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 m