of a simple vertical engine, unbalanced and working with moderate expansion, these variations of speed may be severally estimated when /, the moment of inertia of the revolving parts, r the half-stroke of the reciprocating parts, and W the weight of these parts are known, together with N the number of revolutions per minute, and U the work done per stroke. For, considering the variations as existing separately, we may assume that the angular motion would be steady but for the particular effect, thus : (1) The moment of effort on the crank being constant, and the resistance constant, and equal to the effort, the energy of motion of all the parts is constant. Putting ft>= 277-^/60, and i = r*W/g, / is approximately whence O^-ij. (2) In the same way, considering the effect of the crank effort alone, with a moderate expansion, the energy that has to be absorbed and given out by the revolving parts is about one-fourth part of the work per stroke, and / 2 - 7 cos 2n (t-T)= C, where nT, say is the angle of the crank at which = + -^- t cos 2 f &> < - - J + ^ cos w^J , from this the approximate joint error can be found. But it is sufficient here to show that the individual errors are negligible. The first gives an error in the mean moment < 0-000024 (Mafi. The second and third are inversely proportional to N\ If N is 800, which is the lowest value, the second error is between < + 0-0000025 (Ma, 4 ). The third < + 0-0000001 (Ma*}. These are all negligible quantities, and, as the corresponding effects in the high-pressure and intermediate engines, owing to the cranks being set at angles of 60, would only be to compensate those of the low-pressure engine, the greatest error would not exceed w k as defined by (ML), J-% -L i) or that the heat required to raise one unit of mass of pure water through the definite interval of temperature is equivalent to U W (ML). SW (T ^ This is the definition of the mechanical equivalent of heat in Manchester, adopted by Joule, if n = 1, and 6 is 1 Fahr. between 50 and 60, as deter- mined on his thermometer. But, since the absolute kinetic value of the unit of force as here defined varies with the latitude and height of the place, while that of the unit of heat is constant, this mechanical equivalent varies from place to place with 1/g, where g is the expression, in kinetic units, for the unit of force (M). Thus, expressing the work in kinetic units, the unit of heat, as already defined, is equivalent to 2-rrNRW _ m where the dimensions of C are 66] ON THE MECHANICAL EQUIVALENT OF HEAT. 04-1 Whence, since g has dimension (LT~-}, 2-n-NRW tf where the dimensions of C/g are (LnO~ l ). The object in this research being to replace the standard of temperature, as defined by the scale on a particular thermometer, by the standard obtained from the states physically defined by melting ice and by water boiling under a standard pressure, 6 is here defined to express this interval, and 8 is, in accordance with the definition already given, used to express the ratio which the heat required to raise unit mass over any interval, per degree of rise, bears to that required to raise pure water over the interval 0, per degree of rise. The Standards Involved. 39. It appears from the dimensions of Cfg, as obtained in the last article, that the only general standards to which reference need be made are those of length and temperature. It is, however, to be noticed that the determination of the work and the heat involve the determination of separate masses, and that the units only disappear on the condition that they are equal. The Measurement of Mass. 40. Since it was not necessary to refer the mass to a general standard, the weights used were only referred to a Board of Trade standard for convenience. Thirteen of the 25 Ib. weights used for loading the brake were adjusted to the Board of Trade weight, then carefully balanced against each other, till, balanced in groups of four in any arrangement, there was less than 01 Ib. difference. Four of these weights were then taken as the standard. The compound lever machine, which had two scales on the same lever, one notched to each 100 Ibs. for the position of the large rider, the other with a flat scale for every 1 Ib. for the position of the small rider, was taken to pieces and the knife edges re-ground and re-set (by Mr Foster) till con- sistent results were obtained to the one-hundredth of 1 Ib. Another rider was made to work on the same scale as the small rider, being adjusted to one- hundredth of the weight, so as to road O'Ol Ib. o. R. n. 41 642 ON THE MECHANICAL EQUIVALENT OF HEAT. [66 The scales were then carefully surveyed by the standard 100 Ib. weight, the original small rider being adjusted till the difference between its extreme positions on the scale balanced the standard to < O'Ol Ib., and the cor- rections for each V-notch into which the feather on the large rider fitted ascertained by balancing the standard to a like degree of accuracy. The dead load on the scales, including the empty tank, came to 340 Ibs., about, and between this and 2200 Ibs. the scales would weigh any quantity with the lever swinging to O'Ol Ib. The weights to which the scales had been adjusted Avere then exclusively used on the brake. Thus the brake was balanced by the same weights as were used as the standard in weighing the water, with a sensitiveness which gave the error less than one forty-thousandth part of the weight of water in the smallest trials, while the casual error, which would not exceed this in a single weighing, would be eliminated in the mean of a large number of weighings. Thus the relative limits of error in weighing would not exceed 000025. The Correction for the Weight of the Atmosphere. 41. The balances being made in air, it is necessary to add the weight of air displaced in each case. As the relative weights only are concerned, if D a is the weight of a unit volume of air, D w that of water, and D { that of cast-iron, the weights in air of unit masses are : 1 D a /D w .................. for water, 1 D a /Di .................. for cast-iron. The load on the brake is therefore subject to the correction expressed by the factor (lD a /Di), while that of the water balanced against cast-iron weights, has the correction factor and the relative correction for the actual weight of water, as against the load on the brake in air, is 1 ( 1 yr ) or approximately 1 + ~ , \ "w/ -L/w for the temperature 67 Fahr., D a = 0'0752, D w = 624. Hence, the relative correction factor for the equivalent is (1 -0-001205). 66] ON THE MECHANICAL EQUIVALENT OF HEAT. 643 The Correction for g in Latitude of Greenwich and 45. 42. Since the latitude of Manchester is 53 29', Greenwich 51 29', the value of g being (Memoires sur le Pendule, Soc. Francaise de Physique) # 450 (1 - 0-00259 cos 2X) = g^ (\ + 0-0007558) at Manchester, = # 450 (1 + 0-0005814) at Greenwich, whence the correction factor is (1 + 0'0001746) at Greenwich, and for 45 (1 +0-0007558). The Specific Heat of the Water. The standard capacity for heat being that of distilled water, the obvious course would have been to have used distilled water in the trials, had this been practicable ; but as it was apparent from the first that the quantity of water which would have to pass through the brakes during the trials would amount to some 20,000 gallons, or, say, 100 tons, all of which would have to be brought down to a temperature of 32 Fahr. ; and that to do this, using distilled water, whether or not the water was used over again, the necessary appliances for producing, storing and cooling the water, were impracticable in the laboratory, the last 40 must be removed with ice, and this would require some 25 or 30 tons of ice. While using the town's water direct from the main, the average temperature, from February to June, would not exceed 45, so that only 12 or 13 would have to be removed by ice, which would require from 7 to 10 tons, with no appliances except the relatively small appliance for cooling. The only practical course, therefore, was to use the town's water. And had it not been for the known purity of this, the research would never have been undertaken. As affording definite assurance of the consistent purity of this water, as delivered in the college, Professor Dixon kindly undertook to furnish the mean results of the analyses which he makes periodically for the Manchester Corporation, of the water drawn from the supply in the college. These show that the impurities are almost negligible, and taking 0'2 as the specific heat of the salts, the relative correction is 0"8.s, where s is the relative weight of the salts. 412 644 ON THE MECHANICAL EQUIVALENT OF HEAT. [66 The Effect of Air in the Water. 43. Even distilled water contains air unless special precautions are taken for its removal ; so that any effect such air may have on the capacity for heat as measured would not have been avoided by using distilled water. The direct effect of the same 0'00323 per cent, of air which water exposed to the atmosphere usually contains at normal temperatures, is so small as to be altogether negligible, and it would seem to be an open question whether the standard condition of water, as regards the capacity for heat, does not involve the inclusion of this air. But the indirect effect of such air on the heat necessary to raise water from normal temperatures to near the boiling-point, is by no means negligible. It does not appear that any definite study has hitherto been made of this effect ; but it is a matter of common observation that as water reaches a temperature some 40 Fahr. below the boiling-point, bubbles appear on the sides and bottom of the vessel, which gradually increase in size and rise to the surface, increasing rapidly in size as they rise. The bubbles are usually referred to as bubbles of gas or air. But, a moment's consideration will show that, although the air or gas is the immediate cause of the premature formation and subsequent expansion of the bubble, it is none the less certain that the space occupied by the bubble is filled with saturated steam at the temperature of the water, the function of the air being merely that of balancing the excess of pressure of the surrounding water over the pressure of the saturated steam. It thus appears that every bubble so formed represents a quantity of heat, which is the latent heat of the volume of the saturated steam in the bubble, over and above the heat of the weight of water in this steam. Thus, if bubbles of air exist in water at a temperature of 212 Fahr., the weight of air per Ib. of water being a, and p the pressure of the water in inches of mercury, then, since the pressure of the air is p - 30, and the volume of 1 Ib. of air at 212 Fahr. under 30 inches of mercury is 16 '9 cubic feet, the volume of air per Ib. of water is T , 16-9 x 30 : T^3o- xa ' or, if p = 40, V= 50-7 x a. This is the volume, in cubic feet, of saturated steam at 212; whence, since the latent heat per cubic foot is 36'6 at 212, the excess of heat will be per Ib. of water Fx :16-6 = 1855 x a, 66] ON THE MECHANICAL EQUIVALENT OF HEAT. G45 and this, divided by 180, gives a relative error 10-31 x a. If a = 0-0000323, the error is 0-00033, or 0'033 per cent. The water, after being exposed to the atmosphere in the service reservoir, where it discharges any excess of air, enters the brake cold with this normal air, there it is heated by work, under the pressure of the artificial atmosphere at pressure p, to maintain which it parts with some of the air, which, in passing out into the flexible pipe, carries out saturated steam, which is condensed by radiation from the pipe. The water, with the remainder of the air, is then carried by the centrifugal pressure into the outer chamber in the brake case, under a pressure of about 50 inches of mercury. It then passes the automatic cock, into the flexible pipe, at 41 inches pressure, thence rising to the thermometer bulbs at 40 inches. In passing the automatic cock with a difference of pressure of 9 inches, the pressure will be further reduced, probably 9 inches below that in the pipe, so that any air that might have been retained would come out at that point, and expand further as it approached the thermometer bulb. In the first instance, it was thought that a pressure of 5 feet of water would prevent the formation of bubbles, and the air gap in the pipe leading from the condenser was placed at this height above the thermometer. But many, and sometimes large, bubbles of air were observed passing up the thermometer chamber ; and Mr Moorby observed that he could detect the passage of a large bubble by a fall in the thermometer before the bubble appeared in the glass chamber. To prevent this, the air-gap was raised till it was 12 feet above the thermometer bulb ; so that the error is limited to three ten-thousandths. Even so, as it is much larger than any of the errors of constant sign, it was important to try, by assimilating the conditions under which the water leaves the brake, to obtain experimental evidence which would narrow the limits. It may appear at first sight as though these losses from the air in the water would, like the radiation, be eliminated in the difference of the large and small trials, but this is not so, since the quantity of heat so lost is pmportional to the amount of water used, or it may be greater in the heavy trials. The Standard of Length. 44. The measures of length that the research involves are (1) The horizontal distance of the centres of gravity of the adjustable loads on the brake from the axis of the shaft. 046 ON THE MECHANICAL EQUIVALENT OF HEAT. [66 (2) The vertical heights of the barometer at which the boiling-points of the water were determined. In order to secure a definite reference of these to the British standard, recourse was had to two carefully-preserved and independent measures derived from this standard. (1) A set of gauges by Sir Joseph Whitworth and Co., consisting of three steel bars, 9, 6 and 3 inches respectively, with parallel plane ends f inch in diameter, adapted to a 20,000th of an inch measuring machine, which constitute the standards used in the engineering laboratory. (2) A brass bar by Elliott and Co., 39 inches long, and graduated in inches, used as the standard in the physical department in Owens College. From the Whitworth gauges, two steel bars, f inch in diameter and 9 inches long, with parallel plane ends, were made by Mr Foster, and compared with the 9-inch Whitworth bar by the measuring machine. With these and the Whitworth gauges, placed end to end, an outside gauge consisting of two surfaced angle-plates on a surfaced cast-iron bed was set out, and then a steel bar f inch in diameter with plane ends fitted to these. Careful comparison showed that this bar did not differ from the sum of the lengths of the gauges by T ofiny P ai> ts of an inch. This length was then carefully laid off by the surfaced angle plates on the surface plate, and was so compared with the scale of the Elliott brass bar, account being taken of the temperature, and found to agree within less than 10 ^ 00 of an inch. The 30-inch bar so obtained was then taken as the standard both for the levers of the brake and the barometer, to be carefully preserved. Lengths of the Levers. 45. The V-groove, in which the knife-edge of the carrier, by which the load on the brake was suspended, rested, was originally made at a distance of four feet from the axis of the shaft at ordinary temperatures, and as, whatever the error might be when the brakes were hot, it would be the same for all the trials, since the temperatures were the same, it was decided to take this as the length of the levers in estimating the loads during the progress of the research, and to treat whatever error there might be as a standing correction on the final results. Such correction to be obtained by laying off four feet, less the radius of the shaft, from the carefully squared end of a steel plate 3 inches broad and T 3 ^ inch thick, then placing this, flat, in a vertical plane perpendicular to the shaft, with its edge horizontal, as near as practicable to the knife-edge groove with the squared end touching 66] ON THE MECHANICAL EQUIVALENT OF HEAT. 647 the shaft. Then by means of a theodolite, set so that its line of collimation was in a vertical plane parallel to the axis of the shaft, and intersecting the vertical line on the plate, to observe the distance of the groove from the line on the plate, while the brake was running under the same conditions of temperature, and load as in the trials ; but with the carrier temporarily displaced further along the shaft, so as to leave the bottom of the V-groove visible through the theodolite, and in this way to obtain the actual distance of the groove from the axis of the shaft, as affected by the expansion of the brake, and any displacement of the bearing on the shaft which might result from the running. By using a scale divided to the one-hundredth of an inch, and taking several readings, this could be determined to a thousandth of an inch, so that the limits of accuracy would be 0-00002. The Standard of Temperature. 46. As the most general standard is the difference between the two physically fixed points of temperature, corresponding to the temperature of ice melting under the pressure of the atmosphere, and that of water boiling under a pressure corresponding to 760 millims. of ice-cold mercury in the latitude of 45, taking account of the variation of g, the standard in Manchester is the interval between melting ice and water boiling under a pressure of 760 x 1 '0001 7 21 millim. of ice-cold mercury, which corresponds to 29'899 inches. And this interval divided by 180 is one degree Fahr. According to Regnault's tables, a divergence of one thousandth of an inch from the boiling point would correspond to an error of 0'0017 Fahr., and this would be less than the one-hundred-thousandth part of 180. In order to obtain this degree of accuracy in comparing the pressure of the vapour of pure water, in which thermometers could be placed, with the height of mercury over a range of two or three degrees above, and two or three below the point, at almost any time, irrespective of what might be tin' actual pressure of the atmosphere, it was necessary that the barometer, or pressure gauge, while in free communication with the vapour chamber, should be shut off from the atmosphere, and at the same time so far removed, that the temperature of the mercury should not be affected by the heat from the gas or boiling water. And, further, although in direct communication with the vapour, this must be such that no moisture could reach the mercury ; and, such as involved no current in the passages which might affect the relative pressures, as would result by the interposition of a condensing vessel. 648 ON THE MECHANICAL EQUIVALENT OF HEAT. [66 It was also necessary that the arrangements for reading the vertical distances between the upper and lower surfaces of the mercury should not only give absolute differences of height, but also that they should afford ready means of at any time determining the presence of vapour or gas, other than that of mercury, in the upper limb of the barometer. The Barometer. 47. To meet these requirements, the barometer shown (Plate 8) was designed. The vessel which holds the mercury consists of a bottle-shaped casting of iron, 3 inches in diameter. Through a stuffing-box in the neck of this, the stem of the barometer tube passes. To admit of reading the level of the surface of the mercury in the bottle, two parallel plate-glass windows are arranged, f inch diameter, having their axis f inch from the axis of the bottle. These are sunk into the casting so as to leave the outer cylindrical surface of the bottle clear, the joints between the glass and the cast-iron being faced and made tight with a trace of beeswax, the other openings into the bottle being one for the admission and abstraction of mercury, fitted with a screwed valve, and one for the admission of air, with a mouthpiece for the attachment of a tube from the vapour chamber. The glass stem of the barometer is drawn down into a neck towards the lower end, and this is bent through 180 so as to bring the mouth upwards, and thus admit of its introduction into v the mercury in the bottle without letting in air. This bend has to be passed through the stuffing-box, then the tube is secured by screwing the gland on to the beeswax stopping. A brass guard tube is then screwed into the neck, to support the glass tube, to a height of 24 inches from the mercury in the vessel. For reading the height of the lower limb, a cylindrical brass curtain, with a conical contraction on the top, the aperture in which is threaded internally at twenty threads to an inch to correspond to the screw on the outside of the neck of the bottle, is screwed on to this neck, the lip or bottom of the curtain being truly turned so that, when screwed down to the level of the mercury, it cuts off the light through the windows from a white sheet behind. To the top of the brass casting, which forms the curtain, a brass cylin- drical tube is rigidly attached coaxial with the curtain which fits over the brass guard round the barometer tube, this extends to a height of 26 inches from the lower lip, the internal diameter for the last inch being a little smaller and internally screwed at twenty threads to an inch. Into this is screwed a brass tube, externally screwed throughout its length, about 6 inches long, with parallel opposite slots J inch wide extending to within 66] ON THK MKCHANICAL EQUIVALENT OF HEAT. 649 an inch at either end, to form windows through which to see the light over the upper limb of the mercury. And on to the upper portion of this tube there is screwed a long cap, capable of screwing down to the bottom of the slot. The lower lip of this cap forms the curtain which cuts off the light when the lip is level with the upper limb of the mercury. By this arrangement the variation of the distance between the lips of the lower and the upper curtains depends only on the change in their relative angular positions. For, since the slotted tube has a uniform thread, it can be turned, screwing into the lower curtain and out of the upper, both of which remain unmoved. Thus the position of the windows may be fixed, while the curtains are moved. So that for reading the distances it is only necessary to measure the relative angle. This angle is measured by dividing the circumference of the cap just above the lip into five equal divisions, from to 5, and these again into ten, then a turn through one of the smaller divisions means an alteration in the distance of one-fiftieth of one-twentieth of an inch. As this angle is measured relatively to the lower curtain, a vertical brass scale, divided to tenths and twentieths of an inch, is fixed externally to the top of the extension of the lower curtain, extending vertically just outside the gradu- ated limb of the upper curtain, and thus serves for reading the angular distance of the index mark on the limb of the upper curtain, on any particular thread, and the number of threads from the index on the scale. The Adjustment of the Indices on the Barometer. 48. The lower curtain, together with the slotted tube and cap, is un- screwed from the neck of the cast-iron bottle and lifted off over the tube. Then the 30-inch standard bar is set on end upright on a surface plate, and the lower curtain, &c., are lowered over the bar until the lower lip of the curtain rests on the surface plate, and the top of the bar is 30 inches from this lip. The cap is then screwed down until light is seen over the top of the bar through the slot just cut off. Then a vertical line drawn on the cap just above the lip, at the edge of the scale, is the index on the cap, and a horizontal line, drawn on the scale level with the lip of the cap, is the index point on the scale. And, when these two lines are brought into this position, the distance between the lips will equal the length of the bar. In order to check this the curtain is raised, and two thin pieces of chemical paper are placed on the surface plate, one on each side of the bar, so as to leave a space between the paper and the bar. Then the curtain is replaced so that it rests on the paper, and light can be seen through the interval between the paper and the bar. Then light should be seen to an 650 ON THE MECHANICAL EQUIVALENT OF HEAT. [66 equal extent over the bar, and by screwing down the cap till the light disappears, the thickness of the paper will be measured by the angle turned through. The construction of this barometer, the first of its kind, was undertaken by Mr Foster, who has produced a very beautiful instrument by which direct reading can be taken to the ten-thousandth of an inch. The mercury having been re-evaporated for the purpose, in an apparatus belonging to Dr Schuster, by his assistant, Mr S. Stanton. This barometer could be used as a pressure gauge for pressure up to 34 inches and down to 26 inches, and by connecting the mouthpiece with a receiver in connection with a mercury or water syphon gauge, with the other limb open to the atmosphere, the differences of reading of the barometer for different pressures in the receiver can be readily compared with the corresponding differences in the syphon gauge, and by such comparisons, taken at intervals till the mercury reaches the closing in of the tube, a test is obtained as to the absence of anything but mercury vapour above the mercury. When the barometer is in connection with the vapour chamber in which the thermometer is immersed, the passage of moisture back into the barometer is prevented by connecting the tube by a branch with an air receiver, in which the pressure is maintained higher than that in the vapour chamber ; the branch pipe communicating with the chamber through a piece of quarter-inch glass pipe, 3 inches long, plugged as tightly as possible throughout its length with cotton-wool, through which the air has to pass from the receiver into the vapour chamber. In this way, an indefinitely slow current of clean dry air can be maintained into the passage from the vapour chamber to the valve which controls the exit of the steam into the atmosphere, so that the air does not enter the vapour chamber in which the thermometers are, but directly passes out with the overflow steam. There is necessarily some resistance to the air passing along the pipe to the vapour chamber, but this could easily be tested by removing the pipe from the vapour chamber, and leaving it. open to the atmosphere, so that the barometer would adjust itself to that of the atmosphere, plus the pressure due to the resistance of the current in the pipe ; then, stopping the current by closing the branch pipe, and reading again, the difference would give the pressure due to the current. With the plug as described this was so small as to be negligible, even when the pressure in the receiver was two atmospheres. As during the testing of the thermometers the pressure in the vapour chamber was generally greater than that of the 66] ON THE MECHANICAL KQUIVALENT OF HEAT. 651 atmosphere, in order to maintain this steady, a governor on the gas burner was necessary, as well as an accurately adjustable exit valve. With these appliances the scale of the high temperature thermometer could be tested at intervals, over a sufficient interval on each side of the boiling point (212 Fahr.), the corrections for surface tension, temperature, and gravitation being applied to within the thousandth of an inch of mercury. This gives the limits of error + O'OOOOl. Correction of the Low Temperature Thermometer. 49. The correction on the thermometer for 32 would be at any time obtained in the usual way by immersing the thermometer vertically in a bath of soft snow, but as there was no ready means, as with the scale about 212", of testing the scale at 32, while this would be used for one or two degrees, this correction could only be made by comparison with a thermometer already corrected with the air thermometer, which comparison Dr Schuster allowed to be made in the physical department. Corrections of the Thermometers for Pressure. 50. The pressures in the thermometer chambers of the brake being both some 10 or 15 inches of mercury above that of the atmosphere, it would be necessary to determine the corrections on each of the thermometers under the pressures and temperatures at which they had to work. Thus, if e lt e. 2 are the corrections per unit of pressure in the initial and final thermometers, the correction for the heat is (e^p^ e. 2 p.^). The Range of Temperature over which the Specific Heat would be Measured. 51. The temperature of the effluent water from the brake can be regulated either up or down to any required extent, and although there would necessarily be some divergence from the boiling-point, with care and experience it would be possible to bring the mean result in a number of trials within a close approximation of 212 Fahr. On the other hand, there has been no means provided of regulating the temperature of the water entering the brake. This is determined by the rate at which the water passes through the iced coil and the temperature at which it entered, as determined by the temperature in the town's mains, which varies from 38 in the winter to 55 in the summer. Thus the temperature in the light trials would be from half to a degree above 32, and that of the heavy trials from a degree to two degrees. 652 ON THE MECHANICAL EQUIVALENT OF HEAT. [66 In calculating the heat of each trial, the actual difference with the correction for the thermometers is taken, but if, as is shown by previous investigations by Regnault and others, the specific heat at and near 32 is less than the mean specific heat between 32 and 212 by something like 0*5 per cent., there would be errors in taking the results so obtained as the mean specific heat between 32 and 212. Owing to the extreme difficulty of determining the specific heat over a very short range of temperature to such high degrees of accuracy as '01 per cent., the experimental evidence as to the exact value of the specific heat within a few degrees of 32 is but vaguely surmised from the general fall of the specific heat with the temperature. The law of the thermal capacity of water between C. and t, as deduced by Regnault from his experiments, is avowedly vague as to the lower tem- peratures. It shows no singular point at the maximum density, as would be expected ; and Rankin deduced another law from these experiments, making the minimum specific heat coincide with the point of maximum density. Also other experimenters have obtained higher specific heats near 32 than are given by Regnault's formula. It would seem probable, therefore, that the difference between the specific heat at 32 and the mean between 32 and 212, as given by Regnault's formula, is too large. In that case, the correction obtained by this formula in order to reduce the specific heat between the observed temperature in the trials to that between the standard points, would probably be too large, and thus afford an outside limit of error. Thus, putting s for the mean specific heat between 32 and 212, s(l+X) for the specific heat between T, and 212, when T is small compared with 180, and, by Regnault, taking s(l O005) for the specific heat at T^, then the total heat from T, to 212 is s (1 + X ) (212 - 2V) = s (180 - (T - 32) (1 - O'OOo)} = s (212 - 2V) (l - ^ - *vo x 0-005) , or, neglecting (T l 32) 2 , T S9 X = 0-005 * = 0-000028 (T, - 32). loU Thus, taking the mean capacity of water between the temperatures of 32 and 212 as the standard capacity, the mean specific heat between TI and 212 would be 1 + X = 1 + 0-000028 (2V - 32) ; and, if 2\ is the mean initial temperature of the water of any number of 66] ON THE MECHANICAL EQUIVALENT OF HEAT. 653 trials, 1 + X is the mean specific heat of the water in all the trials. The mean specific heat of the difference of two trials would be 1 + X ; this appears as follows : Suppose 1 4- TI to be the mean specific heat for a set of heavy trials, and ir, the mean weight of water, and (1 + X 2 ) to be mean specific heat of a corresponding set of light trials, and W 2 the mean weight of water, 2\, T 2 being respectively the initial temperatures of W 1 and TF 2 , the difference of the total heats would be (1 + Z,) (212 - T,) W, - (1 + X 9 ) (212 - T 2 ) W s , and the mean specific heat would be approximately (212 - TV) F, -(212 - TV) W 3 +lSO(X l W 1 -X t W 9 ) (212 - 2\) Wi - (212 - T 2 ) W, ISO(X 1 W 1 -X,W. 2 ) and, as in the heavy and light trials TFj = 2 W approximately, the mean specific heat by Regnault's formula would be 1 + 2X, - X 2 = 1 + 0-000028 [2 (Z\ - 32) -(T z - 32)]. This result is obtained by merely summing the trials, but counting the water in the light trials as negative, JT- 2 (IT) The Gradual Rising of the Indices of the Thermometer. 52. Where, as is generally the case, the indices of the thermometers are gradually rising, if they are used between the intervals at which they are corrected, the last observed correction being applied, there will be an error which will be negative, and of magnitude equal to the rate of rise during the interval multiplied by the interval. Thus, if the trials are uniformly distributed between the intervals of correction, the correction would be 0'5, where a is the observed rise in the interval, hence the relative cor- rection on the equivalent, taking a^ arid , as the mean rises between the intervals of correction of the initial and final thermometers, would be 0-5 The Work done by Gravity on the Water. 53. The difference of pressure on the bulbs of the initial and final thermometers which are at the same level, expressed in feet of water, is 654 ON THE MECHANICAL EQUIVALENT OF HEAT. [G6 the work done by gravity per Ib. of water. If p l and p., express these pressures in inches of mercury, the work done by gravity is which gives as the relative correction for the equivalent, approximately, + 0-000008 S [ W ( P! - pj]/ 2(W). The Work absorbed in Wearing the Metal of the Bushes and tihaft. [54. During the six years the brake had been in use, before the trial commenced, the shaft and bushes were occasionally lubricated with oil, chiefly to prevent oxidation of the shaft when standing, and, up to the commencement of the trials, there was hardly any appreciable sign of wear. After the closing of the bushes by the stuffing-box and cap, when the use of oil was purposely discontinued, there was no means of observing the wear of the metal as long as the brake worked satisfactorily, as it did during all the trials. But when, after the completion of the trials, the stuffing-box and cap were removed, in order to return to the original manner of working, the excess of leaking through the bushes showed that there had been considerable wear. At that time it did not occur to me that the proportion of this wear, which took place during the actual running of the trials, would represent a certain amount of work absorbed in disintegrating the metal, or a certain amount of heat developed by the oxidation of the metal, and no attempt was then made to form a definite estimate of the amount of metal which had disappeared. As, however, the worn metal was replaced by a coating of white metal, the thickness of this (less than ^nd of an inch) and the extent of surface (less than 124 square inches) subsequently showed that it could not be more than 1 Ib. This was after it occurred to me that however small might be the effect of this wear, since it was definitely observed to have taken place during the twelve months when the bushes were closed for the purpose of the trials, it was desirable, in order to complete the research, that some outside estimate should be obtained of the limits to its possible effect, whether from disintegration or from oxidation. In as far as the loss of metal was due to the abrasion of the clean metal surfaces, it would be proportional to the number of revolutions, while in as far as it was owing to the oxidation of the metal surfaces, left bright after each run, it would be probably proportional to the number of runs. 66] ON THE MECHANICAL EQUIVALENT OF HEAT. 655 The number of revolutions with the bushes closed, counting ordinary work as well as the trials, is found from the records to be less than 300 x 60 x 360, and the number of runs to be 80, the mean time being 4'5 hours. The revolutions during any one of the accepted trials were 300 x 60. And the trials were made in threes, so that the coefficient for oxidation would be -%fa. Hence, the metal worn by abrasion in a single trial would be less than s^yth of 1 Ib. = 0'0028 lb., and the metal oxidised in one trial less than siiyth = 0'004 lb. So far the estimate is fairly definite, but, for its completion, it is necessary to arrive at some conclusion as to the work absorbed in dis- integrating the metal, and of the heat developed by its oxidation. There does not seem to be any reason why there should be more oxidation of the bright surfaces in a light trial than in a heavy trial, so that there would have been no error from this cause in their difference. As regards the abrasion and the oxidation of the abraded metal, there would be a difference, as the weight on the shaft in a heavy trial is T23 of the weight in a light trial. Thus the differences of abrasion would have been 0-0006 lb. The work necessary to produce a state of disintegration, such as exists in the vapour of the metal, would be the total heat of vaporization, less the kinetic energy and work [fc v /(T 32) + PV~\, and, although the heat of vaporization of the metal is not known, it would seem that it cannot greatly exceed, when subject to the deductions mentioned, the heat of vaporization of ice subjected to like deductions (1,000,000 ft.-lbs.). Assuming this, since the difference in the work of two trials is about 70,000,000 ft.-lbs., the correction would be - 0-00001, which, considering that the disintegration would be very imperfect, may be taken as an outside limit, while the effect may have been even reversed by the oxidation of the degraded metal. Nov. 9, 1897.] Accidents. 55. In contemplating such an extensive and complex research, the result of which depends on the mean of a number of experiments, it was impossible to overlook the question as to how such accidents, as would probably occur, should be dealt with. 656 ON THE MECHANICAL EQUIVALENT OF HEAT. [66 It was clear that, whatever the rule might be, it must be definite and rigorously applied. Two other things were also clear, that, as in surveying, accidents might occur, say in reading the counter or the scales, which would only be apparent from the reduction of the results after the trial was finished. Also, that in these experiments there would be no such rigorous check on the results as in surveying ; so that, without danger of sorting the results, anomalous results, the cause of which was not noted during the trial, could only be rejected when the results themselves contained evidence of the cause of the anomaly, say an abnormal difference between the mean speeds by the counter and the speed gauge. It was therefore, from the first, decided to reject all trials in which there was definite evidence either during the trial or in the results, of uncertainty to which no definite limits could be assigned, in any one of the measurements, without regard for the apparent consistency of the results, and in the same way to retain all other trials. 56. The following table contains a summary of all those circumstances on which the accuracy of the result of the investigation depends, together with references to the several Articles in which they have been discussed. In line with each circumstance is placed the formula for the relative correction in the equivalent, necessary in consequence of the observed deviation from the conditions of equality between the heavy and light trials. In the same line with each circumstance are also given, to the millionth part, the limits of relative error as deduced in the corresponding Articles. ON THE MECHANICAL EQUIVALENT OF HEAT. 657 so .2 &.'* z sat, 5<5 6 ooooooooooooooooooooo -5 s* S 2 ; : 1 ;j G: :::::: fc .g fe ! 1 5 'M C) G 5 cS 's'^ a S " 1 e automatic-gear on the balance of the bn limination of the error of balance heat conducted by the s radiated from the ment of the counter . . tt'erences in speed and temperature the stiiffinf-box . i "~ > C2 .~x I i ~ limination of the water lost by evaporatk .'curacy in weighing the water tie atrnosnhere . br gravity at the latitude of Greenwich ved in the water. . . ' 9 essure on the thermometers, limits of errc standard readings of the thermometers of correction between the initial temperature of wat L :. *- i * 0. "Z^ i c ~ I > \ 1 > _r 3 I ^ ^ -^ r t ? a -^ . r 2 ^ i! -^ n O ffi . "liifj -ii'lll - s - O. R. II. 658 ON THE MECHANICAL EQUIVALENT OF HEAT. [66 PART II. ON AN EXPERIMENTAL DETERMINATION OF THE MECHANICAL EQUIVALENT OF THE MEAN SPECIFIC HEAT OF WATER BETWEEN 32 AND 212 FAHR., MADE IN THE WHITWORTH ENGINEERING LABORATORY, OWENS COLLEGE, ON PROFESSOR OSBORNE REYNOLDS' METHOD. BY WILLIAM HENRY MOORBY, M.Sc. 1. In view of the frequent and extremely careful and accurate deter- minations of the value of the mechanical equivalent of heat which have been made of late years by different experimenters using different methods the present series of experiments may on first thoughts seem superfluous. There did, however, seem to be sufficient disagreement between the results pre- viously published more particularly between values of the equivalent, as derived from the direct methods described by Joule, Rowland, and Miculescu, and the indirect electrical methods of Griffiths, and Gannon, and Schuster, to warrant a new investigation into the value of this important constant, if the proposed new method of working should carry with it advantages not available in previous investigations. I was accordingly very glad to fall in with the wishes of Professor Reynolds that I should undertake a research bearing on this point on lines which he suggested to me in July, 1894. 2. In Part I., par. 3, a full description is given of the apparatus whose existence in the Whitworth Engineering Laboratory led up directly to the institution of this research into the value of the mechanical equivalent of heat. The advantages which the proposed method offered were briefly : (1) The possibility of obtaining a result which in no way depended for its accuracy on the value of the scale divisions of the ther- mometers used in the measurements of temperature (Part I., par. 11). This was done by supplying a stream of water to the brake at a temperature of 32 Fahr., and there raising its temperature to 212 Fahr. before admitting it to the discharge pipe where its temperature was again taken. (2) A means of eliminating from the result all losses of heat due to radiation and conduction from the calorimeter employed (Part I., par. 32). The manner in which this elimination was accomplished is indicated below. 66] ON THE MECHANICAL EQUIVALENT OF HEAT. 659 Let U and u represent the quantities of work done in two trials which differed only in the moment of resistance offered by the brake the number of revolutions of the engine shaft and the duration of the trials being the s;ime in each case. Also let //' and h' be the apparent quantities of heat generated in the brake in these trials. These quantities will be less than the true equivalents of the works U and u by quantities which represent the losses of heat from the brake by conduction, radiation, &c. These losses were made as nearly as possible equal by keeping the temperatures of the brake and its supports and surroundings at the same levels in the two trials. Then the quantity of work (U - u) should be exactly equivalent to the quantity of heat (H' h'), and by dividing the h'rst of these by the second, a value of the constant required is obtained. The power available for the purposes of the investigation enabled me to deal with quantities approaching the following values in trials of one hour's duration : Revolutions, 18,000. Total work done, 135,000,000 ft.-lbs. Total weight of water raised 180Fahr. = 960 Ibs. Total apparent heat generated = 170,000 B.T.U. In quantities so large as these some of the small errors inevitable to all physical experiments became quite or nearly negligible. Preliminary Apparatus and Trials. 3. It will, perhaps, be sufficient to indicate the general arrangement of the apparatus as first set up. This is illustrated in the annexed sketch. The water was supplied from the mains through the iron stand-pipe, A, and the regulating cork, B. Before it entered the brake its temperature was measured by means of the thermometer, C, inserted through a cork in the stand-pipe, the part of the stem on which readings were taken being exposed to the atmosphere. After being discharged from the brake, D, the water entered a flexible rubber pipe, E, bent through an angle of 90, which con- nected a horizontal nipple at the bottom of the brake with a vertical one forming the lower end of a fixed line of copper piping, F. The temperature of discharge of the water was indicated by the thermometer, G, which was enclosed in a glass tube opening through a stuffing-box into the discharge pipe, the whole length of the stem being therefore kept at the temperature of discharge. On leaving the copper discharge pipe the water was directed 422 660 ON THE MECHANICAL EQUIVALENT OF HEAT. [66 at will by the two-way tipping switch, K, either to the left to waste or to the right into the tank, L, standing on the platform of the weighing machine, M. Fig. 1. Preliminary Apparatus. Course of water shown by arrows. A series of trials were made with this apparatus, the water being raised through varying intervals of temperature between 35 Fahr. and 100 Fahr. For obvious reasons the results were not satisfactory, and are therefore not published. Experience was gained, however, which helped very materially in the design of the final apparatus. Common thermometers were used, and calibration errors on the com- paratively small range of temperature through which the water was raised were of sufficient importance to vitiate all results. Again, the exposure of the stem of the thermometer, 0, was a weak spot in the apparatus. I was much troubled also with leakage of water from the two bushed bearings of the brake. In so far as could be judged, the bent rubber pipe, E, was found to be a satisfactory connection between the brake and the copper discharge pipe, and this has been retained in the subsequent apparatus. DETAILS OF THE CONSTITUENT PARTS OF THE FINAL APPARATUS. Artificial Atmosphere. (Part I., par. 23.) 4. To prevent loss of water by evaporation at the centres of the vortices formed in the brake, the ports in the vanes of the outer casing were connected 66] ON THE MECHANICAL EQUIVALENT OF HEAT. 661 through a flexible rubber tube some 4 feet long, with an artificial atmosphere formed in a tin receiver, the pressure in which was maintained by means of a cycle tyre inflator at about 9 inches of mercury, as measured on a U -gauge. The shape of this vessel is made clear in the sketch (Part I., Fig. 8). The ends were made conical for greater strength. The receiver was also provided with an air valve, with which to relieve the pressure when too high, and a cock, with which water accidentally lodging inside could be drained away. The Ice Cooler. (Part I., par. 19.) 5. Some preliminary experiments indicated that a length of about 200 feet of f-inch diameter lead piping would, when immersed in a mixture of ice and water, be sufficient to cool a stream of some 16 Ibs. of water per minute very nearly to 32 Fahr. The ice cooler was accordingly made as follows: A wooden box, 4' 0" x 2' 3" x 2' 0", and lined inside with waxed cloth, was fitted with a horizontal wooden shelf about 2 feet 6 inches long, and on this was laid a flat oval coil of j|-inch composition piping nearly 200 feet in length, the left-hand end of the coil and shelf stopping short at a distance of 1 foot from the end of the box, the right-hand end of the coil reaching the end of the box, but the shelf stopping some 6 inches short of that point. The coil was about o inches diameter, vertically, and over it were placed the wooden guide plates shown (Part I., Fig. 7). An 8-inch diameter paddle, having 6 wooden floats, was placed about the middle of the box, at a height just sufficient to ensure the lower edges of the floats clearing the coil of pipe below it. A galvanized iron wire netting, extending from the shelf upwards to the top, separated the well at the left-hand end of the box from the com- partment to the right containing the coil and paddle. When working, the well and space beneath the shelf contained broken ice, well rammed in ; while the level of the water was automatically kept at about 3 inches above the top of the coil. The paddle, driven by a cord from the line shafting in the engine-room, revolved in the direction shown by the arrow, and caused a circulation of water up through the ice in the well, and then horizontally through the coil and back to the ice under the shelf. Circulating Pump. (Part I., par. 20.) 6. In order to supply sufficient water to the brake against the resistance offered by the 200 feet of pipe in the cooler and the augmented pressure in the brake itself, it was necessary to use a circulating pump. This was a small Mat her- Reynolds centrifugal pump with four 1^-inch wheels, driven 662 ON THE MECHANICAL EQUIVALENT OF HEAT. [66 by a turbine available for this purpose in the engine-room. This pump was capable of supplying 16 Ibs. of water per minute, against a pressure of 25 Ibs. per square inch at the supply valve. Some difficulty was encountered in the summer of 1896 with this com- bination, because the excessive demand for condensing water for the engine hardly left sufficient flow in the falling hydraulic main to work the turbine at the requisite speed to maintain the above pressure. On the whole, however, the combination was exceedingly efficient, and with a graduated supply valve afforded a very delicate means of regulating the flow of water into the brake. Water-tight Joints between the Brake and the Engine Shaft. 7. In Part I., par. 24-29, the necessity of obtaining control over the leakage of water at the bearings of the brake, and the methods by which this was accomplished, are fully discussed. The bearing on the up-shaft end of the brake was provided with a stuffing-box, while the shaft end was covered with a cap. The annexed sketches show the general design of the stuffing- box and cap : A The engine crank shaft. B The outer skin of the brake. C Conical brass bushes screwed into the outer skin of the brake. D Lock nuts on these bushes. E, F, and G Stuffing-box, ring and cover. K Set screws fastening stuffing-box to the lock nut. L Cap covering the end of the shaft. M Small spindle driven by a pin on the end of the engine shaft, passing through a stuffing-box on the cap, and required to drive the revolution counter. The cap completely stopped all leakage from the bearing to which it was fixed, and, when the stuffing-box had worked for a short time, only a few drops of water escaped from the up-shaft bearing. The brass bush bearings needed lubricating, and this was accomplished by supplying a small stream of water to each bearing through the pipes N and P, each provided with a regulating cock. This water carne from the supply pipe between the ice cooler and the regulating valve controlling the main supply to the brake. It was consequently under considerable pressure and at a temperature very little over 32 Fahr. The water thus supplied 66] ON THE MECHANICAL EQUIVALENT OF HEAT. 663 had, of course, to enter the brake, and the amount supplied afforded a very convenient means of controlling the temperatures of the bearings. Fig. 2. Joints between brake and shaft. At a distance of 2f inches from the cap of the stuffing-box was the end of one of the main bearings, R, carried on the cast-iron pedestal, S. It was important that I should have some control over the loss of heat by conduction along this length of shaft. Accordingly, two pieces of brass pipe were soldered on to the cap of the stuffing-box, while two others were screwed, the one in the upper and the other into the lower brass forming the main bearing. Thermometers were placed inside the tube affixed to the stuffing-box cap, which happened to be uppermost at the time, and into the two pipes screwed into the main bearing. It was then assumed that the loss of heat along the shaft would vary with the difference of temperature between the stuffing-box cap and the bearing. In order that the losses of heat occurring in this way in any two trials should be identical, it was sufficient under the above assumption that this difference of temperature should be the same in both trials, and the temperature of the stuffing-box was regulated to this end by means of the amount of cold water passing into it.