PAPERS ON MECHANICAL AND PHYSICAL SUBJECTS Sonfcon: C. J. CLAY AND SONS, CAMBEIDGE UNIVERSITY PRESS WAREHOUSE, AYE MARIA LANE. laagoto: 50, WELLINGTON STREET. TO ILetpjtg: F. A. BROCKHAUS. #eto lorft: THE MACMILLAN COMPANY. Bombag anfi Calcutta : MACMILLAN AND CO., LTD. [All rights reserved."} PAPERS ON MECHANICAL AND PHYSICAL SUBJECTS BY OSBOENE REYNOLDS, M.A., F.R.S., LL.D., MEM.INST. C.E. PROFESSOR OF ENGINEERING IN THE OWENS COLLEGE, AND HONORARY FELLOW OF QUEENS' COLLEGE, CAMBRIDGE. VOLUME III THE SUB-MECHANICS OF THE UNIVERSE CAMBRIDGE: AT THE UNIVERSITY PRESS. 1903 CatnftrtJge: PRINTED BY 3. AND C. F. CLAY, AT THE UNIVERSITY PRESS. PREFACE. f I^HIS memoir " On the Sub-Mechanics of the Universe " was com- municated to the Royal Society on February 3, 1902, for publication in the Philosophical Transactions ; it was read in abstract before the Society on February 13. It was under criticism by the referees of the Royal Society some five months. I was then informed by the Secretaries that it had been accepted for publication in full. At the same time the Secretaries asked me if I should be willing, on account of the size and character of the memoir, which seemed to demand a separate volume, to consent to what appeared to be an opportunity of making a substantial reduction in what would otherwise be the expense. The Cambridge University Press had already published two volumes of my Scientific Papers and were willing to share in the cost of publishing this as a separate volume to range with the other two, special copies being distributed by the Royal Society as in the case of the Philosophical Transactions. To this proposal I readily agreed. OSBORNE REYNOLDS. January 23, 1903. EKRATUM. p. 5, line 22: for 2 read q. TABLE OF CONTENTS. SECTION I. INTRODUCTION. ART. PAGE 1 8. Sketch of the results obtained and of the steps taken .... 1 8 SECTION II. THE GENERAL EQUATIONS OF MOTION OF ANY ENTITY. 9. Axiom I. and the general equation on which it is founded ... 9 10. The general equation of continuity ........ ib. 11. Transformation of the equations of motion and continuity for a steady space 10 12. Discontinuity ib. 13. Equation for a fixed space 11 14. Equation for a moving space 12 SECTION III. THE GENERAL EQUATIONS OF MOTION, IN A PURELY MECHANICAL MEDIUM, OF MASS, MOMENTUM, AND ENERGY. 15. The form of the equations depends on the definitions given respectively to the three entities .......... 14 16. Definition of a purely mechanical medium ib. 17. The properties of a purely mechanical medium necessitated by the laws of motion 15 18. The equation of continuity of mass 16 19. The position of mass ib. 20 21. The expression of general mathematical relations between the various expressions which enter into the equation, into which the density enters as a linear factor 17 18 22. Momentum 19 23. Conduction of momentum by the mechanical medium .... 20 a 5 Vlll CONTENTS. ART. PAGE 24. The actions necessary to satisfy the condition that action and reaction are equal 22 25. The conservation of the position of momentum ...... 23 26. Conservation of moments of momentum 24 27. Bounding surfaces ib. 28. Energy 25 29. The general equation of energy in a medium in which there are no physical properties 27 30. Simplification of the expressions in the equations of energy . . . ib. 31. Possible conditions of the mass in a purely mechanical medium . . 28 32. The transformations of the directions of the energy and angular redis- tribution ............. ib. 33. The continuity of the position of energy ....... 30 34. Discontinuity in the medium 31 SECTION IV. EQUATIONS OF CONTINUITY FOR COMPONENT SYSTEMS OF MOTION. 35. Component systems of motion may be distinguished by definition of their component velocities or their densities 32 36. Component systems distinguished by distribution of mass ... 36 37. Component systems distinguished by density and velocity ... 37 38. The distribution of momentum in the component systems ... 38 39. The component equations of energy of the component systems distinguished by density and velocity .....:.... 39 40. Generality of the equations for the component systems .... 40 41. Further extension of the system of the analysis 41 SECTION V. THE MEAN AND RELATIVE MOTIONS OF A MEDIUM. 42. Kinematical definition of mean motion and relative motion ... 42 43. The independence of the mean and relative motions ..... 44 44.. Component systems of mean and relative motion are not a geometrical necessity of resultant motion ......... 45 45. Theorem A * ...... ib. 46. Theorem B 46 47. General conditions to be satisfied by relative velocity and relative density . 48 48 49. Continuous states of mean and relative motion ..... 50 50. The instruments for analysis of mean and relative motion ... 51 51. Approximate systems of mean and relative motion * ib. 52. Relation between the scales of mean and relative motion ... 53 CONTENTS. ix SECTION VI. APPROXIMATE EQUATIONS OF COMPONENT SYSTEMS OF MEAN AND RELATIVE MOTION. ART. PAGE 53 54. Initial conditions 55 55. The rate of transformation at a point from mean velocity per unit of mass ............. 56 56. The rate of transformation at a point from relative velocity . . . ib. 57. The rates of transformation of the energy of mean velocity ... 57 58. The component systems of the energies of the mean and relative velocity per unit mass may be separately abstracted into mean and relative component systems ib. 59. The rate of transformation from mean to relative energy ... 59 60. The transformation for mean and relative momentum .... 60 61. The rates of transformation of mean energy of the components of mean and relative velocity .......... ib. 62. The expressions for the transformation of energy of mean to relative motion ............. 62 63. The equations for the rates of change of density of mean and relative mass ............. 65 64. The equations for mean momentum ........ ib. 65. The equations for the rates of change of the density of mean energy of the components of mean motion and of the mean energy of the components of relative velocity ib. 66. Equation for the density of relative energy 66 67. The complete equations ib. SECTION VII. THE GENERAL CONDITIONS FOR THE CONTINUANCE OF COM- PONENT SYSTEMS OF MEAN AND RELATIVE MOTION. 68. The components of momentum of relative velocity, as well as the relative density, must respectively be such that their integrals with respect to any two independent variables, taken over limits denned by the scale of relative motion, have no mean values ... 69 69. The existence of systems of mean and relative motion depends on the property of mass of exchanging momentum with other mass . . 70 70. Conclusive evidence as to the properties of conduction and distribution of mass for the maintenance of mean and relative systems . . 71 71. The mass must be perfectly free to change in shape without change of volume or must consist of mass or masses each of which maintains its shape and volume absolutely ib. 72. Evidence as to the conducting properties for the maintenance of com- ponent systems ib. X CONTENTS. ABT. PAGE 73. The differentiation of the four general states of media which as resultant systems satisfy the conditions of being purely mechanical from those which also satisfy the conditions of consisting of component systems of approximately mean and relative motion ...... 73 74. A perfect fluid although satisfying the conditions of a purely mechanical medium as a resultant system cannot satisfy, generally, the condition of consisting of component systems of approximately mean and rela- tive motion 74 75. Purely mechanical media consisting of perfectly conducting members which have a certain degree of independent movement 76 76. The distinction of the purely mechanical media arising from the relative extent of the freedoms ib. 77. The case of uniform spherical grams, smooth and without rotation or motion 77 78. Logarithmic rates of decrement of mean inequalities in the component paths of the grains are necessary to secure that the rates of dis- placement of the momentum shall be approximately equal in all directions 78 79. The inequalities in the mean symmetrical arrangement of the mass are of primary importance and distinguish between the classes of the granular media ........... 80 80. Effects of acceleration in distributing all inequalities are independent of any symmetry in the mean arrangement of the grains . . . ib. 81. The definite limit at which redistribution of the length of the mean path ceases is that state of relative freedom which does not prevent the passage of a grain across the triangular plane surface set out by the centres of any three grains 81 82 83. The fundamental difference according to whether the freedom is within the limit, and the time of relaxation will be a function of the freedom ib. 84. Independence of the redistribution of vis viva on the fundamental limit . ib. 85. The limitation imposed by the methods hitherto used in the kinetic theory . ib. 86. The relative paths of the grains may be indefinitely small as compared with the diameter of a grain 82 87 88. Although media, in which each grain is in complete constraint with its neighbours, cannot consist of systems of mean and relative motion, if there is relative motion there is no limit to the approximation . ib. 89. The symmetrical arrangements of the spherical equal grains ... 83 90 91. Limiting similarity of the states of media with and without relative motion ............. 84 5 92. Summary and conclusions 85 SECTION VIII. THE CONDUCTING PROPERTIES OF THE ABSOLUTELY RIGID GRANULE, ULTIMATE-ATOM OR PRIMORDIAN. 93. The absolutely rigid grain is a quantity of another order than any material body 87 94. The mass of a grain and the density of the medium .... 88 CONTENTS. SECTION IX. THE PROBABLE ULTIMATE DISTRIBUTION OF THE VELOCITIES OF THE MEMBERS OF GRANULAR MEDIA AS THE RESULT OF ENCOUNTERS WHEN THERE is NO MEAN MOTION. ART. PAGE 95. Maxwell's theory of hard spheres 89 96. Maxwell's law of the probable distribution of vis viva is independent of equality in the lengths of the mean paths ...... 90 97. The distribution of mean and relative velocities of pairs of grains . ' . 91 98. Extensions and modifications which are necessary to render the analysis general . 93 SECTION X. EXTENSION OF THE KINETIC THEORY TO INCLUDE RATES OF CONDUCTION THROUGH THE GRAINS WHEN THE MEDIUM is IN ULTIMATE CONDITION AND UNDER NO MEAN STRAIN. 99 100. The determination of the mean path of a grain .... 95 101. The probable mean striking distance of a grain ...'.. 96 102. Further definition of /(r,X") 97 103. Expressions for the mean relative path of a grain, &c. .... 98 104. The probable mean product of the displacement of momentum in the direction of the normal encounter by conduction multiplied by the component of \/2 \\ in the direction of the normal .... ib. 105. The probable mean component conduction of component momentum in any fixed direction at a collision 99 106. The number of collisions between pairs of grains having particular relative velocities in a unit of time in unit space . . . . 100 107. The mean velocity of grains, the mean relative velocity of pairs of grains, and the mean velocity of pairs of grains ib. 108. The mean path of a grain, taking V2A for the mean path of a pair of grains ............. 101 109. The mean path of a pair of grains 102 110. The number of collisions of pairs of grains having relative velocities between V2FV and A/2 (JY + dVY) ib. 111. The mean rate of conduction of component momentum in the direction of the momentum conducted ib. 112. The mean normal stresses in the direction of the momentum conducted and the mean tangential stresses in the directions at right angles to the direction of the momentum conducted 103 113. The mean rate of convection of the components of momentum in the direction jc having velocities \\' for which all directions are equally probable 104 114. The total rates of displacement of mean momentum in a uniform medium . . . ib. Xll CONTENTS. ART. PAGE 115. The number of collisions which occur between pairs of grains having mean velocities between V l \/2 and ( F t ' -f d F/) \/2 .... 105 116. The mean velocity of pairs having relative velocities \/2 F/ and F//V2 ib. 117. All directions of mean velocity of a pair are equally probable, what- ever the direction of the mean velocity ib. 118. The probable component of mean velocity of a pair having relative velocity r 2 = \/ZV l 106 119. The probable mean transmission of vis viva at an encounter in the direction of the normal ib. 120. The mean distance through which the actual vis viva of a pair having rela- tive velocity \/2 F/ is Fj'/ \/2 and the actual vis viva of such a pair is 2(V 1 2 + / |-) = 4(F 1 /\/2) 2 ib. 121. The probable mean component displacement of vis viva at a mean collision by conduction 107 122. The probable mean component displacement of vis viva by convection between encounters by a grain having velocity between Fj' and Vi+dVi ib. 123. The mean component flux of vis viva ....... ib. 124. The mean component flux of component vis viva ..... 108 125. The component of flux of mass in a uniform medium .... ib. 126. Summary and conclusions 109 SECTION XI. THE REDISTRIBUTION OF ANGULAR INEQUALITIES IN THE RELATIVE SYSTEM. 127. Two rates of redistribution analytically distinguishable as belonging to different classes of motion 110 128. The logarithmic rates of angular redistribution by conduction through the grains as well as by convection by the grains Rankine's method 111 129. There are always masses engaged in each encounter . . . . ib. 130. When two hard spheres encounter, &c. . . . . . . . 112 131. The fundamental effects neglected in the kinetic theory hitherto . . 113 132. The concrete effects of encounters between grains ..... ib. 133. Variations in the complex accident ib. 134. The effects which follow from the three instantaneous effects. . . 114 135. The instantaneous and after effects of encounters before the next encounter of either of the grains ........ 115 136. Theorem ib. 137. The theorem, Art. 136, includes the redistribution of the actual vis viva 116 138. The redistribution of inequalities of the angular distribution of normals ib. 139. The redistribution of the rates of limited conduction .... ib. 140. The analytical definition of the rates of angular redistribution of in- equalities in the direction of the vis viva of relative motion . . 117 141. The energy of component motion in any direction cannot by its own effort increase the energy of the component motion in this direction ib. CONTENTS. xiii ART. PAGE 142. The active and passive accidents 118 143. The active accidents are the work spent by the efforts produced, &c. . . 119 144. The angular dispersion of the relative motion ib. 145. The mean angular inequalities 120 146. The angular inequalities in the mean relative motions of pairs of grains have the same coefficients of inequality as the mean actual motions . ib. 147. The mean squares of the components of relative motion of all pairs are double the mean squares of the components of actual motion . 121 148. The rate of angular redistribution of the mean inequalities in actual motion is the same as the rate of redistribution of the angular inequalities in the relative motion of all pairs ..... ib. 149. The rate of angular dispersion of the mean inequalities in the vis viva 122 150. The time mean of mean inequalities in the vis viva .... ib. 151 152. The rates of angular dispersion refer to axes which are not necessarily principal axes of rates of distortion . . . . . 124 153. The analytical definition of the rates of angular redistribution of in- equalities in rates of conduction through the grains .... 125 154. The rate of angular redistribution of the mean inequalities in the position of the relative mass in terms of the quantities which define the state of the medium ........... 126 155. The limits to the dispersion of angular inequalities in the mean mass . 127 156. The rates of probable redistribution of angular inequalities in rates of conduction 128 SECTION XII. THE LINEAR DISPERSION OF MASS AND OF THE MOMENTUM AND ENERGY OF RELATIVE MOTION BY CONVECTION AND CONDUCTION. 157. Linear redistribution requires the conveyance or transmission of energy from one space to another '. 131 158. The analysis to be general must take account of all possible variations in the arrangement of the grains, but in the first instance it may be restricted to those arrangements which have three axes at right angles . 132 159. Mean ranges 133 160 162. Component masses 133 4 163 165. The mean characteristics of the state of the medium . . . 134 5 166. Rates of convection and conduction by an elementary group . . . 136 167. The rate of displacement of vis viva by an elementary group referred to fixed axes ib. 168. The inequalities in the mean rates of flux of mass, momentum, and vis viva resulting from the space variations in the mean characteristics 137 169. Conditions between the variations in the mean characteristics in order that a medium may be in steady condition with respect to all the characteristics 138 170. The equation for the mean flux 139 171. The conditions of equilibrium of the mass referred to axes moving with the mean motion of the medium , . ,.,.< 141 XIV CONTENTS. ABT. PAGE 172. The coefficients of the component rates of flux of the mean component vis viva of the grains .......... 143 173. The rates of dispersion of the linear inequalities in the vis viva of the grains ib. 174. The expressions for the coefficients G and D 144 175. Summary and conclusions as to the rates of redistribution by relative motion ib. SECTION XIII. THE EXCHANGES BETWEEN THE MEAN AND RELATIVE SYSTEMS. 176. The only exchanges between the two systems 146 177. The institution of inequalities in the state of the medium . . . 147 178. The institution of angular inequalities in the rates of conduction . . ib. 179. The probable rates of institution of inequalities in the mean angular distribution of mass 150 180. The initiation of angular inequalities in the distribution of the probable rates of conduction resulting from angular redistribution of the mass 153 181. The rates of increase of conduction resulting from rates of change of density 154 182. The rates of increase of angular inequalities in the rates of convection resulting from distortional rates of strain in the mean system . . 157 183. The institution of linear inequalities in the rates of flux of vis viva of relative motion by convection and conduction ..... 158 184. The institution of inequalities in the mean motion ib. 185. The redistribution of inequalities in the mean motion . . . . 159 186. The inequalities in the components of mean motion typical expressions of accelerations to rates of increase in inequalities in mean motion . 160 187 188. The initial inequalities in the mean motion and accelerations to the dispersive condition 161 3 189. The conservation of the dispersive condition depends on the rates of redistribution of relative motion . . 163 190. Inequalities in relative vis viva and rates of conduction maintained by the joint actions 164 191 192. Steady, periodic institutions in all the eight equations . . . 165 6 193. Approximate solutions of the equations 168 194 195. Expressions for the resultant institution of inequalities of mean motion 1701 196. The equations of motion of the mean system in terms of the quantities which define the state of the medium 173 197. Equations of motion to a first approximation 175 198. Equations of the components of energy of the relative system in steady or periodic motion 176 199. The rates of irreversible dissipations of energy resulting from each of the several actions as expressed in the first approximation causing logarithmic rates of diminution in the linear inequalities of mean motion 178 200. The determination of the mean approximate rates of logarithmic decrement 179 201. Rate of decrefnent of normal wave, also of the transverse wave . . 180 CONTENTS. XV SECTION XIV. CONSERVATION OF INEQUALITIES IN THE MEAN MASS AND THEIR MOTIONS ABOUT LOCAL CENTRES. ART. PAGE 202. Local abnormal disarrangements of the grains, when so close that diffusion is impossible except in spaces or at closed surfaces of disarrangement depending on the value of G, under which conditions it is possible that about local centres there may be singular surfaces of freedom which admit of their motion through the medium in any direction by propagation, combined with strains throughout the medium, which strains result from the local disarrangement, without change in the mean arrangement of the grains about the local centres, the grains moving so as to preserve the similarity of the arrangement . . 183 203 204. (1) Such permanence belongs to all local disarrangements of the grains from the normal piling which result from the absence of any particular number of grains at some one or more places in the medium which would otherwise be in normal piling. The centres of such local inequalities in the mean mass are called centres of negative disturbance or centres of inequalities in the mean density. (2) In the same way inequalities resulting from a local excess of grains institute a positive local inequality which is permanent. (3) Also a mere displacement of grains from one position in the medium to another institutes a complex inequality in the mass, which corresponds exactly to electricity. And (4) the last class is that which depends on rotational displacement of the grains about some axis, which corresponds to magnetism 183 6 205207. (1) The coefficients of dilatation. (2) The normal pressures when a"=0 1867 208. (1) Inward radial displacements from infinity throughout the medium by the removal of any number of grains. (2) The sum of the normal and tangential pressures would equal the mean pressure in the medium ............. 189 209 210. The dilatation resulting from a negative inequality is SB xlO" 1 in c.G.s. units 229 249. From the value of v 1 " the general equation 22Q X 1-8574X 10 11 ( - 2 = V220 x M72 x 10 12 is obtained, and from this the value of X" is obtained as 8'612 x 10~ 28 and the values of or and a" are obtained in terms of ^/22Q . . 230 250 251. The conclusions to be drawn from the absence of the evidence of any normal waves in the medium of space until recent times . 2312 252. The X-rays ............. 232 253. The rate of decrement of the normal wave in terms of \/22Q . . 233 254255. The density of the medium in c.G.s. units is 2212 = 10,000 . . 234 256. The inferior limit obtained from the evidence of Rbntgen rays agrees with the superior limit as obtained from the size of the molecules . 236 257 258. Further analysis the explanation of the blackness of the sky on a clear dark night ........... 238 259. The fundamental dissipation of energy of mean motion to increase irre- versible energy of the grains in the medium ..... ib. 260. The number of grains, the displacement of which through a unit distance represents the electrostatic unit ........ 239 261. The coincidences between the periods of vibration of the molecules and the periods of the waves , ........ ib. 262. Dissociation of compound molecules proves the previous state to have been a state of instability ......... 240 263 264. Light is produced by the reversion of complex inequalities . . ib. 265. The reassociation of compound molecules results from reversion of complex inequalities ............ 241 266. The absorption of the energy of light by inequalities .... 242 267. Negative inequalities affect the waves passing through .... 243 268. Refraction is caused by the vibrations of the inequalities having the same periods as the waves ......... 245 269. Dispersion results from the greater number of coincidences as the waves get shorter ............ 246 270. Polarisation of light by reflection is caused only by that component of the transverse motion in the medium which is in the plane of coincidence and results from the passage of the light from a space without inequalities through a surface into a space in which there are inequalities. Metallic reflection results from the relative smallness of the dimensions of the molecules compai'ed with the wave-length, and the closeness of the piling ........ ib. 271. Aberration of light results from the absence of any appreciable resistance to the motion of the medium when passing through matter . . 249 INDEX . 253 SECTION I. INTRODUCTION. 1. BY this research it is shown that there is one, and only one, conceivable purely mechanical system capable of accounting for all the physical evidence, as we know it, in the Universe. The system is neither more nor less than an arrangement, of indefinite extent, of uniform spherical grains generally in normal piling so close that the grains cannot change their neighbours, although continually in relative motion with each other ; the grains being of changeless shape and size ; thus constituting, to a first approximation, an elastic medium with six axes of elasticity symmetrically placed. The diameter of a grain, in C.G.S. units, is 5-534 x 10- 18 = a: The mean relative velocities of the grains are 6-777x10 = 0". The mean path of the grains is 8-612xlO~ 28 = X. These three quantities completely define the state of the medium in spaces where the piling is normal ; they also define the mean density of the medium as compared with the density of water as 10 4 = 22fl The mean pressure in the medium, equal in all directions, is 1172 xlO 14 =2?. The coefficient of the transverse elasticity resulting from the gearing of the grains, where the piling is normal, is 9-03 x 10 24 = n. The rate of propagation of the transverse wave is 3-004 xl0 10 =r or Vn/j. ^R. 1 2 ON THE SUB-MECHANICS OF THE UNIVERSE. [2 The rate of propagation of the normal wave is 7-161 x!0 10 = 2-387 XT. The rate of degradation of the transverse waves, i.e. the dissipation resulting from the angular redistribution of the energy, or viscosity, is 5-603 x 10- 16 = tt or such as would require fifty-six million years to reduce the total energy in the wave in the ratio 1/e 2 , or to one-eighth ; thus accounting, by mechanical considerations, for the blackness of the sky on a clear dark night ; while the degradation of the normal wave, i.e. the dissipation resulting from the linear redistribution of energy, is such that the initial energy would be reduced to one-eighth in the (3'923 x l()~ 6 )th part of a second, or before it had traversed 2200 metres ; and thus would account by mechanical considerations for the absence of any physical evidence of normal waves, except such evidence as might be obtained within some metres of the origin of the wave; as in the case of Rontgen rays. 2. In spaces in which there are local inequalities in the medium about local centres, owing to the absence or presence of a number of grains, in deficiency or excess of the number necessary to render the piling normal, such local inequalities are permanent ; and are attended with inward or outward displacements and strains, as the case may be, extending indefinitely throughout the medium, causing dilatation equal everywhere to the strains but of opposite sign, i.e. dilatation equal to the volume of the grains, the presence or absence of which cause the inequality. When the arrangement of the grains about the centres is that of a nucleus of grains in normal piling on which grains in the strained, normal piling rest, the nucleus in normal piling cannot gear with the grains outside, in strained normal piling; so that there is a singular surface of misfit between the nucleus and the grains in strained normal piling. Such singular surfaces are surfaces of weakness and may be surfaces of freedom or surfaces of limited stability with the neighbouring grains. These singular surfaces, when their limited stability is overcome, are free to maintain their motion through the medium, by a process of propagation, in any direction ; the number of grains entering the surface on the one side being exactly the same as the number leaving on the other side; so that when the inequalities are the result of the absence of grains they correspond to the molecules of matter. If the singular surface of a negative inequality is propagating through a medium which is at rest, the grains forming the nucleus will have no 2] SKETCH OF THE RESULTS AND SOME OF THE STEPS. 3 motion, whatever may be the motion of the singular surface : but the strained normal piling, which surrounds the singular surface and moves by propa- gation with the singular surface, being of less density than the mean density of the medium, represents a displacement of the negative mass of the inequality, i.e. of the grains absent. And in whatever direction the singular surface is propagated the motion of the medium outside is such as represents equal and opposite momentum ; as when a bubble is rising in water. In exactly the same way, for inequalities resulting from an excess of grains, the momentum resulting from the displacement of the medium would be positive. The principal stresses in the medium outside the singular surface of a negative inequality are to a first approximation two equal tangential pressures equal in all directions ; and a normal pressure p r = %p, the mean of these pressures being everywhere the mean pressure of the medium p equal in all directions. Efforts, proportional to the inverse square of the distance, to cause two negative inequalities at finite distances to approach are the result of those components of the dilatation (taken to a first approximation only) which are caused by the variation of those components of the inward strain which cause curvature in the normal piling of the medium. The other components of the strain being parallel, distortions which satisfy the condition of geometrical similarity do not affect the effort. If the grains were inde- finitely small there would be no effort. Thus the diameter of a grain is the parameter of the effort ; and multiplying this diameter by the curvature of the medium and again by the mean pressure of the medium the product measures the intensity of the effort. The dilatation diminishes as the centres of the negative inequalities approach, and work is done by the pressure in the medium, outside the singular surfaces, to bring the negative inequalities together. The efforts to cause the negative inequalities to approach correspond, exactly, to gravitation, if matter represents negative mass. Taking the mean density of the earth as 5*67, as compared with water (-!)> the reciprocal of the density of the medium being 10" 4 , the mean pressure of the medium 1'172 x 10 14 , a the diameter of the grain 5*534 x 10~ 18 , the mean radius of the earth 6 '3709 x 10 8 ; 12 4 ON THE SUB-MECHANICS OF THE UNIVERSE. [3 the effort to cause approach between the earth and a unit of matter on the surface ( 1) is the product of these quantities multiplied by 47T/3, or pa- x 10- 4 x f TT x 5-67 x 6-3709 x 10 8 = 9'81 x 10 2 . The inversion is thus complete. Matter is an absence of mass, and the effort to bring the negative inequalities together is also an effort on the mass to recede. And since the actions are those of positive pressure there is no attraction involved ; the efforts being the result of the virtual diminution of the pressure inwards. 3. If instead of the negative inequalities, as in the last article, the inequalities are positive, the efforts would be reversed, tending to separate the positive inequalities, and the analysis would be the same, except that the curvature would be negative. And it is important to notice that if such positive inequalities exist, the fact that they repel each other i.e. they would tend to scatter through space together with the evidence that the number of inequalities either positive or negative occupy an indefinitely small space as compared to the total volume of the medium, places any importance such positive inequalities might have on a footing of indefinitely less importance than that of the negative inequalities which are caused to accumulate by gravitation ; and thus we have an explanation of the lack of evidence of any positive inequalities, even if such exist. 4. Besides the positive and negative inequalities there is another inequality which may be easily conceived, and this is of fundamental im- portance whatever may be the cause, it is possible to conceive that a number of grains may be removed from some position in the otherwise uniform medium, to another position. Thus instituting a complex in- equality, as between two inequalities, one positive and the other negative ; the number of grains in excess in the one being exactly the same as the number deficient in the other. The complex inequalities differ fundamentally from the gravitating inequalities, inasmuch as the former involve an absolute displacement of mass while the latter have no effect on the mean position of the mass in the medium ; and in respect of involving absolute displacement of mass the complex inequalities correspond with electricity. Apart from the displacement of mass the complex inequalities differ from the gravitating inequalities. In the complex inequalities the para- meter of the dilatation is not the diameter of a grain but one half the linear dimension of the- volume occupied by the grains displaced, taken as spherical. The effort to revert in the case of the complex inequality is the product of the pressure multiplied by the product of the volumes of the positive 4] SKETCH OF THE RESULTS AND SOME OF THE STEPS. 5 and negative inequalities and again by the parameter r . This is ex- pressed when the positive and negative inequalities are at finite distance apart by R being essentially negative and the dimensions of the effort ( R) are mtt~ 2 which express an effort to the displacement of mass. The complex inequality which corresponds to the separation of the positive and negative inequalities is one displacement, not two. This fact admits of no question and might have been recognised long ago had it not been for the general assumption that positive electricity repels positive electricity, the fact being that the apparent repulsion of the positive electricities is the result of their respective efforts to approach their re- spective negative inequalities. By the assumption it became apparently possible to express the potential V, and the electricity q as rational quantities, when, as it now appears, the potential V and the electricity q are re- spectively (- e 2 )* - and (&*)%, which are both irrational. Their product being the rational quantity which, differentiated with respect to the distance, is - e --R ? mj b and the mechanical explanation of these is, I and for the effort to revert, we have Then for the electrostatic unit we have, since r = l, and R = l, and from the known value of p the number of grains displaced through unit distance necessary to cause the unit effort is 1-615 x 10 48 , and r = G'493 x W~ 3 , from which we have the ratio of the effort to reinstate the normal piling, to the effort of gravitation, from the same number of 6 ON THE SUB-MECHANICS OF THE UNIVERSE. [5 grains absent in each inequality as are displaced in the complex inequality, the distances being the same, 1-2 x 10 15 , so that the effort of attraction between two inequalities, the grains absent about each of which is the same as the grains displaced in instituting the complex inequality, is eighty-one thousand billions less than that of the electric effort. 5. Cohesion between the singular surfaces of the negative inequalities results from the terms which were not taken into account in the first approxi- mation which correspond to gravitation. These secondary terms involve the inverse distance to the sixth power, and therefore have a very short range, and so correspond to efforts of cohesion of the singular surfaces as well as surface tensions having no effect unless the singular surfaces, or molecules, are within a distance very small compared with the diameter of the singular surface. 6. Transverse undulations in the medium, corresponding to the waves of light, are instituted by the disruptive reversion of the complex in- equalities. The recoil sets up a vibration which is exhausted in initiating light. 7. Thus far the sketch of the results has included only those for which there exists sufficient evidence to admit of definite quantitative analysis. Nevertheless these quantitative results show that the granular medium, as already defined, accounts by purely mechanical considerations for the evidence, and affords the only purely mechanical explanation possible. If then the substructure of the universe is mechanical, all the evidence, not already adduced, is such as may be accounted for by an extension of the analysis, and this is found to be the case. The results of the further analysis afford proof: Of the existence of coincidence between the periods of vibration of the molecules and the periods of the waves; that dissociation of compound molecules proves the previous state to have been one of limited stability; that the reassociation of compound molecules results from the reversion of complex molecules ; of the absorption of the energy of light by inequalities ; that negative inequalities affect the waves passing through ; 8] SKETCH OF THE RESULTS AND SOME OF THE STEPS. 7 that refraction is caused by the vibration of inequalities having the same periods as the waves; that dispersion results from the greater number of coincidences as the waves get shorter ; that the polarization by reflection is caused only by that component of the transverse motion in the medium which is in the plane of incidence and results from the passage of the light from a space without, or with few, inequalities, through a surface into a space in which there are more inequalities; that the metallic reflection results from the relative smallness of the dimensions of the molecules compared with the length of the wave and the closeness of their piling when the waves pass from a space without inequalities across the surface beyond which the inequalities are in closest order ; that the aberration of light results from the absence of any appreciable resistance to the motion of the medium when passing through matter. 8. It may be somewhat out of the usual course to describe the results of a research before any account has been given of the method by which these results have been obtained ; but in this case the foregoing sketch of the purely mechanical explanation of the physical evidence in the universe by the granular medium has seemed the only introduction possible, and even so it is not with any idea that this introduction can afford any pre- liminary insight as to the methods by which these results have been obtained. Certain steps, as it now appears, were taken for objects quite apart from any idea that they would be steps towards the mechanical solution of the problem of the universe. The first of these steps was taken with the object of finding a mechanical explanation of the sudden change in the rate of flow of the gas in the tube of a boiler when the velocity reached a certain limit perhaps this would be better described as a step towards a step*. The second step was the discovery of the thermal transpiration of gas together with the analytical proof of the dimensional properties of matter f. The third step was the discovery of the criterion of the two manners of motion of fluids}. * Manchester Lit. and Phil. Soc. 18745, p. 7. t Royal Soc. Phil. Trans. 1879. Royal Soc. Phil. Trans. 1883. 8 ON THE SUB-MECHANICS OF THE UNIVERSE. [8 And it was only on taking the fourth step, namely, the study of the action of sand, which revealed dilatancy as the ruling property of all granular media*, which directed attention to the possibility of a mechanical explanation of gravitation. In spite of the apparent possibility, all attempts to effect the necessary analysis failed at the time. There was however a fifth step ; the effecting of the analysis for viscous fluids, and the determination of the criterion!, which led to the recognition of the possibility of the analytical separation of the general motion of a fluid into mean varying motion, displacing momentum, and relative motion ; and this suggested the possibility that the medium of space might be granular, the grains being in relative motion and at the same time being subject to varying mean motion. And this has proved to be the case. At the same time it became evident that it was not to be attacked by any method short of the general equations of a conservative system starting from the very first principles; and it is from such study that this purely mechanical account of the physical evidence has been obtained. * Phil. Mag. 1885. t Boyal Soc. Phil. Trans. 1895, A. SECTION II. THE GENERAL EQUATIONS OF MOTION OF ANY ENTITY. 9. AXIOM I. Any change whatsoever in the quantity of any entity within a closed surface can only be effected in one or other of two distinct ways : (1) it may be effected by the production or destruction of the entity within the surface, or (2) by the passage of the entity across the surface. To express this general axiom in symbols I put ; Q for the quantity required to occupy unit volume, as an indefinitely small element of volume, 8S, at any point within the surface is occupied. Q is thus the density of the entity at the point, and however it may vary from point to point is a single valued function of the position of the point : S (QBS)= llQdxdydz is put for the quantity within a space $ enclosed by the surface s at the instant considered, S (oQBS) is the quantity enclosed at a previous instant. 2 (pQBS) is the quantity which has been produced within s during the interval, and 2 (cQBS) is the quantity which has crossed the surface inwards during the interval. Then 2 (Q&Sf) = 2 ( QBS) + 2 ( P QSS) + 2 ( C Q8S) is a complete expression for the Axiom. Using B [ ] to express any change effected in the time St this may be written 8R(gSS)]**[S(,a/8)]-fS[S< a Q&^3 ............... (1). And this equation (1) is the general equation of motion of any entity as founded on Axiom (I.). 10. General equation of Continuity. AXIOM II. When the entity considered is some particular form or mode of an entity which, like matter, momentum, or energy, can neither be 10 ON THE SUB-MECHANICS OF THE UNIVERSE. [11 produced or destroyed, any production or destruction of a particular form of the entity at a particular place and instant of time involves the destruction or production, at the same place and time, of an equal quantity of the same entity in some other form or mode. To express this in symbols let Q refer to the general entity without distinction of form or mode and Q 1} Q 2 > &c. respectively refer to the several particular forms or modes of the entity. Then since (2), which is a general expression for the law of conservation, and is the general equation of continuity in terms of the several distinct actions of exchange between the different modes of the entity. 11. Transformation of the Equations of Motion and continuity for a steady surface. Equations (1) and (2) hold however large or small the space S and the interval Bt may be and whatever may be the motion of the surface s enclosing the space S ; for the S covers the S ( ). If however the surface s be steady or fixed in space the S may be covered by the 2 ( ) and the equations written ............ (3), (4). Since these equations hold for indefinitely small spaces and indefinitely small intervals of time in the limit, when dx, dy, dz and dt are severally zero : (5), and 2[S(QSS)] = - t (Q)dtdxdydz ..................... (6). In cases where Q is not a continuous function of t the meaning of such differential coefficients as that in the right member of equation (6) become unintelligible without further definition, and it seems desirable here to point out, once for all, in what sense they are used in this paper. 12. Discontinuity. If Q is any function of xyz and t, which is single valued at every point of space at every instant, but which at a particular time t is discontinuous at a surface which is expressed by = (x, y, z, t)=0. 13] THE GENERAL EQUATIONS OF MOTION OF ANY ENTITY. 11 Where has positive values on one side of the surface and negative values on the other, then putting Qj for the continuously varying value of Q where is negative and Q 2 for Q where < is positive, Q is at all times expressed by the limiting value of the function when n is infinite*. For any finite value of n F is a continuous function of the variables, as are also the derivatives of F; and substituting F for Q, the limiting values, when n is infinite, of any functions derived from F by any mathematical process are taken as the values of the function expressed by the same mathe- matical process performed on Q"f"- 13. Having regard to the foregoing definition of the interpretation to be put upon the meaning of the differential coefficients in cases of discon- tinuity, the expressions obtained by equations (5) and (6) for the rates of convection into and production in such indefinitely small spaces may be treated as continuous functions of the coordinates. Thus taking u, v, w for the component velocities of the entity, to which Q refers, passing a point x, y, z, relative to the surface of the elementary space docdydz at rest or in steady motion, since u, v, w are single valued at each point at any instant of time the convection into the space in the interval dt is expressed by dt ^ (,Q) dxdydz =-dt \^ (uQ) + J- (vQ) + ^ (wQ)\ dxdydz \ v. O or at a point the rate of change by convection is y ...... (V), M> a. d M) , d ( w \ dx dy dz } * Electricity and Magnetism, Maxwell, 8. t Electricity and Magnetism, Maxwell 8. , dF dt dt _ . dF dQ, dF dO n From which, taking n infinite, when is negative = -~ , when

negative to positive over an interval St, indefi- nitely small, gives 12 ON THE SUB-MECHANICS OF THE UNIVERSE. [14 whence substituting in equations (1) and (2) for the indefinitely small element dxdydz and the indefinitely small interval of time dt, these become : dt ^ dxdydz = dt \~ t ( P Q) - (uQ) - ~ (vQ) - ( W Q)| dxdydz ...... (8), ^ ~r+(pQi)dxdydz= dt\-j.( p Q 2 + &c.)> dxdydz ......... (9), dt \jdt J or at a point the rate of change is (11). Equation (10) expresses the rate of change in the density Q at a point in terms of the densities of the actions of production and convection at that point. While equation (11) expresses the relation which holds between the densities of the several actions of exchange between the different modes of Q. 14. Moving Surface. i In the equations (5) to (11) the surfaces of the element of space (BS or dxdydz) are steady, and in equations (3) and (4) the closed surface over which the summation is taken is also steady the 8 being covered by the 2. If, however, the motion of every point of the surface be taken into account it is possible to sum the results of equations (7), (8), (9) over the space enclosed by a surface in any manner of continuous motion. Putting u, v, w for the component velocities of the surface at the point x, y, z, then the component motions of the entity represented by Q relative to the surface at this point are respectively u u, v v, w w, and although u, v, w are only defined at the surface, since the motion of this surface is continuous, u, v, w may be taken as continuous function of x, y, z throughout the enclosed space. Then the rate of convection across the surface is expressed by as K -">]+! K - + f [(w- w)Q]\ dxdydz ............ (12). CtZ ) 14] THE GENERAL EQUATIONS OF MOTION OF ANY ENTITY. 13 The instantaneous rate of production within the surface is not altered by the continuous motion of the surface. Therefore equation (1) becomes and integrating equation (10) over the surface, the rate of change in the space instantaneously enclosed as by a fixed surface is dt whence substituting in equation (13) for 4 s ! from equation (14), or as it may be written SECTION III. THE GENERAL EQUATIONS OF MOTION, IN A PURELY- MECHANICAL-MEDIUM, OF MASS, MOMENTUM AND ENERGY. 15. THESE equations are obtained by taking Q in equations (1) to (16) to refer successively to the density of mass, the density of the component, in a particular direction, of the momentum, and the density of the energy. The forms of the equations so obtained, as well as the circumstances to which they are applicable, depend on the definition given, respectively, to the three entities. If this definition is limited, strictly, to that afforded by the laws of motion as distinct from any physical or kinematical properties of matter, the equations will be the most general possible and applicable to all mechanical systems. In which case by introducing separately and step by step farther definition of the entities the effect of each such definition on the form of the equations and of the expressions for the resulting actions, to be obtained by integration of the equations, will be apparent ; so that the individual effects of the several particular physical properties of matter may be analysed. While on the other hand if the definition is, in the first instance, such as that on which the equations of motion for fluids and elastic solids have been founded the equations so obtained will be essentially the same. And, although the significance of the several expressions in the equations as relating to accu- mulation, convection and production will be more clearly brought out they will afford no opportunity of analysing the several effects resulting from particular physical definition. In this investigation the object sought, in the first instance, has been to render the equations the most general possible. Only introducing restrictive definition where the effect, of such definition, on the form of the expressions which enter into the equations and define the limiting circumstances to which the equations are applicable, becomes clearly defined. 16. A mechanical-system implies the existence, in the space occupied by the system, of an entity which possesses properties which distinguish the space so occupied from that which is unoccupied. If this entity includes everything that can occupy space, within the space occupied by the system, it is the mechanical-medium in which the system exists. 17] GENERAL EQUATIONS OF MOTION IN A PURELY-MECHANICAL-MEDIUM. 15 The sense in which mechanical-medium is here used is not that in which the term ' medium ' or ' medium of space ' is generally used in mechanical- philosophy, nor yet that for which "matter" is used. For although that which is recognised as matter is the only entity included in the equations of motion which has the property of occupying position in space, it is found necessary in order to account for experience to attribute to matter properties extending through spaces which are not occupied by matter, and to reconcile such extension with the absence of any mechanical properties as belonging to space itself it has been recognised that there exists in space some other entity, besides matter, which has the property of occupying position and is recognised in mechanical philsophy as the medium of space or the ether. To the ether are attributed such mechanical properties, whatsoever these may be, as are necessary to account for the observed properties of matter which are not defined by implication in the laws of motion, as well as to account for all the properties extending outside the space occupied by the matter. This amounts to an admission that these physical or extended properties are not inherent in the matter nor yet in the ether, or in other words that they are not the properties of the entity which occupies position in space, but are the consequence of the mechanical actions and of the arrangement of the mechanical system of the Universe. If then everything that occupies position in space is included by definition in the mechanical-medium, experience affords no reason for attributing to such medium inherent properties other than those required by the laws of motion and the law of conservation of energy, and so defined, the medium is here designated a Purely-Mechanical-Medium. 17. The properties of a purely -mechanical-medium necessitated by the laws of motion are (1) The property of occupying definite position in space ; (2) The continuity or continuance in space and time ; (3) The property of definite capacity for momentum, i.e. definite mass; (4) The property of receiving and communicating momentum in accordance with the laws of conservation of momentum and energy. Since the mass of any particular portion of the medium measures the quantity of that portion of the medium and has identically the same position in space as that portion of the medium, this mass is identified with the particular portion of the medium. The density of the mass at every point in space is thus a measure of the density of the medium at every point ; and the equations of motion and continuance in time and space of the mass are the equations of motion and continuance of the medium. 16 ON THE SUB-MECHANICS OF THE UNIVERSE. [18 18. The equations of continuity of mass. Putting pSS for the capacity for momentum or mass in the indefinitely small space SS and substituting p for Q in equation (2) the equation for conservation of mass becomes S[2( rf &Sf)]-0 ............................... (17); and by equations (1) and (17) the equation of motion of mass becomes (18). Whence for the indefinitely small element of space dxdydz and the inde- finitely small interval of time dt it follows by equations (7) that dp dpu dpv dpw _ , . ~j7 T -- 7 P T I -- j - v .......................... \ li '/> at dx ay dz which is the general equation for density of mass or medium at a point. 19. Position of mass. Taking x, y, z as defining the position of the indefinitely small steady space Bs, and putting px, py, pz successively for Q in equation (2), the equa- tions for the conservation of the position of the mass become respectively The equations for the rate of change of position of the mass within space over which the summation extends, become by equations (1) and (20) &}}, &c., &c ................... (21). Since x, y, z are not functions of the time, it follows by equation (19), if x, y, z define the position of the centre of gravity of the mass in the steady space over which the summation is taken, that f ([ JJJ _ ) ( & + H + !?) dxdydz '\dx dy^dz) y , &C., &C .......... (22). -TT r 7 j -, dt $j)p dxdydz For in a fixed space, Also S (pds) = -+ &c - dxdydz. 20] GENERAL EQUATIONS OF MOTION IN A PURELY-MECHANICAL-MEDIUM. 17 For a space moving with the mass by (15) y...(22A). whence since x is not a function of t, / ax ~ \ v / \ p o 2, i p -j os I = 2t (puos), &G., sue. \ l&v / 20. Before proceeding to the consideration of momentum and energy it will be found convenient to express certain general mathematical relations betiueen the various expressions which enter into the equations for quantities into which p enters as a linear factor. When Q is put for pq, where q is a factor which has only one value at each instant for each point in mass, but which value for the point in mass is a function of the time, then the derivatives of discontinuous functions having the meaning ascribed in Art. 12, dt And since by equation (17) dt dt .(23). dt Also dt dQ _ dp dq dt ~ q dt +p dt ^ P .(24). dt and d dx whence subtracting and having regard to equation (19) .(25); dt dt (dt dx ) therefore by equation (8) } (26). i d( p Q) _ (dq , u dq dt (dt dx Again, if Q = pq = pq^ and Q 1 = pq l} Q 2 = pq z , by equations (26), d( p Q)_ (dqty dq^z ) \ p ^ J4 + u j^ + \ d( p p") t dt dt dt dQ_d ( C Q) _ dt ~W~~ From which it appears Vd 2 +u d* + &c *St das dq,q z l ' ' .(27 A). dQ d( c Q) _ dt dt d(M = dt dQ, dt dt dt J + dt dt + dt dt .(28 A). 23] GENERAL EQUATIONS OF MOTION IN A PURELY-MECHANICAL-MEDIUM. 19 22. Momentum. The definition of momentum afforded or required by the laws of motion is, that the momentum in any particular direction is the product of the mass multiplied by the rate of displacement, in the particular direction, of the mass in which it resides. Since at each instant mass has position and capacity for momentum, and the rate of the displacement at the instant has magnitude and direction, momentum has position, magnitude, and direction. Taking ns before u, v, iv to represent the component velocities of the mass passing a point at any instant, and p for the density of the mass at the same instant, the densities of the respective components of momentum are respectively M x = pu, M y = pv, M z = pw. Substituting M x for Q in equation (1) it becomes S[2(M X SS)] = S[2( P M X SS)] + 2[( C M X 8S)], &c., &c ....... (29). By equation (2) substituting P M X for P Q 1 , ,&Sf + &c.)], .fee., &c ............. (30), where 8 [2, ( P Q 2 BS + &c.)] expresses the rate of destruction of momentum in direction x, in all other modes than that represented by M X SS within the space of S. 23. Conduction of momentum by the mechanical medium. As 2 (M X 88) represents the sum of all the momentum in direction x within the space 8, there is difficulty in realising how momentum in direction a; can be produced or destroyed in any other mode. If, as in this research, p&S is defined as including the total capacity for momentum within the in- definitely small space, 8$, the production or destruction of momentum in direction x in any other mode than M X 8S, at a point within the space SS, requires that momentum should have entered the space without having been conveyed by the motion of the mass across the surrounding space. The difficulty thus presented naturally raises the question as to whether such production or destruction is necessarily implied in the laws of motion ? as to whether the entire exchanges of momentum cannot be accounted for as the result of convections by the moving mass ? That it is possible for momentum to be conveyed across a finite space by the mass within the space, and at the same time the momentum of the mass within the space to be zero, has long been recognised, and follows directly as a geometrical consequence of the fact that momentum possesses the property of being negative in exactly equal degree with that of being positive ; just as does electricity ; so that a stream of negative momentum in any direction, 22 20 ON THE SUB-MECHANICS OF THE UNIVERSE. [23 crossing a surface in a negative direction, has exactly the same geometrical significance as an equal stream of positive momentum crossing the same surface in a positive direction. The result being the convection by both streams of positive momentum in the positive direction and negative momentum in the negative direction at equal rates, while the sum of the momenta of the masses in the two streams taken together within the space is zero. In such streams of momentum the action at a surface is, though purely kinematical, that of exchange of momentum between the spaces on the opposite sides of the surface, such exchange proceeding at a definite rate, which rate has a definite intensity at each point of the surface, and the direction of the momentum exchanged is the direction of the motion of the mass at each point. The condition that action and reaction are equal and opposite is thus completely satisfied that is to say, not only is the action one of exchange of momentum, but it is also one of exchange of moment of momentum about every axis. Hence, where the boundary conditions of the medium admit of such opposite streams of momentum in different directions through the same space in the same interval of time, exchanges of momentum in any direction across any surface may be effected while the aggregate momentum is zero. In this way, in the kinetic theory, the stresses in gases at any instant are completely accounted for, as the result of the convection of momentum conveyed by the molecules amongst which the motion is distributed uni- formly in all directions. But even in the case of gas such convection does not account for the maintenance of the distribution of velocities amongst the molecules. This requires that the molecules should exchange momentum, and such exchange as appears by equation (13) cannot be accounted for as the result of kinetic convection by moving mass, but requires mechanical action between the molecules. In the kinetic theory, therefore, it is assumed that ' forces ' exist between the molecules, when within certain distances of each other, either as the result of varying stresses in the matter, or as exerted through intervening space. From these and like considerations it appears that, to whatever extent the transmission of momentum from one portion of space to another may be accounted for as tlie result of convection by moving mass, the communication of momentum from one portion of mass to another requires either that it be transmitted through space occupied by mass otherwise than as moving mass, or that it be destroyed in one place and produced in another. Unless, therefore, it is assumed that, while mass has continuous existence in time and space, momentum can cease to exist in one place and, at the same time, come into existence, in the same quantity, at another place, that is 23] GENERAL EQUATIONS OF MOTION IN A PURELY-MECHANICAL-MEDIUM. 21 unless we accept action at a distance, and thereby preclude all further definition and explanation, it is necessary that the purely-mechanical- medium, in addition to the properties of occupying position, and having capacity for momentum, should have the property of transmitting or con- ducting momentum through the space it occupies otherwise than by the convection consequent upon the motion of the mass ; and, to completely satisfy the condition that the direction in which the exchange is effected is the direction of the momentum exchanged, it is necessary that the direction of conduction should everywhere be the same as, or the opposite to, that of the momentum conducted that the conduction should be by streams, real or imaginary streams, of real or imaginary momentum in the same direction as that of the momentum, just as in the case of convection, except that in the latter case the streams and the momentum are real; so that if I, m, n refer to the direction in which h is measured, which is that of such a stream, of which p is the intensity, positive or negative, of the rate of exchange across a surface normal to h, the intensities of the rates of exchange of momentum, in direction h, across the surfaces yz, zx, xy are respectively pi, pm, pn, and the intensities of the rates of exchange of the components of momentum, in the direction of x, y, z, respectively, are across yz pi 2 , plm, pin, zx pml, pm"*, pmn, xy pnl, pnm, pn 2 . This property of conducting momentum (on which all mechanical action depends), necessitated by the laws of motion as inherent in a purely- mechanical-medium, must be continuous in time and space if the medium is continuous in time and space. As possessed by the medium, therefore, the property differs from the property of strength or that of resisting stress possessed in various degrees by matter in respect to the limits to the strength, which limits depend on the physical condition of the matter and have no existence in the medium. This difference as regards limits, however, does not affect the correspondence, in character, between the property of conduction of momentum by the medium and the property of sustaining stress in matter. The magnitude of stress being nothing more nor less than a measure of the intensity of the flux of the component of momentum, in the direction of the stress across the surface on which the stress acts, if the intensity of stress at a point on a surface is defined to be the intensity of the flux of momentum conducted, as distinct from that conveyed by the motion of the mass across the surface, the notation used for the expression of the stresses in matter becomes applicable for the expression of the components of 22 ON THE SUB-MECHANICS OF THE UNIVERSE. [24 momentum conducted, as distinct from that conveyed, in a purely-mechanical- medium. Thus PXX> Pyx> PZX> Pxyt Pyy> Pzy> PXZ> Pyz> PZZ) the expressions, used by Rankine for the component intensities of the stress, in which the exchange of momentum is in the direction indicated by the second suffix and is across the surface perpendicular to the direction indicated by the first suffix, may be defined to express the intensities of the rates of conduction of the components of momentum in which the momentum is in the direction indicated by the second suffix and is conducted in the direction indicated by the first suffix. Whence, at any instant, the rates of conduction of the component of momentum from the outside into the indefinitely small steady element dxdydz are respectively expressed by the left members of the equations (30 A), -{^ + -df + %r} ******** dp xy y_ = F y dxdydz doc dy dz } .(30 A), >o* dp yz dp zz \ ,77 + --.,-- -f -^ } dxdydz = . /> fiii fi v \ \AJ \J(j II \J(j4/ I F x , F y , F z being merely contractions for the expressions in the left member. 24. Since, in order to satisfy the condition that action and reaction are equal, accumulation of momentum in the mode in which it is conducted is impossible, the expressions for the rate of conduction into the mass in the space dxdydz must also express the rates at which momentum in the mode in which it is conducted, is produced in the mass in the .space outside the element and destroyed within the element. Whence it follows that F x , &c., respectively represent the rates at which the densities of the respective components of momentum, in other mode than that of M x , &c., are destroyed within the element, and as these are the only rates at which momentum within the element is destroyed F x , &c. define the values of ( p Q. 2 + &c.) in equations (30), and the equations of continuity of the densities of the respective components of momentum in a purely-mechanical-medium be- come by equation (11) *=-?- = F x , &c., &c (31), and substituting in equations (29) we have by equation (10) d Jj^ = F x +^( c M x ), &c., &c (32), which are the equations of density of momentum in a purely-mechanical- 25] GENERAL EQUATIONS OF MOTION IN A PURELY-MECHANICAL-MEDIUM. 23 medium expressed in terms of general symbols expressing the separate effects of the distinct actions of conduction and convection. Substituting for F x equations (30 A) and d ( c M x )ldt from (7) we have the full detailed expressions for the equations of the densities of the com- ponents of momentum at a point dM x ( d d d j = i ~j~ \P%x ~^~ P'My) ~l~ ~r~ (Pyx ~^~ pii'v) ~f~ ~r (PZX at \dx dy dz The equations (32) and (33) are the equations of conservation of mo- mentum in a purely-mechanical-medium, at a point, iu which the first terms in the brackets on the right of (33) express the rates of change by con- duction, and the second the rates of change by convection. The integrals of the right members of these equations transform into surface integrals, and thus they express the condition that the change of momentum within any space S is solely the result of the passage of momentum across the surface of 8. 25. The conservation of the position of momentum. It appears from the previous article that the condition of conservation of momentum requires that action and reaction should be equal and opposite, but this is all ; so far p xx , p yx , &c. may be independent of each other, and there is no indication that exchange must take place in the direction of the momentum exchanged. This is however expressed by the equations of conservation of the position of momentum. Taking x, y, z and pu, &c. as referring to a fixed point. Then multiplying each of the equations (33) by x, y, z, successively, we have \ j~ (Pxx + P uu ) + &c. [, &c., &c (34), \Ci*Xs \ ' dt or transforming, since x, y, z are not functions of t, -j- (xpu) p xx puu = \ -j- x ( p xx + puu) + &c. [ -^- (ypu) p ux puv = \ -j- y ( p xx + puu) 4- &c. 1- 1 (35), jj -f \*7 1 / JL a* I J /V/y 7 \ JT *"*- t ' , C V V ( \JuJif -jjr (zpu) - p. x - puw = - \~r- z (p xx + puu) + &c. I at (ax ) and corresponding equations, for xpv, &c. and xpw, &c. The right members of these equations integrated over any space S repre- sent surface integrals. The integrals of p xx , &c. on the left of the equations represent the respective rates of the displacement by conduction of the respective com- 24 ON THE SUB-MECHANICS OF THE UNIVERSE. [26 ponents of momentum within 8, while those of puu, &c. represent the rates of displacement of momentum by connection within 8. Hence what these equations express is that the whole rate of displace- ment of momentum in S, less the internal rate of displacement, is equal to the rate of displacement of the momentum across the surface. This, it appears, follows directly from the condition that action and reaction are equal i.e. the equations of motion and implies no relation between the components of conduction. Such conditions however follow from the further condition that the direction of exchange is the direction of the momentum exchanged. 26. Conservation of moments of momentum. Subtracting equation (35) for ypw from that for ypv, - ypw) - (Pzy - \Tx whence in order that the rate of change in the moment of momentum about the axis of x may be expressed by a surface integral we have the condition, as previously obtained (Art. 23), Pzv=Pvz, and similarly, i,h&tp xz = p zx &ndp yx =p xy ............ (36A). 27. Boundary Surfaces. The conditions at the bounding surfaces of spaces continuously occupied by the medium may be of two kinds, according to whether the surface divides the medium from unoccupied space, or separates two continuous portions of the medium which are in contact at the surface. Taking r, s, t for distances measured from a point in the surface in direc- tions at right angles to each other, that in which r is measured being normal to the surface and l r , m r , n r , l s , m s , n g , l t , m t , n t for the direction cosines of r, s, t respectively, then since Pxy = p y x, &c., &c., i r n r + 2p zx n r l r + 2p xy l r m r + 2p zx n s l s + 2p xy l s m s Ptt p*Jj/3/w s m t -f p zz n s n t -f p yz (m s n t + n s m t ) +PKC (n g lt + l s nt) + Pxy (ls> n t + mk) ptr =pMr + p yy m t m r +p zz n t n r +p yz (m t n r + n t m r ) + Pzx (ntlr + It^r} +Pxy Prs =pxJ>rl>s + p yy m r m g + p zz n r n s +p yz (m r n s + n r m s ) 28] GENERAL EQUATIONS OF MOTION IN A PUR ELY -MECHANICAL-MEDIUM. 25 Where the surface separates the medium from unoccupied space the stresses p^ &c., are all zero at the surface, but where the surface divides two portions of the medium in contact, then the intensity of the flux across the surface at a point is the intensity of the rate at which such momentum is received by the one portion and lost by the other across the surface at the point, and by the foregoing notation p^, p rg , p rt respectively express the intensities of the rates of flux across the surface of the components of momentum in the direction in which r, s, t are respectively measured. These rates are the limiting values at the surface of the respective com- ponents of flux within the medium on either side of the surface in the directions in which r, s, t are measured, and are thus the limiting values, at the surface, of the expressions on the right side of the equations (1). 28. Energy. Although the half of the vis-viva (that is half the rate of the displace- ment of the momentum, or half the product of the momentum multiplied by the rate of displacement of the mass) now called kinetic energy, has long been recognised as the general measure of the mechanical-effect of mechani- cal-action through space, the recognition of energy as a physical entity has resulted from the discovery of the reversibility of actions by which mechanical-action produces physical effects, and of the linear relations which exist between the physical measures of the physical effects so produced, and the kinetic energy which has been expended in producing them. The discovery of these relations and the reversibility of the actions having led to the recognition of the existence in the Universe of physical entities which could be changed to and from the mechanical entity kinetic- energy, these physical entities, although not otherwise mechanically definable, have become recognised as modes of the general physical entity of which kinetic-energy is one mode and the only mode which is subject to strict mechanical definition ; and hence followed the recognition of the law of con- servation of energy. Taking p xx , &c. to have the significance ascribed to them in Art. 23, the intensities of the components of mechanical action that is the intensities of the components of the flux of momentum, by conduction, from the negative to the positive side across a surface of which the direction of the normal is defined by I, m, n are respectively expressed by Pxxl + PyxM + PzxK, &C., &C. These are the expressions for the time-measures of the intensities of the components of mechanical action, in the directions of the perpendicular axes of reference, of the mass on the negative side of the surface, on the mass on the positive side of the surface, at a point in the surface. 26 ON THE SUB-MECHANICS OF THE UNIVERSE. [28 Multiplying these time-measures respectively by u, v, w, the component velocities of the mass at the point, we obtain u (p xx l + p yx m + p zx n), &c., &c., which are the corresponding space-measures of the respective components of the intensity of mechanical action at the point. Adding these and multiplying by Ss, the element of a closed surface, the integral over the surface is expressed by JJ [(up u + wp xz ) I + (up yx + vp yy + wp yz ) m + (up zx + vp zy + wp zz ) n] which is the space-measure of the mechanical action of the mass outside the closed surface on that within. This (if there are no purely physical exchanges) is by the law of conser- vation of energy equal to the rate of change of energy in all its modes, within the surface that is if there is no change by convection across the surface, Avhich will be the case if the surface is everywhere moving with the mass. The changes of energy may be partly in kinetic-energy and partly in other physical modes, according to the expression which is obtained by transforming the equations of momentum (33) by equation (26) ; multiplying respectively by u, v, w, integrating over the surface and adding, the equation becomes, when transformed by equation (15), taking U = u, &c., and assuming the actions continuous in space and time, f 1 ri C C C -r llnp (u? + v" + w 2 )} dxdydz A dt J J J du du du Pxx dx +J ^UX 1 i Pzx ~~7 * dy dz fff dv dv dv ~]Jr + P **dx + Pvv Ty +Pzy dz dw dw dw ( " dx J V yz ~dy' + Pzz dz l > dxdydz + vpxy + wp xz ) I + vpw + wp yz ) m\$S... (38). + vp zy + wp zz ) n The right member is here the measure of mechanical action over the surface moving with the mass ; so that the left member expresses the rate of change of energy, resulting from the mechanical action within the surface. The first term in the left member is the rate of change in kinetic energy, within the surface, and the second term expresses the rate of change of 30] GENERAL EQUATIONS OF MOTION IN A PURELY-MECHANICAL-MEDIUM. 27 energy in other or physical modes within the surface as resulting from the mechanical action on the surface. 29. In a purely-mechanical-medium (including everything that has position in space and possessing no physical properties other tlian are required by the laws of motion) the kinetic-energy must include all the energy in the space over which the integration extends, hence as applied to such medium the second term on the left of equation (38) must be zero, however large or small the space over which the integration extends. Whence putting < 2E = p(u* + v 2 + w' 2 ) and transforming equation (38) by equation (15), the equation of energy for a fixed space becomes rd f ~\ r r -^- &S = 1 1 [(up + vpxy + wpxz + uE) I + (up y x + vp yy + wp yz + vE) m + (up^ + vp zy + wp zz + wE) n} dS . . . (39). Whence since this holds whatsoever may be the size of the space en- closed, we have for the rate of change of the density of energy at a point, by differentiating the left member of equation (3.9) with respect to the limits dE d , . d , . d , ~dt = ''~dx ^ Upxx + Vpxy + Wpxz ' ~ dy - Pyx + Vpyy Wpyz ' ~ dz d(uE) d(vE) d(wE) dx dy dz .(40). 30. In order to simplify the expressions N may be put for the rate at which density of the energy, in whatsoever mode, is produced by the mechanical action at any fixed point in space, and N x , N y , N z for the densities of the energies which have been produced by the components in the directions in which x, y, z are measured respectively, so that Then Whence substituting in equation (40) it becomes dE dN d , 28 ON THE SUB-MECHANICS OF THE UNIVERSE. [31 which may be obtained from (1) and (2) together with the condition that E is continuous and is the equation for the density of energy in terms of genera] symbols expressing the densities of the distinct actions of conduction and convection at a point. 31. The condition of a purely-mechanical-medium. Equations (40) and (43) are the equations of continuity of energy in a purely-mechanical-medium in which the relation between the stresses and strains is continuously, that the second term in the left member of equation (38) is everywhere and continuously zero. Transposing the expression under the integral in the second term in the left member of equation (38) by (36A) and equating to zero we have du dv dw /dv dw\ /dw du Then, for convenience, expressing equation (44) as dR/dt = 0, equation (44) defines the action in the medium as being purely kinematical. From the definition of p xx , &c., &c. as components of intensity of a flux of momentum it follows geometrically that the value of the expression which forms the left member of equation (44) is independent of the direction in which the axes are taken. Hence, if i, j, k, are measured in the directions of the principal axes either of the rates of distortion or of the stresses at a point p and u, v, w are the components of the velocity in these directions, respectively, transforming to these axes we have by equation (44); since either ; ^ + ^ = 0, &c., &c.; or p jk = 0, &c., &c .............. (45), Ctfo CLj du dv dw From these three conditions it appears that no energy is transformed in distorting the medium. And we have as the three possible conditions in a purely-mechanical-medium Pn = Pjj = Pkk = ^\ which is the condition of empty space (40 A), , du dv dw . a ., pa = PJJ =pkk > and -T- + -j- + -j- = ; perfect fluid. di dj djf du dv dw dw dv du dw dv du . . ..^ -j^ + -r + j- = ; or ^- + j- = j- + T-=j--fj- = 0; perfect rigidity. di dj die dy dz dz dx dx dy 32. The transformations of the directions of the energy, and angular redistribution. = " JJJ \dx 32] GENERAL EQUATIONS OF MOTION IN A PURELY-MECHANICAL-MEDIUM. 29 Kinetic energy has direction at every point, although not a vector, and the equations obtained by multiplying equations (33), respectively, by u, v, w are, respectively, the equations of energy in the directions of x, y, z. For an element in a closed surface within the mass Tx + * Ty + V + dy ( pyxU ^ + dz &c., &c. In these equations the members on the right represent work, in the directions x, y, z, respectively, done on the surface within which the in- tegration extends. And as these efforts are all in the direction of x, y or z, respectively, they involve no change from one direction to another. But the second terms on the left of each of the equations represent production of energy in the directions x, y, z respectively, at the expense of the energy in the other directions. It is thus shown by condition (44) which is that the sum of these terms, from the three equations, is zero that, putting R x , &c., &c. for the densities of the rates of angular dispersions at a point, from the directions a?, y, z respectively, these are dR x ( du du du\ du\ Tz) ' &C " It is to be noticed that in a medium such that u, v, w do not represent the velocities of points in mass, R x does not represent angular dispersion only, unless equations (44) are satisfied ; and if not so satisfied dR x /dt would represent the work done against the apparently physical actions in the medium, as well as the angular dispersion. The analytical separation of this action is obtained by transforming the general equation, which becomes dR 1 , /' du dv dw du dv\ /du dw P -(dz--dx fdu dv dw 30 ON THE SUB-MECHANICS OF THE UNIVERSE. [33 From the member on the right of equation (47) it at once appears that the two first terms express angular dispersion only, while the second two terms express distortional motions only, which, by the conditions (45), are zero. 33. The continuity of the position of energy. Kinetic energy has position ; and hence, putting x, y, z for the point at which the density of energy is E, by equation (1) S [2 {JMSf}] = 8 &{ c (Ex)SS}] + S &{ p (Ex) SS}], &c, &c. ...(48), in which x, y, z are not functions of time. And if x, y, z are put for the centre of energy, u, v, w for the component velocities of the surface, as in equations (12) to (16), Art. 14, we have at any instant, x2{E8S} = 2{Ex&S], &c., &c ...................... (49), whence, differentiating with respect to time, ~ 2 (ESS}=-x[2{ESS}'] + [2{EasSS}], &c., &c ....... (50). Then, by equation (15), these equations become f 2 (08) = - at L(dt dx dy dz 2 + > d(Exv) d (Easwy dt dx dy dz - S - dx dy dz + Z(EuSS),&c.,&c ........................................ (51). Whence, for a fixed surface, since u = v = w = 0, dx ,<-, (, _.dE sc<\ 2 \(ae - x) -j- SS\ dt } ^7 V/F^OX ' -> ................... - dt 2 (ESS) For a surface moving everywhere with the mass so that u=u, &c., equation (51) becomes , 2 {(* - x)l ( P E) SS\ + 2 {EuSS}, &c., &c. ^_ _1 _ _ ) _ /K0\ dt 2 {ESS} or, [i{(Jfo)&8}] = 2 ar (,^) Sflf + S(tfte&Sf) ......... (54), where, as in equation (42), differentiating with respect to the limits ^- ( P E) = - |^- (^^M + j^y + p^w) + &c. + &c. . . . J ...... (55), dN = ~- 34] GENERAL EQUATIONS OF MOTION IN A PURELY-MECHANICAL-MEDIUM. 31 34. Discontinuity in the medium. It is to be noticed that the expressions in equations (37) to (55) are adapted to the cases in which the medium is continuous, so that for the complete expression of the actions where the medium is continuous within closed surfaces, only, it is necessary to express the conditions at the bounding surfaces by using the expressions in equations (37). These complete expressions might very properly be introduced at this stage. But as the necessity for the definite use of these does not arise until a much later stage in this research, and then arises in a comparatively simple case which has already been much studied in some of its aspects, it is convenient to proceed as if the medium were continuous until this stage is reached. See equation (132), Section IX. SECTION IV. THE EQUATIONS OF CONTINUITY FOE, COMPONENT SYSTEMS OF MOTION. 35. Component systems may be distinguished by definition of their com- ponent velocities or their density. By a component system of motion distinguished by velocity is here understood a system of motion, howsoever defined, in which the velocity at any point is not necessarily the velocity of the mass at that point either in direction or magnitude. Taking, as before, u, v, w, to express the components of the actual velocities of the mass at the point ac, y, z and time t, and p for the density of the mass, and u", v", w" as expressing the components, with respect to the same axes, of the velocity of a component system, there exist at each point the residual components u'=u u", v' = v v", w' = w w" (56). The sums of these components u" + u, &c. satisfy the equations (33) Section III., and the following equation, for the resultant system, and if one of these systems is subject to any definition, actual or conditional, the equation for the resultant system becomes the equation for the residual system. It is a very general method in mechanical analysis to separate the motion of the mass at each point into two component systems, whenever the condi- tions are such that the independence of these systems is obvious. As, for instance, the motion of the mass at each point at any instant is considered as consisting of the motion of the centre of gravity of the whole mass at the instant together with another component system which is the motion at the point relative to the motion of the centre of gravity. But such instances have hitherto been considered as depending on special theorems, and do not appear to have suggested the study of the method which they involve as a general system of analysis apart from the existence of conditions which render the component systems completely independent, 35] THE EQUATIONS OF CONTINUITY FOR COMPONENT SYSTEMS OF MOTION. 33 It appears, however, that the manner in which the rates of increase of the momentum and kinetic energy of the one component system depend on convection by and transformations from the other may be subjected to general analytical expression, even when the definition is arbitrary and only conditional. This is accomplished by equating the expressions for the rates of increase of u" ', &c. at a point moving with the mass to arbitrary functions which, multiplied by p, express the rates at which density of momentum is trans- formed from the system pu' into the system pu" and represent the only rates of production of momentum in that system, so that the equations of motion of either of the component systems may then be obtained from equations (1) and (2) or (10) and (11) Section II. The equations so obtained will differ in form from the equations of the resultant system in five particulars. (1) The equations for the component system will differ from that of the resultant system from the fact that u", v", w" do not represent the whole causes of convection, which are u, v, w: so that the rate of increase of Q by convection is not d , lir .. d , . d , ,,~J , (duQ dvQ dwQ] sW>-|0" + 5&ftc ................... (57) - where the pre-suffix c" indicates convections by u" and c indicates the con- vections by u', inwards across the bounding surface of the element. (2) A difference in the form of the equations also results from the fact that pu", pv", pw" are not the only modes in which densities of momentum in the directions x, y, z exist at a point in the medium. The rates of increase of density in the modes pu", &c. by conduction, into the steady element of space dxdydz are not the only increases other than by convection ; since there are the further possibilities of exchanges of densities of momentum between the modes pu", and pu', &c. existing at the same point in the same mass. That such abstract exchanges, without mechanical action, must result from the definition by which the component systems are distinguished is at once seen, for to this definition u", v", w" are subject at each point and each instant. And therefore the rates of increase of u", v", w", the defined com- ponents of acceleration of the moving mass, expressed by du" du" du" du" -rr + U -j + V -j + W -j- , &C. &C. dt dx dy dz are subject to arbitrary definition independent of the actual accelerations of the mass. And dp idem dpv dpw\ _ , (dp\ , , fdpu \ A w"- + w' (4-+ J- + HJ ) = u (- } +u J +&c. =0. dt \ das dy dz J \dtj \dx J R. 3 34 ON THE SUB-MECHANICS OF THE UNIVERSE. [35 Taking ^/, &c., &c. as arbitrary expressions for these defined rates of increase and multiplying by p we have as the equations of continuity for the components of momentum pu", &c., &c. by equation (28) Section III. = ""> + ^">> &C " &C ................ (58) ' and again by the equations for the resultant system j t (pu" + pu') = j t ( c pu" + c pu)+F x , Subtracting equation (58) we have for the other system (u" + pu') = ( c pu" + c pu)+F x , &c., &c ............. (59). (60). It thus appears that p -fa (X0 &c -> &c -> express rates of transformation of density of momentum from the component system pu' to the system pu" , &c., &c., consequent on the geometrical conditions by which u", v", w" are defined. The arbitrary rates of increase of density of momentum represented by these transformations may be considered as variations either in an arbitrary system of stresses or an arbitrary system of convections to be determined by the actual definition. (3) The equations of the component systems differ from that of the resultant system on account of the expression for the transformation of energy to and from each of the component systems in consequence of the definition to which they are subjected. The densities of each of these rates of transformation of energy are by equation (28), putting u" for q lt &c. respectively, the sums of the products of the densities of the component ratios of transformation of momentum to the particular component systems (dppu'/dt, &c.) respectively multiplied by the component velocity (u", &c.) of the same system. Thus expressing the density of energy so transformed at a point as p T (E"), &c., respectively, since there is no transformation of mass, 35] THE EQUATIONS OF CONTINUITY FOR COMPONENT SYSTEMS OF MOTION. 35 From these equations it will be seen, at once, that the sum of the trans- formations to the two component systems is not necessarily zero or that the transformation is not wholly between E" and E'. (4) The equations of energy for the component systems differ in form from that of the resultant system in consequence of the fact that the sum of the densities of the energies of the component systems, at a point, is not equal to the density of energy of the resultant system at that point, or that : v- {u" 2 + v" 2 + w" 2 + u 2 + v' 2 + w' 2 + 2 (w'V + v"v + w"w')} whence p (E E" E') = p (u'u 1 + v"v' + w"w') Whence it appears that the transformation of energy is not simply between the systems E" and E', but also between each of these and the system (u"u' + &c.); so that besides the equations of energy of the component systems there is the equation of energy of the residual system to be considered. The density of the rate of transformation to the residual system is by definition equal in value and opposite in sign to the sum of the rates of transformation to the energies of the component systems Another expression for the transformation to the residual system is obtained by multiplying each of the rates of transformation of component of momentum to the component system, by the corresponding component of velocity of the other system and adding, as in equations (28). The density of the rate of production into residual energy may be obtained in the same way by equation (28); then by equations (10) we obtain expressions for 4 ("'-*') and | ( (,V'). (5) In the equation of motion for the resultant system of motion in a purely-mechanical-medium, d (R)/dt, the density of the rate at which energy is produced in other modes than E, is defined as zero; and hence the expression for this production disappears from the equation of energy. It does not however follow as a geometrical consequence that the expressions for d (R')jdt and d(R R')/dt, obtained from the equations of momentum by equation (28), are respectively zero. But it does follow that whatever these 32 36 ON THE SUB-MECHANICS OF THE UNIVERSE. [36 values may be, they are pure abstractions resulting from the definition of the systems of motion, and are therefore transferences of such energy from the one system to the other. Therefore while it is necessary to retain these expressions in the equations of energy for the three systems, it is convenient to indicate that they express a transference by a pre-suffix T as d( T R')/dt. 36. -Component systems distinguished by distribution of mass. Taking, as before, p for the density of the mass at xyzt and p" for any defined density of mass at the same point, there exists the residual mass P=P-P" (63). The sum p" + p' satisfies equations (33) Section III. for the resultant system, also equations (58) and (60), Section IV., for the component systems distinguished by the distribution of velocity, and if p" is subjected to any definition, actual or conditional, the equation for the resultant density defines the equation for residual density of mass. The equations so obtained will differ in form from the equations for the resultant mass in one particular. The fact that the integrals of p" and p' do not, either of them, taken by themselves, represent the only mass included in the space over which the integrals extend, entails a difference in the form of the equations from that of the resultant system. The rate of increase by convection of p" is not necessarily the only rate of increase, since there are possibilities of exchanges between the densities p' and p" at the same point. That such exchanges must result from the definition is at once seen, for dp'Jdt is subject to these exchanges at each point at each instant, and there- fore the defined rate of increase of the component density p" at a point moving with the mass is subject to arbitrary definition independent of the rate of increase of the actual density. Taking as in equations (24 A) Section III. d T p" = 37] THE EQUATIONS OF CONTINUITY FOR COMPONENT SYSTEMS OF MOTION. 37 And by the equation for the resultant system dt dt ' dp d e p'^ d T p" dt dt dt Then, since by equation (24), d p (pu)/dt = pd p u/dt, substituting in equation (32), the equation at a point for the resultant system is du du du du d p u . -f~ + u -= I- v-j h w -7- = j (oo). dt dx dy dz dt Then multiplying by p" and adding u -~ u ~~- to the left member and the equivalent ud p p" jdt to the right member, we have for the equation of momentum of the defined density : dp'u d c (p"u) __ dpU d T p"\ dt dt dt dt / HTX j / // \ f ( 67 )' P dt } and in precisely the same manner dp'u d c p'u __ , dp (u) d T p' dt dt dt dt , ^ dt 37. Component systems of motion distinguished by density and velocity. Again substituting u" and u successively for u in equations (67) and (68) we have the four equations dp"u" d c (p"u") _ dp (p"u") = ,,d T u" d^p" dt dt dt dt dt > (69), dp'u" d c (p'u"} _ d p (p'u"} , d T u" d ' P " ^~. T". 7": t . i~ U ^ . dt dt dt dt dt dp"u d c (p"u') _ d p (p"u) d T u" , d c 'p" p"F x - dt dt dt ~ p dt dt p" dp'u' d c (p'u'} _ d p ( P 'u') _ , d T u" , drf p^_F x - dt dt dt p dt dt p" together with corresponding equations for v", w", v', w. Adding the last three of equations (69) together, it appears that d (pu - p"u") d c (pu - p"u") = d p (pu - p"u") dt dt dt d T p"u" dt (70), 38 ON THE SUB-MECHANICS OF THE UNIVERSE. [38 whence putting M x " for p"u", M x ' for pu p"u" , &c., &c., we have dM x " d c M x " ^dpM*" = ,,d T u" d T p" dt dt dt P dt ' dt dM x ' d c M x d p M x ' _ d T u" ~dt dt ~dT x dt It is to be noticed, however, that these last equations might be obtained by the simple definition of (pu)", so that they do not express all the definition which results from the separate definition of p", u". The importance of this appears at once on proceeding to derive the corresponding equations of energy by multiplying the equations respectively by u" and u, and trans- forming, which process since u", v" have defined values, gives definite results, whereas the mere definition of the product (pu}" which leaves the definition of either factor incomplete would not admit of such derivation. 38. Distribution of momentum in a component system. The condition imposed by the laws of motion, as the result of experience of physical actions, that action and reaction are equal and opposite, and that the exchanges of momentum take place in the direction of the momentum exchanged, will not of necessity be fulfilled by an arbitrarily defined component system. But should this not be so within all sensible spaces and times, the effects of one component system on the other will not accord with any physical action ; so that for purposes of analysis the general expression for this condition in a component system is of the first im- portance. It has already been shown that the first of the conditions requires that the integral rate of increase in each component of momentum, in a resultant system, shall be a surface integral, however small may be the limits (Section III, Art. 24). The same holds for a component system within defined limits ; so that we must have, within such limits, + -j- (pM x ")\ dxdydzdt Cut J o &c - where so far q xx , q yx , &c. are arbitrary. As in a resultant system it is necessary, in order to satisfy the second condition, that the integrals of the rates of increase of the moments of momentum should be surface integrals and that this may be the case within defined limits, it follows, as in Art. 26, that (q zy q yz ) dxdydzdt = 0, &c., &c .................. (73), of 39] THE EQUATIONS OF CONTINUITY FOR COMPONENT SYSTEMS OF MOTION. 39 which is the general condition to be satisfied by the component system pu", &c. if the analysis is confined to physical properties. If this condition is satisfied by the system p"u", &c. it follows that since it is satisfied in the resultant system the same condition will be satisfied by the residual system pu p"u". 39. The component equations of energy of the component systems as distinguished by density and velocity. Multiplying the first of equations (69) by u" and transforming by equations (28 A), Section III., and putting p"E x " for p" (ii") 2 /2, we have d(p"E x '} d c (p"E x ') _ dp (p"E x ') _ ,,d T u" , u" 2 d T p" , s Jj. ~~ "" J.t ~J ^ P Jj. ~* ~n ~7* "^ Ofc'C. at at dt at 2 at Also multiplying the third of equations (69) by u' and trans- forming (28 A) we have dp E x d c (p E%) dp {p EX) dt dt dt p dt J ' dt dt Then multiplying the first by u' and the third by u" and adding, &c. d (p"E x ) d c (p"E x ) _ d p (p"E x ) _ d T p"u'u" , ,, p"F 1 , 7 , 7, T7 T P w T OtC. = uu + p" (u' - u") at at p Again, multiplying the second by u", &c. dp' E x d c (p'E x ") _ d p (p'E x ") _ , d T u" u"* d T p dt ~~dt~ ~dT Up ~dT"2~dT' Multiplying the fourth by u, &c. dp'S. d c (p'E x ')_d p (p'E x } &c. dt dt dt dt 2 dt Then multiplying the second by u and the fourth by u" and adding, &c. dp'E x dt pu dt dt dt + &c. dt dt ,...(74). ON THE SUB-MECHANICS OF THE UNIVERSE. [40 The first of these equations is the equation of the component system p", u". Then adding together the several corresponding terms of the five equations following the first, we have d( P E- P "E") _ d c ( P E-p"E"} = d^pE-^ET) ? dt dt dt for the energy of the system of momentum pu p"u" dp(pE ~f E " } = uF x + vF y + wF z - ^P .................. (76). 40. Generality of the equations for the component systems. As the actions which are respectively expressed by the several terms in the equations (68) to (72) (remembering -TJ-* = ~^T~^ H j ) are mechanically \ at at dt J distinct, these equations are perfectly general and may be applied to the analysis of any resultant system of motion existing in a purely-mechanical- medium, into any two component systems which are geometrically distinguish- able. The motions in the two systems are not necessarily independent but the effects of the one on the other are generally expressed in the equations. Thus it may be that neither of the component systems is a conservative system, since one system may be subject to displacement of momentum by and may receive energy from the other system, although they both exist in a purely-mechanical-medium. And it thus appears that there may exist a non-conservative system of motion in a purely-mechanical-medium; that is to say, it appears that, so far as one abstract system of motion is concerned, a purely-mechanical-medium may be possessed of physical properties in consequence of the simultaneous existence of another system of motion. Thus where the only motion apparent to our senses is that of a component system, (the other component system being latent,) although this exists in a purely-mechanical-medium, the apparent system will not of necessity follow the laws of a conservative system, but is expressed by equations involving terms expressing the effects of the latent system on the apparent system, which apparent effects depend on certain physical properties in the medium. Such apparent physical properties however receive mechanical explanation when the complete motion of it is known; or, on the other hand, the experimental determination of these properties may serve to define the latent component motion so as to account, in the equations of the recognised system, for the terms expressing its effect; as for instance the potential energy. 41] THE EQUATIONS OF CONTINUITY FOR COMPONENT SYSTEMS OF MOTION. 41 41. Further extension of the system of analysis. So far the complete expression of the equations of motion has been confined to the case of two component systems of motion. But by a precisely similar method either of the two component systems of motion may by further definitions be again abstracted into two or more component systems of motion which in virtue of the definition are geometrically distinguishable from each other and from the remaining component system. If instead of taking u", v", w" to express the defined components of the motion after the abstraction of the residual motion, we take and for C Q put ^Q + C -Q + ^"Q + &c., for T M' put P M" + P M'" + &c., and so on for the other functions, expressions are obtained for the equations of as many component systems of motion as are distinguishable by definition. SECTION V. THE MEAN AND RELATIVE MOTIONS OF A MEDIUM. 42. Kinematical definition of mean motion and relative motion. By the mean motion of the medium is here understood an abstract component system of motion of which the mass and the components of the velocity respectively satisfy certain conditions as to distribution ; (1) The condition of continuous velocity, that the mean component velocities are continuous functions of x, y, z and t, however discontinuous the mass may be, Art. 12. (2) The condition of being mean velocities, that the quadruple integrals, with respect to the four variables, of the respective densities of the mean-components of the momentum (the components of the mean velocity multiplied by the density of the mass at each point) taken over spaces and times, the measures of which exceed certain defined limits, shall be the same as the corresponding integrals of respective components of the density of the resultant momentum. (3) The condition of momentum in space and time of the components of momentum of mean-velocities, that the integrals of the momentum of the mean velocities taken over the same limits as in (2) shall be respectively the same as in the resultant system. (4) The condition of relative energy, that the quadruple integrals with respect to the four variables, taken over limits, of the products of the differences of the respective components of the actual, or resultant, and mean velocities, each multiplied by the density of the corresponding components of momentum of mean velocities, as defined in (2) shall be zero. By the relative velocity of the medium is here understood the velocity which remains in the medium after the mean-velocity is abstracted from the resultant motion when this velocity satisfies certain conditions besides those entailed by the abstraction of the mean-velocity. The conditions entailed by the abstraction of the momentum of mean- velocities are, besides the condition (4) 42] THE MEAN AND RELATIVE MOTIONS OF A MEDIUM. 43 (5) The condition of the momentum of relative-velocity, that the mean densities of the components of momentum of relative velocity are zero. (6) The condition of distribution in space and time of the momentum of relative velocity, that, taken over the same limits as the mean velocity, the means of the products of the respective components of the momentum of the relative velocities multiplied by any one of the measures of the variables are all zero. The further condition that must be satisfied by the velocity left after abstracting the mean motion in order that this may be relative- velocity is: (7) The condition of position of energy of mean and relative velocities, that the mean values of the products of relative energies, as denned in (4), multiplied by measures of any one of the variables, shall be zero, or that the mean position of the energies of the mean-velocity, together with the energy of relative-velocity, shall be the mean position in time and space of energy of the resultant system. By the mean density of mass is here understood an abstract system of mass which satisfies certain conditions as to distribution. (8) The condition of continuous density, that the mean density is a continuous function of the variables. (9) The condition of mean density, that the quadruple integrals witli respect to the four variables of the mean-density taken over spaces and times which exceed certain defined limits shall be the same as the corre- sponding integrals of the actual density. (10) The condition of distribution of mean-density, that mean position in time and space of the mean-mass shall be the same as the mean position of the resultant mass. By the relative density of the medium is here understood the density (positive or negative) which remains in the medium afterithe mean-density has been abstracted, when this residual density satisfies certain conditions besides those entailed by the abstraction of the mean-density. The conditions entailed by the abstraction of the relative density are": (11) The condition of relative density, that the mean of the relative density is zero. (12) The condition of distribution of relative mass, that the product of relative density multiplied by the measure of any one of the variables has no mean value when taken over the defined limits. The further conditions which have to be satisfied by the relative density of mass are : 44 ON THE SUB-MECHANICS OF THE UNIVERSE. [43 (13) The condition of momentum of relative mass, that the products of the components of mean velocity multiplied by the relative density of mass have no mean values over the defined limits. (14) The condition of distribution of momentum of relative mass, that the products of the components of mean velocity multiplied by the relative density of mass and again by the measure of any one of the variables have no mean values over the defined limits. (15) The condition of energy of relative mass, that the products of the squares of the components of mean velocity multiplied by the relative density have no mean values when taken over limits. (16) The condition of position of energy of relative mass, that the products of the squares of the components of mean velocity multiplied by the relative density and again by the measure of any one of the variables have no mean values. By the mean motion of the medium is here understood the product of the mean-velocity multiplied by the mean density, which is also the density of the mean momentum. And by the relative motion of the medium is understood the density of the resultant momentum less the mean mo- mentum. In the same way by the density of energy of mean-motion is understood the product of the square of mean-velocity multiplied by the mean-density of mass ; and by the density of energy of relative motion is understood the density of energy of resultant motion less the density of energy of mean- motion. 43. The independence of the mean and relative motions. It will be observed, that according to the foregoing definitions, in any resultant system which consists of component systems of mean- and relative- motion, satisfying all the conditions, all the motion which has any part in the mean momentum or in the mean-moments of momentum is, by integra- tion, separated from the relative-motion in such a manner that the motion of each component system is subject to the laws of motion. Action and reaction being equal and opposite and the exchanges of momentum taking place in the direction of the momentum exchanged. And that the relative motion, separated out by integration, is confined to motions of linear and angular dispersion of momentum the effects of which on the mean-motion are such as correspond to the effect of observed physical properties of matter. It also appears that all the conditions must be satisfied in the resultant motion in order that such separation may be effected. 45] THE MEAN AND RELATIVE MOTIONS OF A MEDIUM. 45 44. Component systems of mean- and relative-motion are not a geo- metrical necessity of resultant motion. A very general process in Mechanical Analysis is to consider motion in a mechanical system for a definite interval of time as consisting, at each point of space at any instant of time, of com- ponent velocities which are the mean-component velocities of the whole mass over the whole time, together with components which are the differences between the actual components at the point and instant, and the mean- components. These systems respectively satisfy the conditions as to con- tinuous and mean-velocity (1) und (2). Also the condition of relative- velocity (5), and that of relative-energy (4), but they do not satisfy the conditions as to distribution of mean-momentum or any of the other conditions ; and hence are not mean and relative, except for particular classes of motion, in the sense in which these terms have been defined. Such component systems of constant mean-motion in a defined space and time are a geometrical necessity in any resultant system. And, although I am not aware that it has been previously noticed, it appears that con- sidering the number of geometrical conditions to be satisfied by the momentum of mean-velocity and of relative-velocity ((1), (2), (3), and as a consequence (5) and (6)), and the opportunities of satisfying them, the latter are sufficient for the former ; so that every resultant system of motion existing in a defined space and time consists of two component systems which satisfy the con- ditions (1), (2), (3), (4), (5) and (6), although they do not, as a geometrical necessity, satisfy all the further conditions required for mean and relative motion as here defined. 45. THEOREM A. Every resultant system of motion consists of a component system of mean motion which satisfies all the conditions of mean-velocity (I, 2, 3), and the condition of relative energy (4), but not, of necessity, that of position of relative energy (7); together with another system which satisfies the conditions of relative velocity (5) and (6), but not of necessity (7), the condition of distribu tion of relative energy. Taking the mean-velocity at a point x, y, z at the time t within the defined limits, to be expressed by u" = A+(x-x)A x + (y-y)A y + (z-z)A z + (t-t)A t , &c., &c....(77), where the barred symbols refer to the mean-position of the mass within the limits, whether time or space, thus jjjjxpdxdydzdt ~ ' the limits being assumed ; the conditions to be satisfied by the component velocity u" are : 46 ON THE SUB-MECHANICS OF THE UNIVERSE. [46 (1),(2),(5); that If I IP ( u ~ w ") dxdydzdt = 0, (3), (6) 1 1 1 1 xp (u - u") dydxdzdt = 0, &c., &c., &c (79). (4) nnp (u - u") u" dxdydzdt = 0. The last of these conditions will be identically satisfied if the others are satisfied. Hence there are only five conditions to be satisfied, while in the expression for u" there are five arbitrary constants, which are determined by putting _ffff(pu) dxdydzdt ff!f(p) dxdydzdt" then integrating the four equations of position and obtaining the values of A x , A y , A z , A t by elimination from the resulting equations. These values must be real since the A x , &c. enter into the equations in the first degree only. The same reasoning applies to the component velocities v" and w" ; so that the first part of the theorem is proved. To prove the second part all that is necessary is to observe that the con- dition (7) requires that r/ff (81), when it is at once seen that this condition is not satisfied as a geometrical consequence of the definition of u", since the terms involve products of the variables a; (y y) pA y , &c., which do not necessarily vanish on integration : so that the second part of the theorem is proved. 46. THEOREM B. In a similar manner it appears that every resultant system of mass consists of a component-system of mean-mass which satisfies all the conditions (8), (9) of mean density, and the conditions of relative density (11) and position of relative density (12), also the condition of momentum of relative mass (13) ; but does not satisfy, of necessity, the condition of distribution of momentum, of relative-mass, or of mean-mass (10), (14), nor the conditions of energy of relative mass, (15) and (16). Taking the mean-density of mass at x, y, z and t to be p =J) + ( X -^D x + (y-.y) Dy+ ( Z -Z)J) 2 + ( t-t)D t (82), where, as before, the barred symbols refer to the mean position of mass between limits of time and space. And putting t 1} x li y l , &c , as referring to 46] THE MEAN AND RELATIVE MOTIONS OF A MEDIUM. 47 the mean position in time and space, not of the mass, but of the time and space between limits. Since the mean value of p" between limits is not the mean value at the centre of gravity or epoch, the conditions to be satisfied are: (8), (9), (11) ...(83), which five conditions determine D, D X) D y , D z and D t whatever may be the distribution of mass, so that putting p = p p" the conditions (11) and (12), ffflp'da:dydz = (10), (12) x(p p") dxdydzdt = 0, &c., &c., &c. rrrr \\\\xp dxdydz = Q, &c., &c., &c. are satisfied. Again, since the constants A and D in the equations (77 and 83) for u" and p" are respectively the values of u", p", at the mean position of mass respectively, and the constants A x , &c. and D x , &c., are the differential coefficients of u" and p", respectively, the equations may be written u" = u" + u, &c., S (85). p" = p " + p, & c ., &c.J w Then multiplying the corresponding members, pu = p 'u" + p 'u" + pu, &c., &c ...................... (86), whence it appears, since the integrals of the last three terms on the right are by definition of necessity zero, that (87), so that condition (13) is of necessity satisfied, which concludes the proof of the first part of the theorem. To prove the second part. Multiplying the equation respectively by x, &c., then, since the integrals of xpu, &c. are zero while those of x 2 p' are not of necessity zero, and the expression of xpu, &c. includes the terms du" x^p -j , &c., it appears that the product p"u" does not of necessity satisfy CLOG the condition of position of mean-momentum for every distribution of mass, which proves the second part of the theorem. 48 ON THE SUB-MECHANICS OF THE UNIVERSE. [47 It has thus been proved that in order that a resultant-system of motion may satisfy the condition of consisting of a component system of mean- momentum which is a linear function of any one or more of the variables together with a component-system of relative-motion which satisfies all the conditions (1) to (15), the relative motion and the relative-mass must, what- ever may be the mechanical cause, be subject to certain geometrical restrictions relative to the dimensions of the limits over which the mean motion is taken. With a view to studying the mechanical circumstances which cause such restrictions, where they are shown to exist by the existence of systems of mean and relative motion, it becomes important to generalise, as far as possible, the geometry of these restrictions. 47. General conditions to be satisfied by relative-velocity and relative- density. The general condition to be satisfied by relative- velocity is that, in addition to the conditions which follow from the definition of mean-velocity, the integrals of the products of the density of relative component energy, pu"u, multiplied by the measure of any variable, are zero, or IJIJz;pu"u'dxdydzdt = 0, &c., &c., &c (88). Hence as u" is a linear function of the variables these conditions will be satisfied if pu, multiplied by any variable, and again by the squares of any power of this variable, all vanish on integration with respect to all four variables, so that the general condition is at once seen to be that pu', &c., the components of momentum of relative velocity, integrated between limits with respect to any two independent variables independent of the variable in whicli u" varies, must have no mean value ; and in the same way for v", w", since v", w" are not necessarily functions of the same one variable, in order to generally satisfy the conditions pu, pv', pw must vanish when integrated with respect to any two variables. Again when the previous condition of relative velocity is satisfied, it appears that the general condition of position of mean-momentum, 1 1 1 \xp"u"dxdydzdt = 1 1 1 \xpudxdydzdt, &c., &c. requires that the products x 2 p', &c. shall vanish when integrated between limits with respect to all four variables. Whence we have for the condition of relative mass that the integrals of p taken between limits with respect to any two independent variables which are independent of the variable in which u" varies &c. must be zero. If both the previous conditions are satisfied it appears that the conditions (15) and (16) will be satisfied for pu-p"u" = p'u" + pu' (89), 47] THE MEAN AND RELATIVE MOTIONS OF A MEDIUM. 49 and since u" is a linear function of the variables (pu- p"u")u"=pu" 2 + pu'u" ..................... (90), whence the integrals of both the terms on the right vanish by the previous conditions. And further, the conditions (ft fa; (pu - p "u") = 0, &c., &c., &c .................... (91) are satisfied ; for by taking u" constant in equation (77), by the definition of u" we have one relation between four independent variables, so that there are three independent variables with respect to which u" is constant. And in exactly the same way there are three independent variables with respect to which p" is constant. Therefore u" z and p" are each functions of one independent variable only. Hence in the expressions xp'u" 2 + xpu'u", &c., &c., since v", w" are not functions of the same variable as u", p'x, &c. must vanish when integrated with respect to any two variables, or u", v", w", must be constant. The factors of p and pu are each functions of two independent variables only, and hence these terms vanish on integration between limits with respect to all four variables by the previous conditions of relative density and relative velocity. Whence it appears that the general conditions, besides those which follow from the definitions of mean velocity and mean density, that must be satisfied by the momentum of relative motion and by relative density, are that these must have no mean values when integrated between limits with respect to any two independent variables independent of the variable with respect to which u" varies, &c. And it is only resultant systems in which these conditions are satisfied that strictly consist of dynamical systems of mean- and relative-motion. That these conditions can be strictly satisfied by any system within finite limits seems to be impossible ; as for this it would require that, in a purely mechanical medium, there should be, in the same space and time, two masses moving in opposite directions, such that at each point the density of the momentum of the one was equal and opposite that of the other. It is how- ever possible to conceive masses with equal and opposite momenta at any finite distance from each other, and in such cases the conditions may be con- ceived to be satisfied to any degree of approximation. R, 4 50 ON THE SUB-MECHANICS OF THE UNIVERSE. [48 48. Continuous states of mean- and relative-motion. The abstract systems of relative velocity and relative density as denned in the previous article must, as a geometrical necessity, be of an alternating character in respect of some of the variables, such that the respective means of the positive and negative masses of relative densities, and the positive and negative momentum of relative velocity, taken over the limits as to any two variables, balance. And as a consequence the distribution of such relative-masses and relative-velocities, whether regularly periodic, as in the case of waves of light or sound, or such as the so-called motions of agitation among the molecules of a gas, involves a geometrical scale of distribution defined by the dimensions of the variables over which the alternations balance. Such scales of relative- density and velocity, clearly, define the inferior limits of the spaces and times over which the resultant system can consist of systems of mean- and relative-motion. But there is no necessity that the defined space and time over which the system of mean-motion extends should be confined to the dimensions of such scales. That is to say the defined space and time, over which the mean-system may be a linear function of the variables, may be in any degree larger than the minimum necessary for the satisfaction of the conditions of relative-density and relative-velocity, since these conditions will be satisfied for the whole space if they are continuously satisfied in every element of dimensions defined by these conditions. 49. Under such circumstances the expressions for the mean-motion admit of another interpretation, one which has already been discussed in a paper on " The Theory of Viscous Fluids*." In this expression the mean-velocity at any point x, y, z, t is defined as the mean taken over an elementary space and time, of dimensions defined by the scales of the relative-velocity and density, so placed that the mean position of the mass within the element is defined by x, y, z, t. Then, since by definition the relative-velocity and relative-density, as defined by integration over the whole space and time, have no mean value in the element, the mean velocity at x, y, z, t (the mean position of mass) obtained by integration over the element will be the same as that at the same point obtained by integration over the whole space and time, as in the first of equations (79) ; and since, by definition, not only the relative density, but also the variations of relative density, with respect to any variable, have no mean values in the element, the mean-density at the mean position x, y, z, t, obtained by integration over the element as in equations (87) will be the same as that obtained (as in the second equation (89)) by integration over the whole space and time. * Royal Soc. Phil. Trans. 1894, pp. 123164. 51] THE MEAN AND RELATIVE MOTIONS OF A MEDIUM. 51 It thus appears that p", u", in equations (89) to (91) may be taken to represent the values of the mean-density and mean-velocity at x, y, z, t, as defined by integrations with respect to two variables over an element having dimensions defined by the scales of relative-velocity and relative-density, so placed that the mean position of the density in space and time is at x, y, z, t. 50. The instruments for analysis of mean- and relative-motion. It further appears that, since in the method of Arts. 43 and 44 u' may be taken to represent any entity, quantities consisting of the squares and products of u, u", u', FJp may by the theorems of those articles be separated into mean- and relative-components which satisfy the conditions Art. 42, (1), (2), (3), (4), (5) and (6), respectively, the mean components being linear functions of the variables, and the relative components having no mean value when integrated with respect to any three independent variables over dimensions determined by the scales of relative-velocities and relative- density. And in the case of the quantities p, pu', &c., subject to the further definition Art. 48, but only in the case where the relative components will have no mean values when integrated with respect to any two independent variables over the same scales. But in either case, if Q expresses the density of any function, integrating over definite limits about any point x, y, z, t as mean position of mass at that point we have MQdxdydzdt Q> , ffffdxdydzdt and ffffdxdydzdt and putting h and k for any two variables, r \ , fjTJA (Q - Q") dxdydzdt = 0, l\\\hk (Q - Q") dxdydzdt = 0, Equations (92) are thus the general instruments of mean and relative analysis. 51. Approximate systems of mean- and relative-motion. The interpretation of the expressions for mean- and relative-motion con- sidered in the last article is adapted to the consideration of systems in which the mean motion, taken over spaces and times which are defined by the scales of relative- density and relative-velocity, is everywhere approximately a linear function of the variables measured from the mean position and mean 42 52 ON THE SUB-MECHANICS OF THE UNIVERSE. [51 time. Thus if p" and u" are any continuous functions of the four variables x, y, z, t, taking x y Z(,t as referring to a particular point and time, then at any other point x, y, z, t, (93), where the differential coefficients are all finite. Therefore as (x # ), &c. approach zero all terms on the right except the first approximate to zero, and the terms of higher order which involve as factors multiples of the variables of degrees higher than the first become indefinitely small compared with the linear terms. It is therefore possible to conceive periodic or alternating functions of which the differential coefficients, continuous or discontinuous, are so much greater as to admit alternations to any finite number being included between such values of (x a? ), &c., as would leave the terms of the second and higher orders indefinitely small as compared with those of the first order, and those of the first indefinitely small as compared with the constant term. Therefore as long as p" and u" are finite and continuously varying functions of the variables it is always possible to conceive systems of relative-density and relative-motion which together with their differential coefficients satisfy the conditions of having approximately no mean values over the limits, and thus to any degree of approximation satisfy the con- ditions necessary to be relative-component systems to the mean system p" u o" + & c - within the limits defined by the scale of relative motion. The method of approximation therefore consists in obtaining p", u", p"u", &c., &c., and the variations of these, Q", when Q is any function of // // // ' / ' p u , p , p, pu, by integrating over the element taken about x, y, z, t, as the mean position, then using these quantities as determined for x, y, z, t, td express by expansion p"u", &c., &c., for any other point within the limits of integration as in equation (93) so as to obtain the mean values of these terms in the equations by integration over the elements, neglecting the integrals of all terms which involve as factors functions of the increments of the variables of degrees higher than the first : and in this way may be obtained any necessary transformations of products of mean inequalities and rates of variation, as u"dp"u" = dp"u"* - u"p"du", &c. 52] THE MEAN AND RELATIVE MOTIONS OF A MEDIUM. 53 It thus appears that the only motions neglected are those which are defined as small by the conditions, being of the second degree of the dimension of the scale of relative motion, while those retained may have any values at a point, and are, within the limits of approximation, linear functions of the variables ; so that within the same limits p, pu', &c., &c., satisfy by the special definition the conditions of having no mean values over the limits of any two variables; and generally Q' has no mean value over three independent variables. As has already been pointed out the maintenance of such a system must depend on the distribution and constraints, and the process of analysis consists in assuming such a condition to exist at any instant, and then from the equations of motion ascertaining what circumstances, as to distribution and properties of conduction, the actions of convection and transformation by and to the relative-motion on the variations of the mean-motions will be to increase or to diminish these variations of the first and second -orders. 52. Relation between the scales of mean- and relative-motion. From the previous article it is clear that the absolute dimensions of the scale of mean-motion, as determined by the comparative values of the terms of higher orders as compared with those of the lower, do not enter into the degree of approximation to which the conditions of relative-mass and velocity are satisfied, except as compared with the scale of the relative- motion. But it does appear that the degree of approximation depends on the comparative values of these scales. And hence it is only under circum- stances (whatever these may be) which maintain distributions of mass and velocity which admit of complete abstraction into two systems widely distinct as to relative scales, that systems of mean and relative motion can exist. Thus, as we have previously pointed out, it is not sufficient that the relative motion, or one class of motions such as the motion of the molecules of a gas in equilibrium, should be subject to superior limits by the scale of distribution. It is equally necessary that the scale of variation of mean motions, such as the mean motions of a gas, should be subject to superior limits (whatever may be the cause) which prevent the scale of these mean- motions approaching that of the molecules. And it is the existence of circumstances which secure both these effects, which is indicated by resultant systems which satisfy the conditions of mean- and relative-motion as defined. It has been already proved that the existence of component systems which satisfy the conditions of mean position of density and of relative energy, as well as those of mean-density and mean-position of momentum of mean-velocity, is not a geometrical necessity of the definition of mean- motion as is the existence of component systems which satisfy the latter 54 ON THE SUB-MECHANICS OF THE UNIVERSE. [52 conditions only. Were it not so there would be no point in the analysis, for then the existence of such component systems would reveal no special circumstances as to the geometrical distribution of the medium, or the motion in the medium, whereas it has now been shown that the existence in such systems of mean- and relative-motion, as indicated by the observed mean- motion and the apparent " physical" properties of the medium or matter, depends (if in a purely mechanical medium) upon circumstances which constrain the geometrical distribution of the motion of the medium. Thus the application of this method of analysis affords a general means of studying the conditions of the medium, either intermediate or fundamental, which would admit of such relative or latent motion as is necessary to account, as a mechanical consequence, for the apparently physical properties of matter and the medium of space. SECTION VI. THE APPROXIMATE EQUATIONS OF COMPONENT SYSTEMS OF MEAN- AND RELATIVE-MOTION. 53. THESE equations must conform to the general equations of component systems as expressed in the equations (61) to (76), Section IV. Thus if in equations (69), (70), (71), together with equations (74), (75), (76), p", u" and p'u' are at any time subject to the respective definitions for mean- and relative-motions, these suffice, for the instant, to determine the rates of transformation (as expressed by arbitrary functions) in terms of the several defined rates of convection and production. Then these rates of transformation, as expressed in the defined symbols, having been substituted in the equations, these equations express the approximate rates of change of the mean and relative component systems at the instant. These equations express, in terms of the so far defined mean and relative quantities, the initial approximate rates of change in the defined quantities and thus afford the means of studying whatever further conditions must hold in the distribution of the medium in order that these rates of change may tend to maintain or increase the degree of approximation to which the conditions of mean- and relative-motion are initially subject. This study of the further definition, however, must of necessity follow the complete expression of the initial equations, to which this section is devoted. 54. Initial conditions. The initial conditions for approximate component systems of mean- and relative-motion, as defined in Arts. 50 and 51, Section V., define all mean quantities as continuous functions of the variables, such that within the limits over which the means are taken they are constant to a first approxi- mation, whether they are the means of density, means of velocity, or means of component momentum ; also the means of any products or derivatives of products, of velocity, or density, the means of any products of mean and relative quantities, while the products of the relative quantities, correspond- ing, multiplied by the density, are such that their means taken over the same limits are zero. 56 ON THE SUB-MECHANICS OF THE UNIVERSE. [55 Thus if Q be any term expressing increase of density of mass, momentum, or of energy for the resultant system, or for either of the component systems at a point, x, y, z, t, at distance 8x, By, Sz, 8t, jjjjdxdydzdt dx } .(94), Q' = Q-Q" satisfy the conditions (1), (2), (3), (4), (5) and (6), Art. 42, of being respectively mean and relative, approximately, that is to say Q" is, approximately, a linear function of the variable, and Q' has approximately no mean value when integrated over any three independent variables. Also if -~- is a derivative of any quantity -- dx) - dx and doc dx dx 55. The rate of transformation, at a point, from mean-velocity, per unit of mass. From equation (58) or the first two of equations (69) transforming by equation (19), du" du" du" du" dt dx dy dz , du' , du" , du" d . p p -j ~j- -j- ^IP - dx dy dz dt v The first four terms in this are all mean accelerations, while the last three terms on the left are such that multiplied by p have no mean values- are entirely relative-accelerations whence by definition it follows that since du"/dt is a mean-acceleration the right member must contain terms which exactly cancel the last three terms on the right, and that these form the only relative terms it can contain. These terms which represent the acceleration at a point per unit mass, due to convection of mean velocity by relative velocity, are the only transformation from mean velocity at a point. Since after abstracting these terms the right member remains wholly mean, we have " <97) - 56. The rate of transformation at a point from relative velocity, per unit of mass. From equations (60), or the last two of equations (69), 58] COMPONENT SYSTEMS OF MEAN- AND RELATIVE-MOTION. 57 In this the term on the left is, by definition, such as has no mean value, hence taking a mean by equation (92), Section V. , &c., &c.; or dividing by p" it appears that the transformation from relative-velocity to mean-velocity, at a point, is expressed by 1 \d c (pu) p" \ dt that is the mean accelerations due to the mean convections of the relative- velocity by the relative-velocity, plus the mean acceleration due to con- duction. Substituting from equation (97) the expression d p u"/dt in equations (58) and (60), Section IV. , du" -j- dy , du" d p u" , du" -5 = u -j- dt dx ,du^_ I d c (pu')" F x " dz p" dt r p" ' d 77 dt , du" .' _ _ n\ _ 7 "? dx dy ,M' 1 d e (pu) F x " ' dz p" dt ~ p" ' (100). A7T IXC. 57. The rates of transformation of the energy of mean-velocity. As already pointed out, Art. 35, Section IV. equation (61), the rates of transformations of energies per unit of mass, of mean-velocity and relative- velocity, are respectively obtained by multiplying the rates of transformation of mean- and relative-velocity, u" and u', &c., &c. respectively ; thus ^ fee., &c. 1 d T (u'J 1 = 2* 1 ,d(u'J &c )' c(pu') r/ 2 dt ld T (uJ dx ' , ,du" , , du" dt , ,du"\ 2 dt UU -. \ U V ax dy ' dz} u' \d c (pu) &c., &c. d T dt ....(101). 1 d(u'J (u"-u')(d c (pu') ~2 M dt ~ P 7r ~~(dt . , du" p + uu , h &c., &c. dx j 58. The expressions for the rates of transformation in equations (100) and (101) include all the rates of transformation of component velocities, and of the squares and products of the component velocities of the component systems of mean- and relative-velocities which enter as arbitrary functions into the equations (69) and (74). But as is pointed out in Art. 35, Section IV. 58 ON THE SUB-MECHANICS OF THE UNIVERSE. [58 any one of these quantities, the rate of increase of which is expressed by one of the equations, may, by definition, be further abstracted into two component systems. The component systems of the energies of the mean- and relative-velocity per unit mass may, therefore, be separately abstracted into mean arid relative component systems. And the importance of this at once appears, since the process of analysis is solely between the mean and relative, and while (u") 2 is mean and (w'V) is relative, (u) 2 , although positive, is not continuously distributed as a continuous function of the variables. The rate of transformation from the mean rate of increase of energy of relative-velocities to relative-energy of relative velocity. Adding the second and fifth of the equations (74) as they stand, and substituting the expression for the transformation-function from the second of equations (101), we have 1 dp (uj = ld[ e (puj + u'F x ] 2 dt 2 dt dx dt .(102). Then putting (103), where ((w') 2 )" is obtained after the same manner as u" ; putting d( T ((u'y 2 )")/dt for the total rate of transformation, we have as in equations (97) and (98), substituting ((u') 2 )" for u" and the three last terms in equations (102) for F x in equations (100), since the last term has no mean values, (d\_d c p((uj}'} dt + and 2 dt 2 dt Then since -~ = 0, (104); d p (u 2 )_d T (u'J d T ((uJ}" ~dT ~dT ~~dT i d p [* - w] i + &e.-'JV u^ (1 d c puf f 7 1 2 ~dT + / ts " ...(105). 59] COMPONENT SYSTEMS OF MEAN- AND RELATIVE-MOTION. 59 The expressions for the production of mean energy of relative motion which form the left members of equations (104) are not transformations from energy of mean motion only. They include the relative parts of the rates of convection and production of energy of relative motion which are being transformed to the system of relative energy. These rates of convection and production of relative-energy are expressed by the first two terms in the equations (104), while the last term expresses the only rates of trans- formation from energy of mean -motion. Whence the only transformations from energy of the component mean motions are -P { , , du" , , du" , , du"\ < u u , h v u -= h w u -f > , &c., &c. ( ax dy dz ) 59. The rate of transformation from mean to relative energy. From equation (64), at a point, d T p" dp" dp"u" dp"v" dp"w" dp"u dp"v dp"w j. TT -1 -- -j -- 1 -- -j -- 1 -- j -- 1 -- -j --- 1 -- -. -- 1 --- -7- at at dx dy dz dx dy dz where the first four terms on the right are all mean, and the last three may be in part mean and in part relative. Hence the relative part of the convection of mean-density by the relative-velocity is the transformation to the relative density at a point, and this must form the only relative of the left member, and d T p" d^p" /d e 'p"\" t (d T p / e 'p\ V dt J dt dt \ dt J \ dt Also from the last of equations (65) dr = d_ dp'u" dp'v" dp'w" .(107). _ _ dt dt dx dy dz dt In the last of the equations (107) the first four terms on the right are relative, and therefore the mean rate of transformation is d T p _ dt dt Then adding the mean and relative parts ; since dt dt and (pu + &c.)" = 0, d T p" d c -p" 60 ON THE SUB-MECHANICS OF THE UNIVERSE. [60 60. The transformations for mean and relative momentum. TTT 1 ^T\P W J // CvrpU .. a/^ P . . We have -^ - = p -^ u - (HO). dt dt Then substituting from the first of equations (101) and (109), and trans- forming, and we have 61. The rates of transformation of mean- energy of the components of mean- avid relative-velocity. From equations (74), (100) and (109) we have dt dt 2 dt "2 dt *' dt +u'F x [ - .// \ // ..(112). In the second of equations (112) it is the last term only that expresses transformation from energy of mean motion. The last terms of equation (112) admit of different expression, by substi- tuting for dt (dpu'u' dpv'u dpw'u'} " its equivalent - | ^ + -^ +-^-| , or and we have u dt 'dp" (u'u 1 )" dp" (v'u)" dp" (w'u'y dx dy dz = - | (P (UU) U ) &c I ^ u , u y, _ { dx ) ( r da; also dp yx dp 2X dx dy dz ' 61] COMPONENT SYSTEMS OF MEAN- AND RELATIVE-MOTION. 61 so that by equation (95), F x " may be expressed by doc \. Then we have w ^ / = ^ + &c ._y>_ also (u'F x )" = (uF x -u"F x )" ...................................... (115), and this may be expressed as ' dx du" Then substituting in the first of equations (112) we have for the rates of transformation to the energy of mean motion ld[ T (p" (u'J}} _ ld[ e (p" (u'J}} \d Q" (?" (uu)"}} } 2 " dt 2 " " dt { dx j \d(u"p" xx ) ) f r , , , ,du" | \ , - + &c.|-4--{[p (uu) +Pxx\-j h&cl ...(lib), { dx ) ( dx ) and again substituting in the second of equations (112) we have for the rates of transformation to the energy of relative motion 2~ dt ~2~ dt Wl" (J ( upxx )" j + &C, d du" J- dx The purpose of this transformation is easily seen on adding the equations. The two last terms in each equation cancel, showing that they represent a transformation between the rate of increase of the mean-energies of relative- and mean-velocities ; while changing the sign of the right members of the resulting equation, which then represent the rate of transformation to 62 ON THE SUB-MECHANICS OF THE UNIVERSE. [62 the energy of residual motion, or of relative energy, these become 1 d T [pu* - p" ()"] = 1 d ^...(118); dt 2 dt 1 d [c (p (uj. >']",(<*[ u (p (u u \"\~\ } ' J 4- &T I 2 dt ' I dx du ; and these are the exact forms in which the rate of transformation to relative- energy, obtained by substituting w 2 , (u 2 )", (u 2 )', uF for u, u", u, F respectively in equation (111) for relative momentum, is expressed. In a purely mechanical medium the last terms in these equations (118) represent the mean rate of angular dispersion both of mean and relative motion of energy, as explained in Art. 32, Section III., while the integrals of the remaining terms are all surface integrals. It is thus seen that the rates of exchange between mean-energy and relative-energy are purely conservative within the limits of the approximation. On the other hand, the integral rates of exchange by transformation between mean-energy of mean-motion and mean-energy of relative-motion as expressed by the integrals of the last terms of equations (116), (117) are not surface integrals, nor are these rates confined to angular dispersion ; so that they express exchanges at each point which are not expressed by a surface integral, and thus appear to represent those actions of the relative-motion on the mean-motion the study of which is the object of the investigation. But this is found on closer examination not to be the case. 62. The expressions for transformations of energy from mean to relative motion. du" The expressions p"(u'u')" -= H &c., which occur in the last terms of equations (116) and (117), are simply transformation terms expressing the mean effect of the convections of relative-momentum by relative motion on the energy of mean motion, and this is the most general and most important transformation. The other transformations are the results of conduction. These are in- cluded in the expressions du as they occur in equations (116) and (117), but they are not explicitly expressed by these. The first of these expressions includes the rate at which the energy of the component of mean-motion is being increased by angular 62] COMPONENT SYSTEMS OF MEAN- AND RELATIVE-MOTION. 63 dispersion from the energy of the other components of mean-motion, as well as the rate at which the energy of the component of mean-motion is being increased by transformation from the energy of the corresponding com- ponent of relative-motion. The second of these expressions includes both the rates at which energy of the component of mean-motion and the energy of the component of relative-motion are increasing, by angular dispersion, at the expense of the other components in their respective systems, together with the rate at which energy of the component of the resultant system is being increased by transformation from energy in some other mode which latter rate does not exist if u, v, w are the motions of points in mass. In the expressions &c., &c., du o o and p xx -= |- &c. , &c., &c., -= dx the analysis necessary to separate out the expressions for the separate actions in either system is furnished by equations (47 A), Section III., the symbols for the mean and the relative motions being substituted for those of the resultant system. Putting p ?= ^p t- t the first two terms in these equations (47 A) o which express the rates of angular dispersion in the directions of x, y, z respectively on the square of the components of the mean and the resultant system, become respectively du" dv" dw"' riv/2 [3 P \ dx dy dz du dv\ (du dw g - &c " The corresponding expressions for the rate of increase of the resilience are [1 (du" L dv" . dw"\ ^ . du" ~ o P -j- + j~ + TT- ) + (P xx ~P }~j- |_3 r V dx dy dz J ' dx du dv dw\ , .du 1 f fdu . dv\ . (du . dw^ 2 64 ON THE SUB-MECHANICS OF THE UNIVERSE. [62 Substituting these for ( p" xx j + &c. ) and ( p xx -7- + &c. ) as they enter \ dx / \ dx f into equations (116) and (117), these equations become 1 d T [p" (u'J] I d c ' [p" (u'J] $dj,[p(u'u')"] , s _{ \du"p" xx } o ~jl ~ ~ o ~jl i j ^ &c - 1 ~ \ r~ ^ o^c- r 2 dt 2 dt [ dx } ( dx du" _ dv" _ dw"\ If,/ /dtt" dw"\ /dw" dw" da; dy &c., &c 1 d [ T p" (u'u')"] = 1 d l' P " (u'u)"] , 1 d [ C ' P (uV)T fd (^>x.) r/ , p ) 2 rf 2 d x "*~2 rf ( da? j [p^ / ^/ _ ^/ _ dw'\" If /dw _ M\" ldu_ du/\" + [p" fdu' dv dw'\" (. .du\" - "o (^r-+ J- + -J- +H/>B-P) j-r [_3 \^ rf^ tw/ ( da;] If /dtt' i .j H" *-" ( dx (d[up xx ] } JC 'I (r rfM V+^ \ dx l&C j f &c.. &c. . 1^3W H ...i (124). 67. Complete equations. 1 d [p" ((u'J+ (v'J+ (w'J)} 2 dt " ((u'J + (v'J + dt dx l &c ' + ^ (125). 67] COMPONENT SYSTEMS OF MEAN- AND RELATIVE-MOTION. 67 dt dt dt * .>> + + I 7>z* -r- .(126). d [p (w 2 + v 2 + ^ 2 ) - p" ((u*)" + (v*)" + (w 2 )")] (ft i d o (p (w 2 + v 2 + w 2 )) - p" (o 2 )" + (v 2 )" + (w 2 yy 2 d 1 2 dt L_J ' J i &p v c? ' ) d[w"p"(wu}"} \ I >.vc. t dx \ I dx dx &c + ! (Pvx -T- + \\Pzx-T- 52 68 ON THE SUB-MECHANICS OF THE UNIVERSE. [67 The equations (119) to (127) are the equations for mean and relative component systems of any resultant system in which the conditions are satisfied, irrespective of the medium being a purely mechanical medium ; that is to say, irrespective of whether or not in the resultant system (p, u, v, w, p xx , &c.) are related to the actual, mechanical-medium, or represent the densities, motions and stresses of a component system of mean-motion of the resultant system. It has already been pointed out (Art. 52) that the absolute scale of the variations of the mean motion has no part in determining the degree of approximation, but only the relative magnitude as compared with the scale of variations of the relative motion. So that any component of mean-motion may be a resultant system if the conditions exist which ensure its satisfying the conditions of mean and relative motion. There is however this difference according to whether the unqualified symbols refer to the purely mechanical medium or not. If they do refer to the mechanical medium, then the last terms in equation (124) and the last but two in (123) represent angular dispersion of energy only, and the last term in equation (127) and the last but one in (126) are zero ; if not, they represent changes of energy. SECTION VII. THE GENERAL CONDITIONS FOR THE CONTINUANCE OF COM- PONENT SYSTEMS OF MEAN- AND RELATIVE-MOTION. 68. THE general conditions for the existence of mean-, and relative- motion, as defined in Art. 47, Section V., are that the components of momen- tum of relative- velocity, as well as the relative density, must respectively be such that their integrals with respect to any two independent variables, taken over limits defined by the scale of relative-motion, have no mean values. By equation (1), Section II., it follows that for the continuance of such states the respective rates of increment of these quantities by all causes, convection and production, must satisfy the same conditions. Therefore as the necessary and sufficient conditions we have, that f'dj^) iHg) r'd^f) rg J o at J o at Jo at Jo at where the limit t may have any value, when integrated between the limits, as initially defined by the relative scales, with respect to any two indepen- dent variables shall be zero within the limits of approximation. The satisfaction of these conditions does not follow as a geometrical consequence of the initial condition. The rate of change in the density of relative-momentum is a consequence of the space rates of the variation of the convections and conductions existing at the instant. And initially the mean- and relative-motions are subject to definition, from which, as a geometrical consequence, their varia- tions, in space, are also subject to definition, which although less complete has been already fully defined, Art. 45, Section V. It therefore follows that the general conditions to which the initial rates of increase, by convections and conductions, are subjected, are defined. And this at once appears on considering the equations of motion for the momen- tum of relative- velocity, which are obtained by substituting in equations (98) the expressions for the rates of transformation from equations (100), Section VI. 70 ON THE SUB-MECHANICS OF THE UNIVERSE. [69 / . du" , dv , dw \ , plu -j h v j I- w j dt r \ dx dy dz ) d , d . d t, &c., &c (128). P In these equations, according to the method of approximation, all the terms in the member on the right are such as have no mean values when integrated over any three variables, as a geometrical consequence of the definition. It therefore appears that it does not follow as a geometrical consequence that , &c., &c., 7, should satisfy the condition of having no mean values when integrated with respect to any two variables, to the same degree of approximation as do the initial values of pu', pv', pw'. And this applies to both rates of increment by convection and rates of increment by relative accelerations. If, then, this condition is to be continuously satisfied it must be as the result of some redistributing effects of the actions of conduction on the convections. For the rates of increase by convection are a geometrical consequence of the initial motions which are subject to the definition as to scale and relative-motion ; while on the other hand, the rates of increase by conduction depend on the conducting properties of the medium, as well as on the distribution of the medium in space and time. 69. The fourth property of mass, necessitated by the laws of motion, is that of exchanging momentum with other mass, Art. 17, Section II., and it now appears that this is the fundamental property on which the existence of systems of mean- and relative-motion depends. For if there were no conduction, that is, if mass were completely pene- trable by mass; so that two continuous masses could pass through each other without affecting each other's motion ; then the only rates of increase would be those by convection, each point of mass preserving its course with no interruption, with constant velocity, and there could be no redistribution. Hence : Certain properties of conduction are necessary for the maintenance of systems of approximately mean- and relative-motion. 72] COMPONENT SYSTEMS OF MEAN- AND RELATIVE-MOTION. 71 70. Notwithstanding the extremely abstract reasoning on which the foregoing conclusion is based it is definite. And it appears possible to carry this reasoning further and so obtain conclusive evidence as to what the general properties of conduction and the general distributions of the medium must be for the maintenance of the mean- and relative-systems, when the resultant system is purely mechanical. 71. The general laws of conduction of momentum by a purely mechan- ical medium, as denned by the laws of motion, have already been deduced (Section III. Art. 24), and the effects of conduction in displacing momentum and in angular dispersion of vis viva have been proved (Section III. Arts. 31 2), and also the effect of conduction on the resilience, if any. However, since there is no resilience in a purely mechanical medium, it at once follows that the medium must be perfectly free to change its shape without changing its volume, or it must consist of mass or masses, whether infinite, finite, or indefinitely small, each of which absolutely maintains its shape and volume ; that is to say, each of which is a perfect conductor of momentum. Thus the class of media in which the general conducting properties satisfy, as a resultant system, the condition of being a purely mechanical system is not large ; being confined to (1) The "perfect fluid"; (2) The perfect solid ; (3) Perfect discontinuous solids ; (4) Perfect discontinuous solids with perfect fluid within their inter- stices. This class of media all satisfy the conditions for purely mechanical media as resultant systems. But it does not follow, as a geometrical necessity, that they all satisfy the conditions of consisting of mean and relative com- ponent systems. For although any medium which satisfies the conditions of consisting of component systems of mean and relative motion must of necessity satisfy the conditions as a resultant system, the converse of this is not a necessity. It therefore remains to obtain from the previous definition the further limitations imposed, as a geometrical necessity, by the conditions of consisting of component systems of approximately mean- and relative-motion. 72. Evidence as to the properties of conduction for component systems. (1) From the equations (128) it appears, as already pointed out, that in order that f Jo 72 ON THE SUB-MECHANICS OF THE UNIVERSE. [72 may satisfy the condition of having no mean values, when integrated between the limits of the scale, in time and space, of relative motion, over any two independent variables to any defined degree of approximation, the time integrals of the members on the right must satisfy the same condition. Whence it follows that the condition for the maintenance of the inequalities steady requires that the rate of increment, as expressed by all terms on the right, in each of the equations (128), shall be such as has absolutely no mean value when integrated over limits, with respect to any two independent variables. This condition, although it applies only in a somewhat particular case, is such as must be satisfied for the maintenance of mean and relative systems to be general, and hence any evidence that may be derived from it must be perfectly general. To apprehend the importance of this evidence we have only to consider, what has already been pointed out, that the first four terms in the right members in each of the equations (128) require, as a geometrical necessity, integration between limits over three independent variables in order that they may have no mean values. Whence it follows that in order to maintain the inequalities steady the fifth term, which expresses relative rates of increment of momentum by conduction, must be such when integrated, over limits, with respect to any two variables, as will exactly cancel the integrals of the other four terms when they are taken over the same limits with respect to the same two variables. Thus we have for a particular case, which however must occur in all general systems consisting of component systems of mean- and relative- motion, an inexorable condition as to the necessary properties of conduction. It will be readily granted that the satisfaction of this condition involves the absolute dependence of the functions F x ', &c., on the condition of the medium and its relative-motion. (2) Evidence as to the necessary properties of the medium is also obtained from the condition that the inequalities must be maintained small. The satisfaction of the condition of equality between the rates of opposite actions resulting from transformation, convection, and conduction, does not define the magnitudes of the inequalities which may be maintained, but only the fact that they remain steady. It therefore appears that the definition of the relative values of the inequalities which are maintained depends on a balance of rates of institu- tion and decrement. And in order that such a balance should institute itself and remain steady, it is necessary that the state of the medium shall be such that integrals of F x ', &c., taken over limits with respect to any two independent variables, shall be such functions of the inequalities that they 73] COMPONENT SYSTEMS OF MEAN- AND RELATIVE-MOTION. 73 increase with the inequalities and are of opposite sign, whereby the in- equalities are subject to logarithmic rates of decrement. Then, whatever might be the rates of institution of inequalities resulting from all the other actions, the inequalities would increase, increasing the rates of decrement by conduction until these balanced the rates of increment, that is until the other actions were cancelled by the actions expressed by F x ', &c., after which the inequalities would remain steady as long as the rate of institution remained steady. (3) Evidence as to the necessary properties is also obtained from the conditions that define the scales of relative motion. Where mean motion is everywhere uniform this condition requires that the scale of relative velocities and relative mass shall approximate to some finite scale at which it will remain as long as the mean motion is everywhere uniform. This does not follow as a geometrical necessity of the initial definition, for if constraining limits were absent from the mass, the actions which insure the logarithmic rates of decrement would continue to diminish the scale indefinitely ; hence inferior limits of relative-mass and relative- motion define the properties of the medium as regards limiting constraints. 73. This evidence, together with the definitions of mean-velocity and mass, suffices to differentiate the four general states of media, which, as resultant systems, satisfy the conditions of being purely mechanical, from those which also satisfy the conditions of consisting of component systems of approximately mean and relative motion. Since continuous mass cannot pass through continuous mass without exchanging momentum, the reciprocal actions between the masses in relative motion will be to cause continual diversions of the paths of points in mass. And by definition of relative motion, if there is no mean motion, the mean component momentum in any positive direction is exactly equal to the mean of the negative momentum in the same direction. Therefore the mean rate of increase of component momentum in the positive direction, by the components of the reciprocal relative accelerations, is exactly equal to the mean rate of increase by the component reciprocal accelerations of the component momentum in the negative direction. The mean motions being uniform, the reciprocal accelerations have no effect on energy of relative motion in all three independent directions. Whence the effects of the component reciprocal accelerations are rates of change in the positive and negative component momenta, in one direction, with the positive and negative momenta in other directions. Such exchanges of positive and negative momenta from one direction to another are possible only when the component accelerations of relative motion are, not resultant accelerations, 74 ON THE SUB-MECHANICS OF THE UNIVERSE. [74 but, are the means of the components of resultant reciprocal accelerations with various degrees of divergence from the direction of the previous motion. And it is thus shown that any angular redistribution of positive and negative components of momenta, or, which is the same thing, of the vis viva of the component velocities, results solely from the impenetrability of the medium. 74. From the foregoing reasoning it might be inferred that the impene- trability of mass together with the definition of relative motion must secure logarithmic rates of decrement of all inequalities provided that the medium were sufficiently mobile. That this is not the case is however at once seen from the theory of a " perfect fluid." (a) For in such media every point in mass is in complete normal con- straint by the surrounding medium, with lateral freedom. So that, while no point can move without affecting the motion of every other point in some degree, there is no lateral action. Thus the continuous finite accelerations do not cause finite diversions of the paths of points in mass from the previous directions at any point of their courses, but cause finite curvature of these paths. And thus the paths of adjacent points are ultimately parallel. There being no finite lateral deviation, there is no lateral exchange of momentum in the direction of motion at any point. Whence such lateral exchange of momentum being necessary in order that there may be general rates of logarithmic decrement of inequalities, it follows that in a perfect fluid there cannot exist logarithmic rates of decrement of all inequalities of relative motion. It thus appears, since, as has already been pointed out, general logar- ithmic rates of decrement of all angular inequalities are necessary for the maintenance of approximate systems of mean and relative motion, that a perfect fluid, although satisfying the condition of a purely mechanical medium as a resultant system, cannot satisfy, generally, the condition of consisting of component systems of approximately mean and relative motion. (6) A perfect continuous solid, that is a continuous mass which conducts momentum perfectly, whether direct or lateral, can only move as one piece, and therefore cannot consist of component systems of mean and relative motion. (c) It thus appears that of the class of media that satisfy the conditions of a purely mechanical medium, neither the perfect fluid nor the perfect solid satisfies the condition of consisting of component systems of approxi- mately mean and relative motion. And as these are the only two continuous media in the class we have the conclusion : that no continuous medium can satisfy the condition of consisting of component systems of mean and relative motion. 74J COMPONENT SYSTEMS OF MEAN- AND RELATIVE-MOTION. 75 (d) If then the conditions for mean and relative systems are to be satisfied it can only be by discontinuous media. These all include perfectly conducting parts and are capable of separation into two classes according to whether or not these parts are or are not in such constraint with each other that each part is in complete constraint with the neighbouring parts ; lateral as well as normal. (e) In media in which the perfectly conducting parts are each in complete lateral as well as normal constraint with their neighbours, there can be no logarithmic rates of decrement. Whence, as in the case of a perfect fluid, such discontinuous media cannot generally consist of com- ponent systems of approximately mean and relative motion. It thus appears that no purely mechanical medium can satisfy the condi- tion of consisting of approximate systems of mean and relative motion unless it includes discontinuous perfectly conducting parts, each of which has certain degrees of freedom with its neighbours. (/) If, therefore, it could be shown that, as in the other purely mechanical media, these discontinuous media, with degrees of freedom, do not admit of logarithmic rates of decrement of the inequalities of relative motion, it would follow that component systems of approximately mean and relative motion are impossible. As it is, however, it can be shown that these discontinuous media, with or without perfect fluid occupying the interstices, as long as the perfectly conducting parts have any degrees of freedom with their neighbours, do admit of, and not only admit of, but entail, logarithmic rates of decrement of all inequalities of relative-momentum. This will be fully proved in the following sections. But it is sufficient at this stage to show how this comes about. (g) The actions between perfectly conducting masses are instantaneous finite exchanges of momentum in the direction of the common normal to the surfaces at contact. The direction of this normal has no necessary connection with the direction of the relative motion of the masses before contact ; therefore the direction of relative motion after contact has no necessary connection with the direction before contact. And thus the actions will be to render the path of the centre of each mass a rectilinear polygon in space, with angles which may be anything from to TT according to the freedoms. Such action entails that mean component, positive or negative, accelera- tion of the relative motion in any direction is not a resultant acceleration, but the mean of the component resultant impulses in all directions, thus 76 ON THE SUB-MECHANICS OF THE UNIVERSE. [75 securing continued angular redistribution in direction and magnitude of the relative momentum of each of the perfectly conducting masses ; so that any mean inequality in the relative motion is subjected to rates of decrement proportional to the inequality, and to the mean of the positive or negative components of relative velocity, divided by the scale of relative motion to a logarithmic rate of decrement. (h) The evidence furnished by the necessity of the maintenance of the scales of relative mass and relative motion has not been drawn upon in the foregoing reasoning, and therefore may now be brought forward as confirming the conclusion already arrived at; that the only media that satisfy the conditions of mean and relative component systems are those which involve discontinuous perfectly conducting parts, since such media are the only media in which limits to the scales of relative mass and relative motion are of necessity maintained. 75. Having thus arrived, for reasons shown, at the conclusions that the only purely mechanical media which can consist of component systems of approximately mean- and relative-motion are those which consist of perfectly conducting members which have certain degrees of independent movement, and that such media of necessity satisfy the condition of securing logarith- mic rates of decrement of all mean inequalities in the positive or negative components of relative-momentum in every direction, the further analysis may be confined to this class of media only. It is still a class of media and not a single medium. Such media may be distinguished according as the interstices between the grains are occupied by perfect fluid or are empty of mass. But this is by no means the only distinction. For the perfectly conducting members may have any shapes, and hence may include any possible kinematical arrangement or trains of mechanism, provided that there is always a certain amount of freedom or backlash, as it is called in mechanism; or they may consist of parts of any similar shape but of different sizes or of parts the same in size and shape, as for instance, spheres of equal size and mass. Nor is this all, for the relative extent of the freedom as compared with the size of the members may introduce fundamental distinctions in the properties of media consisting of similar members. 76. This last source of distinction, arising from the relative extent of the freedoms as compared with the dimensions of the grains, being perfectly general however the media may otherwise be distinguished, is a subject for general treatment, the outlines of which may with advantage be drawn at this stage from the evidence, already adduced, as to the conducting properties of the media consisting of component systems of approximately mean- and relative-motion. 77] COMPONENT SYSTEMS OF MEAN- AND RELATIVE-MOTION. 77 In this preliminary discussion of the effect of the extent of the freedoms, relative to the dimensions of the perfectly conducting members, the latter may be considered as being spherical grains of equal size and mass. In the first place it must be noticed that, so far, in this section, no account has been taken of any transformation of mass or of the displacement of momentum by conduction, so that the logarithmic rates of decrement by accelerations refer only to changes in the direction of the vis viva, leaving out of account the fact that there is displacement of momentum by con- duction at each encounter, and, thus, the reasoning, so far, does not touch on the possibility of redistribution of inequalities of rates of conduction of component momenta. It has, however, been shown that, owing to the fact that the directions of the normals at contact are independent of the directions of relative motion before contact, in a granular medium, there must exist rates of redistribution of all mean angular inequalities in vis viva of the components of relative motion, whatever may be the inequalities in rates of conduction of momentum in different directions. Thus far, then, for anything- that has been shown in the previous reason- ing, the actions which determine the rates of displacement of momentum by conduction may be independent of any effect of the independence of the direction of the normals at contact, and the direction of the relative motion of the grains before contact, which, as shown, secures angular dispersion of the momentum of relative motion. 77. In the simple case of uniform spherical grains, which may be conceived to be smooth, without rotation, whatever may be the relative paths of the grains as compared with their diameters, if the state of the relative-motion is without angular inequalities, since this state is maintained by the continual finite exchanges of momentum lateral to their paths, the mean component of the aggregate momentum in an interval of time, deter- mined by the time scale of relative motion, must be the same in all directions, as also must be the aggregate component paths traversed in a positive direction, and also those traversed in a negative direction. But it in nowise follows as a necessity of complete angular dispersion of components of momentum, within the limits of relative motion, that the mean length of the component paths traversed in one direction shall be the same as the mean of those in another direction. The clear apprehension of this fact is of extreme importance, when we come to consider the rates of displacement by conduction of momentum ; this is easily seen : If each grain traverses the same aggregate, positive and negative, com- ponent paths in the same time, but their mean component paths in one 78 ON THE SUB-MECHANICS OF THE UNIVERSE. [78 direction differ from those in another, since the paths are limited by en- counters, and the displacement, by conduction, of momentum in the direction of the component is the mean of the product of the diameter of the grain multiplied by the component of the relative momentum ; then, if the mean component conductions are the same in all directions, the number of the conductions in any direction must be inversely proportional to the component mean path in that direction. And thus the rate of displacement of momen- tum in any direction must be inversely proportional to the mean component path in any direction. 78. In order to secure that the rates of displacement of the momentum shall be approximately equal in all directions, it is not sufficient that there should be logarithmic rates of decrement of the mean inequalities of the relative components of momentum, positive or negative, but requires in addition that there should be logarithmic rates of decrement of mean inequalities in the mean component paths of the grains. The length of the path of a grain in any direction depends only on the positions of the surrounding grains ; and if the mean distance between the grains is such that the probable length will carry its centre through several surfaces set out by the centres of these other grains, then, since all possible arrangements of the grains would be probable, all directions of the normal at encounter would be equally probable, whatever might be the directions of the paths. And hence continual encounters would lead to such distribution of the grains that the probable length of the path would be equal in all directions; and, so, there would be logarithmic rates of decrement of inequalities in the lengths of the mean paths in different directions. 78 A. Evidence of the necessity of such logarithmic rates of decrement of inequalities in the arrangement of the mass is furnished by the equations of relative-mass; in a manner similar to that furnished by the equations of relative-motion as to the necessity of logarithmic decrement of the inequalities of vis viva. This at once appears from the equations of relative-mass (119), which may be expressed : d (p') (d (p'u) ) (d (pu') j. = \ -j h &c. f { ; f- &c. at ( ax J ( ax In this equation, according to the limits of approximation, the terms in the right member are such as have no mean values when integrated over the denned limits with respect to three independent variables. Therefore it does not follow as a geometrical consequence of the definition of relative mass that dp' ~dt 78 D] COMPONENT SYSTEMS OF MEAN- AND RELATIVE-MOTION. 79 should satisfy the condition of having no mean value, when integrated over definite limits with respect to any two independent variables, to the same degree of approximation as do the initial values of p ; and this applies both to the rates by convection and the rates by transformation. If then the conditions are to be continuously satisfied, it must be as the result of the redistributing actions on the rates of convection by the mean- velocity, which alone institutes inequalities. 78 B. Inequalities in the integrals of relative mass, over defined limits, with respect to any two independent variables, correspond to inequalities in the products and moments of relative mass. And it thus appears that these inequalities have no connection with inequalities in the mean-mass, which is a mean over all four variables. Therefore these inequalities are inequalities in the symmetry or angular arrangement of the relative mass. This significance of the inequalities becomes apparent on multiplying both members of the equation of relative mass by the square of any variable, as a?, or by the product of two variables, as yz, and taking the mean over all four variables ; as ft/ j a/ -\ . ait ( ote Then if a?p integrated over all four variables satisfies the conditions to any degree of approximation, the maintenance of the same degree of approxi- mation requires that ~dt should satisfy the identical conditions to the same degree of approximation. Hence we have the necessity, in order to maintain the inequalities steady, that, whatever may be the rate of institution, resulting from distor- tional mean motions, as expressed by the first term in the right member, the rate of rearrangement resulting from the transformation expressed by the second term must be such as exactly counteracts the rate of institution. 78 c. It thus appears, as in the case of Art. 72, that this condition of equality between the rates of institution and rearrangement can be satisfied only when the rate of rearrangement, as expressed by the second term, depends on, and is proportional to, the inequality instituted. 78 D. From this evidence it appears that the logarithmic rate of decre- ment of inequalities in the mean arrangement of the grains, which has been shown (Art. 7 8 A) to follow as the result of diffusion in granular media, is a necessity for the maintenance of systems of mean and relative motion. 80 ON THE SUB-MECHANICS OF THE UNIVERSE. [78 E And thus it appears that granular media may satisfy the condition of consisting of component-systems which are mean and relative in respect of conductions as well as convections. 78 E. It also appears, and perhaps this is of greater analytical import- ance, that the two rates of logarithmic decrement, that of inequalities of vis viva, and that of rearrangement of mean inequalities in the symmetry of the mean arrangement of the grains, which also secures the redistribution of angular inequalities in the rates of component conduction of momentum, are in a measure independent and are analytically distinct. 79. The inequalities in the mean symmetrical arrangement of the mass, although, being the most remote, they have presented the greatest difficulties to recognition and analytical separation, are of primary importance and distinguish between classes of granular media. It has been shown that logarithmic decrement of these inequalities results from diffusion among the grains. 79 A. It does not, however, follow that such logarithmic rates of decre- ment would exist when the grains were in such close order that no grain could break through the closed surface which might be drawn through the centres of its immediate neighbours. For then, whatever might be the order of arrangement of the grains, notwithstanding the existence of a certain extent of freedom, it could undergo no change. If in this last case the general state of the medium were such that the mean freedoms of each grain were equal in all directions, so that there were no inequalities in the mean component paths in different directions, the relative-motion would be in a state of mean equilibrium without inequalities and the rates of displacement, by conduction, would be equal in all directions. But if, from the last condition, the medium were subjected to a mean distortional strain, however small, the mean component paths of the grains would no longer be equal in all directions ; and the rates of displacement of the momentum, by conduction, would be no longer equal in all directions, but would be such as tended to reinstitute the former condition ; that is to say, the rearrangement of the grains within the limits of freedom would be such as to balance, not the external mean stresses by which the strains were brought about, but the stresses necessary to maintain the strain steady. And thus the logarithmic decrement would not be to a state in which the mean paths were equal in all directions, but to a state in which the in- equalities in the mean paths were such as to maintain the necessary inequalities in the rates of displacement, by conduction, to secure equili- brium under the external stresses. 80. It thus appears that, while the effect of relative accelerations to redistribute all mean inequalities, in the angular distribution of relative 85] COMPONENT SYSTEMS OF MEAN- AND RELATIVE-MOTION. 81 vis viva, is independent of any symmetry in the mean arrangement of the grains, and, hence, of mean angular inequalities in the mean component paths of the grains, and is therefore subject to no limits. Whatever the relative freedoms of the grains may be, the angular redistribution of in- equalities in the mean component paths depends solely on the rate of redistribution of the mean inequalities in the symmetry of the arrange- ment of the grains and is subject to limits depending on the relative lengths of the mean component paths of the grains, taken in all directions, as com- pared with the diameters of the grains. 81. It also appears that the definite limit, at which redistribution of the lengths of the mean paths ceases, is that state of relative freedoms which does not permit of the passage of the centre of any grain across the triangular plane surface set out by the centres of any three grains which are neighbours. This definite limiting condition obviously corresponds to that at which all diffusion of the grains amongst each other ceases. 82. It thus appears that there is a fundamental difference in media, otherwise similar, according to whether or not the freedoms are within or without this limit. This difference amounts to discontinuity in the media, for within the limit there will be no rearrangement of the grains however long a time may elapse or whatever the state of strain may be. While outside the limit, in however small a degree, any state of mean strain must ultimately be relaxed however long the time. 83. The time taken for such relaxation will in some way be a function of the degree in which the freedoms are without the limit of no diffusion which will range from infinity to zero, so that there are continuous degrada- tions in the properties of the media according to the degree in which the freedoms exceed the fundamental limit. 84. The independence of the redistribution of relative vis viva on this fundamental limit to redistribution of the arrangement of mass in media consisting of perfectly hard spheres, or of masses of any rigid shapes, does not appear to have formed a subject of study by those who have developed the kinetic theory of gases : so that however complete this development may be with respect to limited classes of granular media which have formed the subjects of this study, the methods employed can have been applicable only to those classes of media in which the extent of the relative freedoms has, in a large degree, been outside the fundamental limit of no diffusion. 85. It seems important that the limitation imposed, by the methods of analysis hitherto used in the kinetic theory, on the class of media to which R. 6 82 ON THE SUB-MECHANICS OF THE UNIVERSE. [86 that theory applies, should be distinctly pointed out here, before proceeding to the further analysis of the general theory. Otherwise confusion might arise in the mind of any reader acquainted with the conclusions already accepted as resulting from the kinetic theory, as to the reason why, after having arrived at the general conclusion that the only media which can consist of component systems of mean and relative motion belong to the class of granular media with some degree of freedom, which is also the class of media to which the kinetic theory has been applied, any further analysis should not simply follow the lines of the kinetic theory as hitherto developed ? This question having been anticipated by the answer which is given in the previous paragraph, in which it is shown that the general class of granular media is subject to fundamental differentiation according as the ratio of the mean paths of the grains to the dimensions of the grains is within certain limits ; and that hitherto the method of the kinetic theory has not been such as to take account of these limits, and is thus only applicable to media in which the relative paths are large as compared with the linear dimensions of the grains*. 86. Besides the fundamental limit of no diffusion there is also another fundamental limit, which appears as soon as a finite relation between the paths and the linear dimensions of the grains is contemplated. This limit is that to which the medium approaches as the paths of the grains approach zero. If the granular medium is in a steady condition, then if the relative vis viva is finite there will be some extent of freedom. But for any given vis viva the mean paths will depend on the rates of conduction or vice versa. Thus it is possible that the relative mean paths may be indefinitely small as compared with the diameters of the grains, and the rates of conduction indefinitely large. 87. It has been shown Art. 74 (a) that a granular medium, in which the grains are in such arrangement that each grain is in complete constraint by its neighbours, cannot consist of mean and relative systems of motion. While from the previous paragraph it appears that granular media in which there is finite relative-energy may approach within any approximation of the condition of complete constraint with their neighbours. 88. The conclusion, as stated at the end of the last paragraph, has a fundamental significance. It clears the way to the recognition of the definite geometrical distinction between the effects of redistribution in media, otherwise similar, in which the mean paths are respectively within and without the fundamental limit of no diffusion. * Phil. Mag. 1860, Vol. xix. p. 19, Vol. xx. p. 21. 89] COMPONENT SYSTEMS OF MEAN- AND RELATIVE-MOTION. 83 When there is no relative motion and each grain is in complete con- straint with its neighbours, if there is no mean motion, it follows, at once, that the directions of the normals, at the points of contact, to the surfaces of the grains, whatever these directions may be, are undergoing no change are fixed in space. If then, as shown in the last paragraph, granular media in which there is vis viva of relative-motion may approach indefinitely to the condition of complete constraint, it follows that in such media, when the mean paths are indefinitely small compared with the diameters of the grains, the directions of the normals at points of contact approximate indefinitely to certain definite directions fixed in space, that is, as long as there is no mean- motion. Thus we have the definite geometrical distinction, that as long as the mean paths are within the fundamental limit of no diffusion, and there is no mean-motion, the normals to the surfaces at encounters are within certain angles of directions fixed in space ; while if the mean paths are without these limits, in however small a degree, the normals continually change their directions so that, if sufficient time is allowed, all directions are equally probable. 89. While within the fundamental limit any one grain can only have contacts with a strictly limited number of other grains, in the case of Fig. l. uniform spherical grains, in regular symmetrical piling, the number of grains any grain can come in contact with is twelve, so that if there is no strain in the medium and the mean paths are indefinitely small, as compared with 62 84 ON THE SUB-MECHANICS OF THE UNIVERSE. [90 the diameter, there are twelve fixed normals in which this grain can have contact with other grains. The twelve normals radiate from the centre of the grain, and when the grains are in the regular formation each normal is in the same line with an opposite normal so that there are six fixed axes symmetrically situated in which encounters take place. And as the resultant accelerations are in the directions of the normals at encounter, these six directions of the normals are six axes of conduction of momentum. These axes pass through the twelve middle points in the edges of a cube circumscribing each grain, if there are no mean strains in the medium, and are thus symmetrically placed with respect to the three principal axes of the cube. This is shown in Fig. 1, p. 83. If, then, the rates of conduction across surfaces perpendicular to these six axes are equal, the momentum conducted being in the direction of the axes, the grains will, of necessity, be in mean equilibrium. This state of equilibrium in no way depends on the mean density of the relative vis viva of the grains. Therefore, in the limit, as the mean paths of the grains become indefinitely small, as compared with their diameters, as regards the direction of the rates of conduction, whatever the relative vis viva may be, the state will be the same. Thus, if there is no relative motion, but the grains are under stress, equal in all directions, by rates of conduction resulting from actions at the boundaries of the medium, the rates and directions of the resultant actions would be the same as if the rates of conduction resulted from the exchanges of momentum of relative-motion. 90. This limiting similarity between the states of media, one of which, having no system of relative motion, is purely kinematical, and cannot satisfy the conditions of consisting of mean and relative systems of motion, while the other, essentially, satisfies these conditions, has a fundamental significance, although (except by the recognition that in the one case the conduction results from mean actions at the boundaries of the medium, while in the other the conductions are between the moving grains) this significance in no way appears as long as there are no mean strains in the media. If these media are subject to any indefinitely small distortional strains the discontinuity between them, as classes of media, appears. In the case of kinematical media without mean strain, the stresses being equal in all directions and finite, no strain will result from indefinitely small stresses, nor will any strain result until the mean distortional stresses arrive at the same order as the mean stress equal in all directions. Thus if p 92] COMPONENT SYSTEMS OF MEAN- AND RELATIVE-MOTION. 85 represents the stress, equal in all directions, and^?^ p is the normal stress imposed in the direction in which x is measured, the stress in the direction at right angles remaining equal to p (and not affected by the strain), there will be no strain until p xx is greater than 2p. Whence it follows that any distortional strain is attended by an increase of mean volume occupied by the medium equal to the contraction in the direction in which x is measured, since there is no work spent in resilience, or in accelerations of relative vis viva. Thus the kinematical medium has absolute stability up to certain limits*. 91. On the other hand, the granular medium with relative motion, however small may be the mean paths, when subject to no distortional strain, and to indefinitely small distortional stresses, yields in proportion to the stress so that such stress is equal to the strain multiplied by a coefficient which is constant if the terms involving the square and higher powers of the strain are neglected; and this medium has the character of a perfectly elastic solid for indefinitely small strains. It has therefore no finite absolute stability, arid no dilatation as long as the squares of the strains are indefinitely small. As the strains increase, however, dilatation ensues, as expressed by the terms involving the squares and higher powers of the strains. Thus, although for small strains the two media are fundamentally different, as the strains become larger the conditions of the two classes of media approximate towards similarity, as regards the relation between stresses and strains; and thus the door opened to mechanical analysis by the recognition and analytical study of the property of dilatancy, as belonging to all media consisting of rigid discontinuous members, is not closed to the analysis of systems of mean and relative motion. So far from this being the case, the recognition of the coexistence of relative motion, by easing off the condition of absolute stability, belonging to the purely kine- matical system, supplying as it were kinetic cushions at the corners, has removed difficulties which otherwise rendered analysis impossible. 92. The primary conclusion arrived at in this section, that the only media which, as purely mechanical resultant systems, can consist of com- ponent systems of mean and relative motion, are those which consist of discontinuous perfectly conducting members with some degree of freedom, while limiting, as already pointed out, the scope of the subsequent analysis necessary for the definite expression of the several rates of action resulting from convections in such media, also indicates the methods by which this analysis may be accomplished. * Phil. Mag. Dec. 1885, "On the Dilatancy of Media composed of Rigid Particles in Contact." 86 ON THE SUB-MECHANICS OF THE UNIVERSE. [92 Given the mean actions across the boundaries of any portion of the medium, the mean action of the grains enclosed is, at any instant, a mean function of the generalised ordinates which define the shapes, positions and dimensions of the members, the intervals of freedom, number of grains in unit volume, their velocities and their directions of motion. Thus the method of analysis is to express the several probable mean rates of action, resulting from convection and conduction, in terms of the mean vis viva of relative velocity, the mean component-paths and mean paths, their number, mean-mass, and any other generalised mean ordinates that the shapes of the grains may entail. Then these expressions may be substituted in the members on the right of the equations, Section VI., since these include general expressions for the several actions. The method thus indicated constitutes a general extension, or completion, of the method employed in the kinetic theory of gases. SECTION VIII. THE CONDUCTING PROPERTIES OF THE ABSOLUTELY RIGID GRANULE, ULTIMATE- ATOM OR PRIMORDIAN. 93. ALTHOUGH the absolutely rigid atom is as old as any conception in physical philosophy, the properties attributed to it are outside any experience derived from the properties of matter. In this respect, the perfect atom is in the same position, though in a different way, as that other physical conception the perfect fluid. Both of these conceptions represent conditions to which matter, in one or other of its modes, apparently approximates, but to which, the results of all researches show, it can never attain, although this experience shows that there is still something beyond. The analysis of the properties of conducting momentum, which must belong to the perfect atom considered as of uniform finite density, is obtained from the principle of conduction defined in Art. 72, Section VII.; from which it appears that it must conduct in all directions at an infinite rate, or that it must be capable of sustaining stress of infinite intensity, tension, com- pression or shearing; while it is shown that the property of conducting negative momentum in a positive direction or vice versd requires that the momentum and the conduction shall be imaginary. In the case of matter (rigid bodies) these imaginary stresses and rates of conduction are held to imply rates of actual conduction, round the outside of the bodies, in the medium of the ether. A conclusion confirmed in the case of matter by the existence of limits to the intensities of these stresses. Such outside conduction is at variance with the conception of fundamental atoms outside of which there is no conducting medium and which atoms do not possess the properties of changing their shapes or of separating into parts. It becomes clear therefore that any fundamental atom must be con- sidered as something outside of another order than material bodies, the properties of which are not to be considered as a consequence of the laws of motion and conservation of energy in the medium but as the prime cause of these laws. 88 ON THE SUB-MECHANICS OF THE UNIVERSE. [94 94. If, for the sake of simplicity, the medium consist of closed spherical surfaces of equal radii this definite expression of the law it will be seen that it is confined to direction only and would apply equally to cases where in some directions the grains were making short paths and in others long paths, as well as to that in which the mean paths are equal in all directions. Q. E. D. 97. The distribution of mean and relative velocities of paws of grains. In Proposition V. of the same paper Maxwell extended the law of probable distribution of vis viva to the distribution of the relative vis viva of all pairs of grains. He does not seem, however, to have further extended it to that of the mean motions of the pairs; which is remarkable as it appears to follow directly from his method and would have saved him much subsequent trouble. These extensions do not in the least involve the arrangement of the grains. It is however convenient to introduce the demonstration of the law of distribution of the mean-velocities here, for the purpose of reference, and it is simpler to demonstrate both at the same time. Taking x, y, z as the components of the mean-velocity of a pair of grains and x', y', z as the relative components of the same pair, and cc ly y lt z 1} # 2 , y-it ^2 as the components of the individual motions, we have = Z-z. Then for the numbers of grains for which x is between ac^ and #j + B^ , y l between y and y l + By 1} z between z^ and z + Bz 1} and ac 2 is between # 2 and # 2 + Bas 2 , &c., &c. N, t(x+at)* , (y+yQ* g+m , \ n x = \,e \ rf a* ' a* / dxdydz .iVo _JV:i_.. -_j-i fLL-i-^z. Z-i-Y -, . -. i -, i \ n 2 = ,,\*e \ ** * > dx dy dz o. (TT)* / The first of these equations expresses the probable number of grains having mean- velocities between x and x + Bx, &c., &c., for any particular value of x', the relative-velocity, &c., &c. And the second equation in the same way expresses the number of grains having relative-velocities between x and x' + Bx', &c., &c., for any value of x, &c., &c. Whence the probability of the double event is expressed by the product T TIT 1 'dydy'dzdz (132). 92 ON THE SUB-MECHANICS OF THE UNIVERSE. [97 Then if r = OP + y* + 2? and r' = x' z + y' 2 + / 2 , the number of pairs having mean-velocities between r and r+ Brand relative velocities between r' and r + Br is w i W 2 = -TZT e~ 2(72+r/2) dxdydzdx'dy'dz' (133). These admit of integration either with respect to x, y, z, or x', y', z . Thus integrating x, y, ~z from x = oo to # = oo we find -tSu^ d *' d ^' <"*> for the whole number of pairs whose components of relative velocities are between x' and x' + Bx', y' and y' + By', z' and z' + Bz'. And integrating for r' instead of r we find 7^^ e~~tfdxdydz ...(135) (/V2) 3 (7T)* for the number of pairs whose mean components of velocity are between x and x + Sx', &c., &c. These may be expressed in a more convenient form by substituting r 3 d cos 0d<}> for dx, dy, dz. And applying this to the three expressions for the number of grains having velocities between r and r + Br, of pairs having relative-velocities between V2r and V2 (r + Br), of pairs having mean velocities between r/V2 and (r + Sr)/V2, since ^V is the number of grains in unit volume and N(N 1) is the number of pairs of grains, AT4/V12 _f i *Br = rh (136), (137), (138> (a/V2) s VTT Q. E. D. The first and second of these laws of angular distribution of vis viva are the same as those given by Maxwell ; and the third, that for the distribution of the mean vis viva of pairs of grains, leads to the same results as Maxwell arrived at in a different manner. Together they constitute the principal means of giving definite quantitative expression to the results of the analysis of the actions in a granular medium. And it is important to notice that they are derived from the probable independence of the preceding and antecedent 98] DISTRIBUTION OF VELOCITIES OF MEMBERS OF GRANULAR MEDIA. 93 directions of the relative velocities of a pair of grains before and after encounter under conditions in which the mean density and constitution of the medium remain unaltered. In Proposition VI. Maxwell has shown the rates at which the several members of the medium exchange vis viva, using arbitrary constants. And in his Proposition VII. he proceeds to the demonstration of the probable length of the path of a grain in terms of N, the number of grains in unit volume, s the diameter of a grain, and v the velocity. He has first shown that if r is the relative velocity of a particle with respect to N particles in unit volume, this particle will approach within the distance s of JVvrrs 2 particles in a unit of time. Thus in Propositions VIII. and IX. he determines the number of pairs moving according to the laws expressed in equations (137) and (138) which will undergo encounters in a unit of time, and in Proposition X. determines the mean path of a particle to be In this result there are two things to be noticed. In the first place the ?rs 2 in the denominator represents the area of the target exposed to the centre of a spherical grain by another grain in the direction of their relative motion; while the \/2 is merely the ratio of the mean relative velocity of the pair to the mean velocity of either grain, equations (136), (137). It is thus seen that, although the dimensions of the grain are, perforce, taken into account as determining the probability of an encounter, no account is taken of the third dimension of the grain in diminishing the actual distance the centres of the grains would travel between encounters. Hence Maxwell's mean path I can only be an approxi- mation when his s is small with respect to his I. The second point to be noticed in Maxwell's deduction of the mean path is that he has tacitly assumed I to be the same in all directions. And has thus assumed not only that the density is constant, which is assumed in the determination of his laws of distribution of vis viva, but also that the arrange- ments of the particles must be such that the mean chance of encounter is equal in all directions, a condition which does not enter into the laws of distribution of vis viva, and consequently limits the application of this mean path to conditions of the medium such that all directions afford equal chance of encounter. A condition which is obviously approximated to as the actual density becomes small compared with the maximum density, when each particle is in continuous contact with twelve neighbours. 98. In pointing out the limits to the application of Maxwell's analysis of the action in a medium of hard elastic spheres, my chief object has been to 94 ON THE SUB-MECHANICS OF THE UNIVERSE. [98 direct attention to those extensions and modifications which are necessary to render the analysis general, and thus to present a clear idea as to how far Maxwell's method may be applied. At the same time it seemed very desirable to show clearly, that in extending the analysis to include conditions of the medium to which Maxwell had not applied his method, there is nothing at variance with the results he had obtained under the condition to which his application of this method extended. Maxwell's laws of the probable distribution of vis viva, and mass, extended to include the mean vis viva of pairs of grains, are, as already pointed out, perfectly general. But it is necessary to obtain expressions in terms of the quantities which define the relative motions of the medium for the rates at which the actions of conduction through the grains displace momenta and vis viva of relative motion, which expressions shall, if possible, be as general as the law of distri- bution of vis viva. In the media considered by Maxwell the distances between the grains are assumed to be large compared with the dimensions of the grains. Whereas in the general theory it is fundamental that cases should be considered in which the distances between the centres of the grains, which are neighbours, approach indefinitely near to the linear dimensions of the grains. Such consideration involves methods of analysis by which the several effects of the action between the grains may be defined whatever may be the relation between a- the diameters of the grains and X their mean path. In the first instance the consideration of these rates is confined to states of the media in which, whatever may be the density as compared Avith the possible density, the arrangements of the grains, however varying, are such that the mean actions in every direction are similar and equal ; the medium being everywhere in mean equilibrium. And afterwards to proceed to the effects of inequalities both angular and linear. SECTION X. EXTENSION OF THE KINETIC THEORY TO INCLUDE PROBABLE RATES OF CONDUCTION THROUGH THE GRAINS, WHEN THE MEDIUM IS IN ULTIMATE CONDITION AND IS UNDER NO MEAN STRAIN. 99. THE mean rates of convection and conduction of momentum, ex- pressed in equations (120) by p xx , p yx , &c., and p" (uu')", p" (v'u)" , &c., admit of expression as where p = % (p xx +p yy -h p a ), p' (v'v)" = p" (u'u' + v'v + w'w')" and in this case p and ^p"(v'v')" represent the mean action, equal in all directions, while p xx p, p" (u'u')" ^p"(v'v')" &c., p yx , &c. and p"(v'u)" repre- sent inequalities. In this first extension of the kinetic theory the object is to express the actions indicated by p and p"(v'v)" only, assuming that the inequalities are zero, in terms of the quantities which define the condition of the medium. 100. To determine the mean path of a grain, The mean path of a grain expressed by X is the distance traversed by its centre between encounters, which is not the component in the direction of its motion, of its distance between the points at which the two actual contacts, which limit the path, have occurred, although it approximates to this as X/o- becomes large. Maxwell has shown that neglecting o-/\ the mean path of a grain and the relative path of a pair of grains are expressed by \ = -=^- - and V2X = -=. .................. (139) * respectively, while both of these are obtained from .............................. (140), 96 ON THE SUB-MECHANICS OF THE UNIVERSE. [101 where N expresses the number of grains in unit volume ; so that either member represents the mean volume maintained free from other grains by the kinetic action of each grain. In this estimate however no account is taken of the striking distance, of the centres of the pair of grains, from the plane, normal to their relative paths before contact, through the point of contact, so that the centres of both grains are assumed to be in this plane at the instant of contact. When X/o- is large we have all positions of the projection, in the direction of relative motion of the striking grains, over the disc 7r be any small area on the surface of the sphere taken so that its mean position is at the point in which the diameter in the direction of the normal meets the surface of the sphere. Then by the law of probability of the striking distance it follows that, at a chance encounter, the probability of the normal meeting the surface in co is Q) COS X (1 A! COS % + &C.) ~~ 105] EXTENSION OF THE KINETIC THEORY. 99 or multiplying this probability by the product of the normal component of the relative velocity ^F/eoe^, and again by a, the normal displacement, integrating over the hemisphere for all values of ^, and dividing this integral by the integral of the probability of an encounter on , &c., which when integrated over the surface of a hemisphere are zero, if all directions of relative motion are equally pro- bable, but have values in a medium with linear inequalities when the axes of reference are other than the principal axes of the inequalities. It is therefore necessary to obtain their integral values over the several groups of pairs having relative velocities in directions in which the sign of the component displacement is the same as that of the component of normal velocity, as * 1 ?/(T) V2 F/ f f 2 cos 6 sin cos < sin eded$ , 3V 7 \\J^ Jo Jo y _4 oJo which multiplied by the mass and the number of collisions and taking the mean is so that to each of these groups of pairs there is a corresponding group for which the normal components of mean-relative motions are of opposite sign, the mean taken over the two groups or over the whole unit sphere is zero ; so that in a medium without linear inequalities p w 7/ = 0, &c, &c ............................ (162). 104 ON THE SUB-MECHANICS OF THE UNIVERSE. [113 113. The mean rate of convection of components of momentum in the direction x by grains having velocities F/, for which all directions are equally probable, is expressed by /JT y 2irp \Vi -~ cos 2 6 sin Odd v/2 Jo A. r i 2-777? I si J o sin dd6 ^- (163), which becomes, taking Maxwell's expression for the mean value of v 2 from to oo , (a 2 . f), when multiplied by the product of the mass into the number of grains, And for the mean rate of momentum conveyed in the direction of the momentum p" = p'(U'Uy',&c.,&xs ...................... (165). For the lateral convections of momentum the expression is ir IT I rz rz (F/F/)" />( I A. cos#sin 2 0c#sin, &c. have the same signs, positive or negative. The groups in which the corre- sponding signs are opposite have integrals with the opposite signs negative or positive, so that for the complete integrals p"(V'W')" = Q, &c., &c (167). 114. The total rates of displacement of mean-momentum in a uniform medium. Adding the expressions for the rates of conduction and convection in the respective members of equations (159) and (165), also (162) and (167), we obtain for the whole rates of displacement of the components of momentum PXX" + P" (U'U'y = p" \~L+- --^f(-}> (U'uy, &c. &c. /-i/? Q \ ( o v2A, \A./j \- ...(lOo^. 2V+p"(tf'n"=o, & c . & c . j 117] EXTENSION OF THE KINETIC THEORY. 105 115. The number of collisions which occur between pairs of grains having mean velocities between F//V2 and ( F/ + d F/)/\/2. Since the mean distance traversed between changes of a pair of grains, irrespective of mean velocity, is A A/2, the mean time of a pair of grains having mean velocity F//V2 in traversing their mean path is \JV. And since the number of pairs of grains in unit volume having mean velocities between F//A/2 and ( F/ + d VV)/\/2 is n(n 1), and each of these pairs changes F/A times in a unit of time, the total number of changes of these mean paths is V n(N-\Y-. And since there are 2 (N 1) changes for each collision the number of collisions of the n (n 1) pairs of grains in unit volume in unit time is which integrated gives the total number of collisions a/\/7r . A. 116. The mean velocities of pairs having relative velocities V2F/ and F//V2. Since the time of existence of a pair between changes, whatever the mean and relative velocity, is the time of existence of both the mean and relative velocities between changes, and the mean ratio of the mean and relative paths between changes is that of l/\/2 to \/2 or 1 to 2, it follows that the mean ratio of the mean and relative velocities is 1 to 2. And hence the mean velocity of all pairs having relative velocities between A/2 F/ and V2 ( V t ' + d F/) is between F//V2 and ( F/ + d FOA/2. Q. E. D. 117. All directions of mean velocity of a pair are equally probable what- ever the direction of the mean velocity. This follows directly from the expression for the number of pairs having particular mean and relative velocities r , - -^ - . e~^ . dr,d (cos r - - . e 2 . dr z d (cos r x being the mean velocity, r 2 the relative velocity and 0^, #2^2 having reference to the angular positions of r v and r 2 . For, taking r^r^ and r 2 8r 2 constant, and ascribing any particular values to # 2 < 2 and S# 2 S$ 2 , the number of pairs, having a mean velocity V l in 106 ON THE SUB-MECHANICS OF THE UNIVERSE. [118 directions such that, referred to the centre of a sphere of unit radius, they meet the spherical surface element dcos Oidfa, is to the total number which meet the sphere as d cos d^dfa is to 4?r. Q. E. D. 118. The probable component of mean velocity of a pair having relative velocity r 2 = \/2 V t in the direction of the normal at encounter. Since r^ = r z /2 and r 2 = V2 F/, r x = F//\/2. In all directions the probable component value is 119. The probable mean transmission of vis viva at an encounter in the direction of the normal. When two equal spheres encounter, the displacement of energy by conduction of momentum is the product of the displacement a- multiplied by twice the product of the components of the mean velocity and relative velocity of a pair in the direction of the normal. Therefore since the probable component of mean velocity in the direction of the normal (last article) is F//2 \/2, and the probable component of the relative velocity as obtained by dividing out the a in equation (147) is 2 *J2 .f(a-/\).V 1 /3, the probable displacement of vis viva in the direction of the normal is If I, m, n are the directions of the normal referred to fixed axes, the component displacements of the vis viva of components parallel to the axes are 120. The mean distance through which the actual vis viva of a pair of grains having relative velocities between \/2 F/ and \/2 ( F/ + 8 F/) is dis- placed at a mean collision. Since the mean velocities of pairs of grains having relative velocity v'2 F/ is F//-V/2 and the actual vis viva of such a pair is we have for the displacement of the total vis viva of a pair of grains And since the displacement of vis viva by convection by a grain having velocities between F/ and F/ + 8F/ between encounters is XF^ and there 123] EXTENSION OF THE KINETIC THEORY. 107 are, in unit time, twice as many mean paths traversed as there are collisions, the relative rates of displacement of vis viva by convection and conduction are as X to cr.y( is X cos 6, and multiplying by the number of mean paths traversed by each of such grains in a unit of time we have IV x X cos 6n . ~- sin 6 . dO . d TO T/ , 2 d sin 2 6.d0.d 47T Then integrating from 6 = to 6 7r/2 and < = to = ?r/2 and from 7 = to F/ = oo / ^I'l * "^ /i tr>\ T-= T -J- (I'O), Jo 4 4 V"" 126] EXTENSION OF THE KINETIC THEORY. 109 and taking account of the mass of a grain a is the flux of mass by the grains for which cos is positive, &c., &c. 126. The extension of the kinetic theory has thus been carried as far as to include the expression of the rate of flux of momentum, vis viva, and mass, by conduction, as well as by convection, in the ultimate state of the medium without mean strain. Q. E. D. It is to be noticed that the analysis effected in this section does not complete the extensions which are desirable, and possible, as these include the extension for the expression of the rates of conduction as well as con- vection, when the medium is subject to mean uniformly varying conditions though still in equilibrium. These form the subject of Section XII. so that their consideration may follow the consideration of the logarithmic rates of redistribution of angular inequalities resulting from the varying condition of the medium on which they depend. SECTION XL REDISTRIBUTION OF ANGULAR INEQUALITIES IN THE RELATIVE SYSTEM. 127. WHEN a granular medium, however uniform and symmetrical its mean initial condition, passes from a state of equilibrium and mean rest into a state in which there are mean rates of strain, there follow, as a consequence, rates of establishment of inequalities in the mean distribution in the relative system, which are expressed by the rates of transformation from mean to relative motion, as in the last term in equations (116) and (117) and in (116 A) and (117 A). The general analysis of the effects of the mean motion on the relative motion for granular media comes later in the research * ; and it is sufficient here to have pointed out the general source of such inequalities, as in this section we are not concerned with the source except in as far as it may be an assistance in realizing the general distinction between the two classes of inequalities. Thus the inequalities which are called into existence by rates of strain partake of the characteristics of the rates of strain. Local volumetric rates of strain, which cause the density to vary from point to point, institute what will here be called linear inequalities, while uniform distortional rates of strain institute what will here be called angular inequalities. The inequalities so instituted, owing to the activity of the relative- motion, are subjected to rates of redistribution proportional to their magni- tudes, and it is the determination of these rates in terms of the constants which define the condition of the medium that constitutes the purpose of this section and the next. These two rates of redistribution, like the volumetric and distortional strains, are analytically distinguishable as belonging to different classes of mean actions. The rates of angular redistribution have the characteristics of production at a point. Their integrals are not surface integrals, and they are included in the expression for angular redistribution in the fourth term, equation (117 A). * Section XIII. 129] REDISTRIBUTION OF ANGULAR INEQUALITIES. Ill The rates of linear redistribution, on the other hand, have the character- istics of a flux. Their integrals are surface integrals, and they are included in the expressions for the linear rates of distribution in the second and third terms, equation (117 A). It thus appears that these rates require separate treatment, and as the analysis for the linear rate depends, to some extent, on the angular rate, the angular rate is taken first as the subject for this section, and the linear for the subject of the next, Section XII. 128. Logarithmic rates of angular redistribution by conduction through the grains as well as by convection by the grains. The necessity of logarithmic rates of angular redistribution in the mean angular inequalities in the vis viva of relative-motion, and of inequalities in the symmetry of the mean arrangement of the grains, for the maintenance of approximately mean- and relative-motion has already been proved in Section VII. ; and the actions on which these rates depend have undergone considerable qualitative analysis (to use a chemical expression) in the same section. What is necessary, therefore, in this section is the application of the definite, or quantitative, analysis for the definition of these rates. The first step in this direction is the definite consideration, in the concrete, of the instantaneous effects of encounters between hard spherical grains of equal mass and dimensions. For this purpose use is here made of the conceptions and the method given by Rankine in his paper "On the Outlines of the Science of Energetics*," a remarkable paper, which seems to have received but little notice. 129. In a purely mechanical medium, since any variation of any com- ponent-velocity of a point in mass can only result from some action of exchange of density of energy with other points in mass, there are always masses engaged in such an exchange. Considering these to include all the mass through which the exchange extends (as between some particular portion of the medium and all the rest) the sum of the energies of the components of motion, in any particular direction that of x immediately before the exchange is the active accident, or the " effort," of the component energy to vary itself, by conversion into some other mode, which, in a purely mechanical system, considered as a resultant system, can only be energy of component motion in some directions y and z at right angles to x. The energy so converted into directions y and z is called the "passive accident." And in the same way the sum of the energies in the directions y and z, antecedent to the action, is the active accident or the effort of these energies to vary the energy in the direction x. * Proc. of the Phil. Soc. Glasgow, Vol. in. No. 1 ; Bankine's Scientific Papers, p. 209. 112 ON THE SUB-MECHANICS OF THE UNIVERSE. [130 It is at once apparent that the result of such accident is, taking account of the dimensions of the grains, to produce three instantaneous effects, while, if the dimensions of the grains are neglected as being small (as has been the case in the kinetic theory), only one of these effects is recognised as the result of the exchanges of energy on the instant. And although this one effect has been taken into account in the kinetic theory its position in that theory has not been generally denned, nor has it been made the subject of separate expression in the equations. The first, and hitherto the only, published mention it has received as a specific effect occurs in Arts. 20 and 21 of my paper " On the Theory of Viscous Fluids *," where reference is made to the " angular redistribution of relative-mean motion." It was not however till some time afterwards that I was able to distin- guish, geometrically, the circumstances on which the existence of angular redistribution of relative motion depend, and obtain separate expressions for their effect. It is included in those terms in equations (47 A), Section III. of this research, which are not surface integrals, although not specifically expressed, being associated with the resilience-effects in these equations for a resultant system ; the specific expressions for the separate effects for a resultant system are however effected in equations (47 A). The instantaneous action of which this angular redistribution is the effect turns out to be the only instantaneous action on the energy of the relative motions of the mass or densities of masses engaged other than the effects on resilience ; so that, when the masses engaged are two equal hard spheres, angular dispersion of the energy of their relative velocities, that is, of their velocities relative to their mean position, is the only instantaneous effect on this relative energy. This theorem may be easily proved. 130. When two hard spheres encounter, their relative-velocities are in the same direction, and their momenta, relative to axes moving with their mean-velocity, are equal and opposite. Suppose the axis of x to be the direction of relative motion. Then at encounter the grains exchange components of momenta in directions of the line of centres, and thus the relative component momentum of each sphere in the direction of the line of centres is reversed ; so that if the line of centres does not coincide in direction with the lines of relative motion, the instantaneous effect (1) of conduction is exchange of energy of component motion from the direction x to those of y and z at right angles to x. This is angular redistribution of the energies of component motion, and is the only change of the energies of the relative motions, measured from the moving axes. For as the relative * Eoyal Soc. Phil. Trans., Vol. 186 (1895) A, pp. 1467. 133] REDISTRIBUTION OF ANGULAR INEQUALITIES. 113 momenta in direction of the line of centres of the respective grains are reversed at the instant there is no change in the position of their energies ; so that at the instant there is no linear displacement of the energy of the relative motions. Q. E. D. 131. The other fundamental effects of the action between the grains those which have been neglected in the kinetic theory are (2) the dis- placement of momentum which results when two spheres encounter, having components of actual momentum (referred to fixed axes), in the direction of the line of centres, which differ in magnitude, causing the instant displace- ment of the difference of the component momenta, in the direction of the line of centres, through a distance v 1 , w lt u 2 , v 2 , w 2 for the antecedent velocities of the two grains, and U lt V lt W ly U z> V z , W 2 , for the subsequent velocities, it follows as a direct result of the.exchange of the components of motion in the direction of the normal that at a single encounter, 1 + [/" 2 2 - ui> - w 2 2 = - 2 (m 2 + ft 2 ) l z (u 2 - Ujf '' (v z - v^ + ft 2 (w z - &c.,&c. ...(177). + 2 (21 - 1) [hn (^ - tO (v a - v,} + nl Then, since for any two spheres with particular relative motion, u^ u^, v 2 v 1 ,w 2 w 1 , the probability of their normal, at the point of contact, having a direction within any small area, sin 0ddd cos ^ ~^~ ~' where % is the angle between two radii, one meeting the surface of the unit sphere in the direction of the point of contact, and the other in the direction of the relative motion, drawn so that ^ is an acute angle, so that % is always between zero and Tr/2. 142. The active and passive accidents. In considering the action resulting from conduction of momentum of two spheres at a single encounter, the problem is greatly simplified by taking the direction of one of the axes of reference to be that of the relative motion of the spheres ; while, as will be seen, it does not lose in generality. Taking ^ to be measured in the direction of the relative motion, v 2 v 1} w 2 w l are each zero, and putting i (MI + O 2 + i (M, - wO 2 for ?V + w 2 2 , &c., &c. in equation (177) we have US + U,? - 1 ( Ul + utf - ( Ma - O 2 = - 2 (m 2 + w 2 ) l*(u> - uj* + + ..(178), in which the ciphers represent the values of the terms having factors (v a - Vj) and (w 2 Wj). Multiplying these equations by the factor of probable positions of the normal and integrating over the sphere of unit radius, since cos % is positive UNIVERSITY or s_ _ - 144] REDISTRIBUTION OF ANGULAR INEQUALITIES. 119 and equal to + cos 0=l, the equations become on transposing the last terms in the left members _ (179), where, since the square of the relative motion, (w 2 u^ 2 , is double the sum of the squares of differences between the actual component motions and the mean component motions, ! + W 2 \ 2 1 MI + 2 /! + j. + tt, - + K- ...(180). 2 ; ' v 2 The left members of equations (178) express, respectively, the effects, both active and passive, of the accidents on the energies of the components of motion in the directions of x, y, z respectively. The first terms in the right members, which are all negative, or zero, express the effects of the active accidents on the energies in these directions respec- tively, while the last two terms, which are positive, or zero, express the effects of the passive accidents in these directions. Q.E.D. 143. The active accidents are work spent by the efforts produced by respectively, in other directions than those of a, y, z v respectively. Thus the effort in the direction of the normal caused by w 2 MI is 21 (u 2 u-i) and the component of the relative velocity u^ ^ in the direction of the normal is I(u 2 Wj); so that the total result of this effort is 2 2 (w 2 Wj) 2 , work spent by energy in direction of x. Of this 2 4 (w 2 %i) 2 i g work returned to the energy in direction of #; so that the portion of the energy in the direction x expended in (passive accidents) changing the energy in directions of y and z is 2 (/ 2 1) I 2 (u 2 - u^, and the passive accidents in the directions of y and z are 2 2 m 2 (u z w^ 2 , 2 2 n 2 (w 2 u^f respectively. 144. The angular dispersion of relative motion. The equations (180) show that considering the chance encounter between two grains, whatever their relative-motion before encounter, all directions of the subsequent relative-motion are equally probable. So that any angular inequality in their relative-motion is virtually extinguished after a single 120 ON THE SUB-MECHANICS OF THE UNIVERSE. [145 encounter; although if the pair have any mean-motion, whatever it may be, the inequality in this remains as before encounter. Q. E. D. 145. The mean angular inequalities. Before we can pass from dispersion of the component relative-velocities of a pair of grains to that of the mean-inequalities of all the grains the demon- strations of several propositions become necessary. For reasons, which will appear, we have here to consider only such mean angular inequalities as are introduced in the relative motion of the medium while the mean system is undergoing mean rates of strain. These inequalities, as Maxwell has shown, for a medium consisting of equal hard spheres, are expressed by, taking N for the number of grains in unit volume, dxdydz where a 2 , f&, 7 2 are double the mean of squares of the respective component velocities. Since the differences between a 2 , /3 2 , / "^"Ti / a 2 (l+2a).vV e dXldx ' /Y> Then, since x l + may have any value from oo to + oo for any value 41 of x', integrating for x l between these limits for any particular value of x, the number of pairs which have component relative-velocities, in direction as, between x' and x' + Sx' is : V2a(l + a)V7r In exactly the same way it is shown that the numbers of component relative-velocities between y' and 3/4- By* and between z' and /+/ are respectively ]\Tl _5_(i_2c) IV *" 2 dz'. V2a(l Multiplying these expressions by x' 2 , y' 2 , z'* respectively and integrating from oo to + oo , and dividing by JV 2 , we have for the mean-squares of the respective components, in the directions x, y, z 2a 2 (1 + 2a), 2a 2 (1 + 26), 2a 2 (1 + 2c), which have precisely the same coefficient of angular inequalities as the mean squares of the components of the actual velocities obtained from equations (181) a 2 (l + 2a), a 2 (1 + 26), a 2 (l + 2c). Q.E.D. 147. The mean squares of the components of relative-motion of all pairs are double the mean squares of the components of actual motion. In the last paragraph of the last article it has been shown that the mean squares of the components of relative-motion of all pairs including the inequalities are double the mean squares of the components of the actual motion, so that no further demonstration is necessary. 148. The rate of angular redistribution of the mean inequalities in the actual motion is the same as the rate of redistribution of the angular inequalities in the relative motion of all pairs. This follows at once from the inequality of the coefficients of inequalities which has already been proved. 122 ON THE SUB-MECHANICS OF THE UNIVERSE. [149 149. The rate of angular dispersion of the mean inequalities in vis viva. It has been shown, equations (180), that the angular inequality in the squares of the relative velocities of any pair of grains is virtually extinguished at a single encounter. From this it follows that the virtual inequality in the motion of any grain exists only from the time of the institution of the inequality to the time of its next encounter. This time is expressed by ^L TV Fj being the actual velocity of the grain, and Xj the distance traversed before encounter. This distance Xj may be anything from to oo . But it is proved by Maxwell to be independent of V and to have a probable mean value, neglecting cr as compared with X, of Taking a- into account, as will be shown, the probable value of A, becomes The probable path being X, the probable time of any grain with velocity V, is A TV It thus appears that, although the mean relative distance traversed between encounters by pairs of grains having the same relative velocities Fj is independent of F a , the mean time between encounters varies inversely as Fj. In order therefore to obtain the probable mean time of existence of inequalities in the angular distribution of the vis viva, it is not sufficient to find the probable value of the mean time ^- , for all values of F 1} since this ^i would only be the probable mean time between encounters during which the inequalities in the mean velocity are sustained. 150. The mean time of mean inequalities of vis viva. The direction of motion of each grain is the direction of its path ; so that if I, m, n are the direction -cosines of the motion, the probable times of the continuance of the components of motion in directions x, y y z are \l \m \n V\l' V\m' V\n^' 150] REDISTRIBUTION OF ANGULAR INEQUALITIES. 123 and since the chance of a collision in a unit of time is "Pi/A, the probability of continued existence is and the probability of continuing for a time ._n 1 \ = TT is e~ n > . Whence it follows that, taking account of all the pairs of grains at different relative velocities, but moving nearly in the same directions, the times for which their continuance is equally probable are , _WiM _ n 2 \l "\ -rr j > kg -ir- -I ) W'^ ....................... V" 10 "/' so that, multiplying Vfl 2 , V 2 H 2 , &c. respectively by ^, t a , &c., and adding, the sum will be equal to 2 {n 1 \l^(V 1 + F 2 + &c.)}, = and similarly for the other two components. And putting Fand F 2 respectively for the mean values of F and F 2 , the mean time of equal probability for the continued existence of F 2 is obtained by dividing the product by F 2 : ~^= , and for the other components V v F 2 ra 2 ' These mean times, it will be noticed, are independent of the directions of the groups, being all expressed by t = ^= , where the probable continuance is e~"> = e *? ...... (186). Differentiating this expression with respect to t, From equation (181) the mean values of w 2 , v 2 , w 2 are found to be In these a 2 is constant, and a -f b + c = 0, and the inequalities are ^(l + 2a)-^ = 2a|-, &c., &c (188). 124 ON THE SUB-MECHANICS OF THE UNIVERSE. [151 Then by equation (187) the probability of continued existence is ex- pressed by Whence if % = 0, or = - 2a 3 , &c., &c. Q.E.F. 151. Translated into the notation adopted in this research for the ex- pression of the velocities of the component system of relative motion, we have for the mean inequalities referred to their principal axes, ')"l &c., &c ............ (190), and for the rates of dispersion with reference to the same axes we have, putting 9 2 /9 2 in place of d/dt to distinguish these as rates of angular dispersion, // 82 r / / 'v/ i //','/, / '\>n 3 "" '/ [~/ ' 'v ( u/u> + v ' v ' ' / '\>n 3 "" '/ [~/ ' 'v ( u/u> ww)] = -^-ap \(uu) -^ &c., &c., ...... (191), where 2a/vV is the time-mean of the velocities of a grain, and \ is the measure of the scale of the system of relative motion. (N.B. These rates are independent of .(192). f = m-ji^o! + m z nj)' + w 3 w 3 c' g = n^a' + nj,yb' + n 3 l 3 c' h = ^Wjd' + I. 2 m 2 b' + I 3 m 3 c' t From these, adding the second, third, and fourth, .(193). Also since the principal axes do not change their position in consequence of the dispersion of the inequalities 9 2 (a') (194). ?., &c. Then substituting from equations (190) for d 2 a'/d 2 t, &c., in (194), and remembering that lju' + 1&' + I 3 w', when referred to the principal axes is the same as u referred to the fixed axes, we have by equation (193), for the rates of dispersion, referred to any axes, Vat w'w')"} 3 ,,*J7T 'S ~\ n V 77 " =-Z p ~ a 4( A< V) /7 ], &c., &c. " &c - 153. The analytical definition of the rates of angular redistribution of inequalities in rates of conduction through the grains. As already proved, Arts. 78 c and 79, Section VII., and the theorem Art. 136 in this section, the angular inequalities in the rates of conduction are the result of unsymmetrical arrangement of the grains. And as, according to the definitions of mean- and relative-mass, Art. 47, the mean-mass is inde- pendent of the arrangement, since the number of grains within the scale of relative-mass is not affected by the arrangement, the inequalities in the rates of conduction are the result of unsymmetrical arrangement of the relative-mass. 126 ON THE SUB-MECHANICS OF THE UNIVERSE. [154 It has also been shown, Art. 77, Section VII., that angular inequalities in the mean conduction result from angular inequalities in the lengths of the mean paths of the grains, and it has been further pointed out that angular inequalities in the lengths of the mean paths are the result of the distortion rates of mean strain. And the number of paths traversed being inversely proportional to their lengths, there are more mean paths traversed in direc- tions in which the relative paths are shortest. It thus appears that, although the rates of conduction are not of the same dimensions as the mean paths or the position of relative-mass, the rates of angular redistribution of the angular inequalities are the same. 154. The rate of angular redistribution of mean inequalities in the position of the relative-mass in terms of the quantities which define the state of the medium. When, owing to the rates of distortional or rotational strain in the mean- motion of a granular medium, there are instantaneous inequalities in the symmetry of the arrangement of the grains, there will be inequalities in the lengths of the mean component paths; and, the number of com- ponent paths traversed being inversely proportional to their lengths, there will be more relative paths traversed in the directions in which they are shortest. Then, since after each encounter all directions of relative paths are equally probable, after each encounter any inequality which may be attri- buted to any pair of grains is virtually extinguished. And, as shown in Art. 1 50, the probability for the continued existence for a time ti = n^ is e- H (196). v\ From this it follows, as in equation (185), ^ = WIT), t 2 = n^ 17 j, &c., &c (197), V in " 2 fr 2 in which expressions the direction cosines 1 1} m lt n 1} &c. are nearly constant and n lf the index of probability, is constant. Therefore taking the products (^ V 1 + &c.) and dividing the mean product by V the mean velocity the mean time of existence of the inequality is found to be * = "i^ ....(198), and the mean probability of continued existence is -It e~ n i = e A (199), 155] REDISTRIBUTION OF ANGULAR INEQUALITIES. 127 which when the inequalities are small becomes If, then, we take a, f, &c., the angular inequalities in the positions of relative mass, we have for the relative rates of angular dispersion, It will be observed that the logarithmic rate of decrement of inequalities in relative mass differs somewhat from that of the vis viva. This is a consequence of the difference in the mean time of probable existence of V and of F 2 . 155. The limits to the dispersion of angular inequalities in mean mass. The numerical coefficient is the only respect in which the rate of angular redistribution of mass differs from that of vis viva as long as \/ & c - \dz dx]}\ Q. E. F. SECTION XII. THE LINEAR DISPERSION OF MASS AND OF THE MOMENTUM AND ENERGY OF RELATIVE-MOTION, BY CONVECTION AND CONDUCTION. 157. THESE actions are expressed by the second, third and fifth terms in equations (123), or more concisely by the second and third terms in (117 A), 2 \dx [(P u ' u ' + P*x)' u '~\ + &c -| > &c -> &c - It has been shown that the actions of the component mean and relative stresses on the space- variations of the relative velocities (p'du'/da + foc.)" are confined to the resilience and the angular dispersion of the energy of the components of relative-motion at the points where the inequalities of angular distribution exist ; and therefore do not account for any linear redistribution from point to point. Linear redistribution requires the conveyance or transmission of energy, &c. from one space to another, and the integrals of these actions must be surface integrals. These actions of linear redistribution are again such that their effects can be studied only by considering the causes which determine the rates at which energy, &c., is carried and conducted across a plane from opposite sides. The relative-velocities at which the grains arrive at a plane, or which come in collision with a grain intersected by the plane, are not determined by any action at the plane, but by the antecedent actions. As far as these actions of redistribution depend on the convections, that is, neglecting the dimensions of the molecules, they have been taken into account in the kinetic theory of gases. Clausius was the first to obtain the true explanation* on the supposition that the mean distance between the molecules was so great, compared with their dimensions, that the latter might be neglected. In this method he takes * Fogg. Ann. 1860. 92 132 ON THE SUB-MECHANICS OF THE UNIVERSE. [158 account of the principle, that after a collision the mean velocity of the pair is the same as before, and of the consequence, that the molecules crossing a plane surface, perpendicular to the directions in which the inequality varies, from opposite sides, must have mean velocities such that their sum, in the direction of the downward slope of the inequality, is equal to V, the mean velocity of the encountering molecules, the same as if they arrived at the plane from uniform gas in motion with this mean velocity, V I ; the uniform gas being discontinuous at the surface in respect of density and velocity, but continuous in respect of mean vis viva] the density and the mean relative- velocity on either side of the plane surface being that of the varying gas at a distance proportional to the mean path of a molecule. Maxwell by a law of force (which he had arrived at from his experiments on viscosity* as the fifth power of the distance) obtained a numerically different, but otherwise, essentially, the same law. In a communication "On the dimensional properties of matter in the gaseous state -f*" I have fully discussed this action, of the linear redistribution by the convections ; confirming and extending Clausius' explanation. In that paper, by making use of the arbitrary constant s for the mean- range, or distance from the plane at which the molecules crossing the plane receive their characteristics as those of a uniform gas in motion with the mean velocity, V, of the molecules which cross in unit of time, the assumption that this distance is proportional to the mean path is avoided, and this is important where the mean path (X) is of the same order as the dimensions, a-, of the molecule or grain. In these analyses account has not been taken of any effects of conduction: so that, neither Clausius' nor Maxwell's, nor yet my own previous method is directly applicable for the determination of the rates of linear dispersion of linear inequalities in a medium in which a- and A, are of the same order, or in which A/cr is small. It thus appears that to render the analysis general these methods must be extended by taking account of the expressions (159), (162), (165), for the rates of flux by conduction of momentum, as well as of vis viva in terms of X, and to = 77/2 and divided by the respective integrals of the flux, between the same limits, the component ranges of momentum in the direction of the momentum, by convection and conduction, respectively, are found to be X and fo-. 134 ON THE SUB-MECHANICS OF THE UNIVERSE. [162 And performing the same operation on equations (160) and (166), the component mean-ranges of momentum at right angles to the direction of the momentum, by convection and conduction, respectively, are fX, and f and dividing by the respective integral rates of flux, the respective mean-ranges are found to be, for convection and conduction, For actual energy f X and fa-, coefficient f. Direct displacement X 0-, ||. Lateral The mean-ranges of momentum and vis viva, inasmuch as they are expressed in terms of X, and a, are general when X has the value expressed in equation (146). It should be noticed that while the mean-range of the grains in an elementary group is X, the mean path from centre to centre, owing to con- duction, the mean-range of the velocities and the squares of the velocities are respectively extended to that is to say the velocity of the grain is not determined by the mean condition at the centre of the grain at which it last undergoes encounter, but at a position further back ; and this becomes of fundamental importance when X/o- is small. 163. The mean characteristics of the state of the medium. The mean quantities which define the state of a (spherical) granular medium in uniform condition are (1) ), is ~ e a ddd.d, n = sin d sin $ for the direction cosines of the normal at contact of a pair of grains referred to axes moving with the mean motion of the medium, in the directions of x, y, z, and remembering that the range of convection is \ while that of conduction is cr, that for momentum the rates of the fluxes are A/2or/( ^ )/3X and for vis viva of(- ) /3\, VA,// \A// and putting d( c Q) xx , &c., and d( p Q) xx for the respective rates of convection 136 ON THE SUB-MECHANICS OF THE UNIVERSE. [166 and conduction of an elementary group in direction defined by d (cos 6) d, with respect to fixed axes; for the flux of mass we have by equation (175) < j>, &c., &c. ...(211). \ , / j \ vTT And by the last Art. 8 GQ,) = 3 (<&)** + B (*) + 8 (u") + B (JV) whence the inequality of flux is -9(c&)** = (s^ Equation (212) is general and Q may represent mass, momentum or vis viva. 166. Rates of convection and conduction of momentum by an elementary group. Substituting the mean-rate of flux of momentum by convection, and noticing that the component mean-path is increased from X cos 6 to X (u" + F/ cos 0)/ F/ while the conduction is not altered by the mean- motion omitting the square of the mean-motion and dividing out the X, we have : For direct action referred to faced axes 3 ( c Qi)xx + 3 ( P Qi)xx = p\(u' + F/ cos 0) 2 +~ ?/ (? J F/ 2 cos 2 ! ^. d ( - cos &c. &c. &c. &c. (213). For lateral action a ( c Qi\x + a ( P Qi)yx = p \(u" + F/ cos 0) (v" + F/ sin B cos + w " sin ^ sin ^) ...(215). 168] LINEAR DISPERSION OF MASS, MOMENTUM AND ENERGY, ETC. 137 The first term within the brackets on the right, which is the convection term, becomes, omitting the terms of second order, 3t*"Fi'* cos 2 6 + F/ 3 cos 3 0. One part of the first of these two terms expresses the rate of displace- ment of mean vis viva by u" ; while the remainder of this term expresses the displacement of the inequality of vis viva (2u" F x ' cos 2 0) by F/. The second of the two terms, which changes sign with cos 8, expresses the displacement ( F/ 2 cos 2 6) by F/ cos 6. The second term within the brackets expresses the displacement resulting from conduction on the mean normal velocity, and this does not change sign with cos 0. For the lateral action U = p JO" + F/ cos 6) (v" + F/ sin cos <) 2 \ ~- /(- )V 1 ' i (u"cos0+v"sin0cos)cos0sm-acos*] o \A/ ^ d (- cos 0) d ) 1 ,. I _| J (216). 168. The inequalities in the mean rates of flux of mass, momentum and vis viva resulting from space variations in the mean characteristics in a medium of equal spherical grains. When the mean state of the medium varies continuously from point to point, so that (X/JV) (dN/dx), da. I T (/, , f(T\ /' \ ) 7 /, , 7 Idas. X + v/zo-/ - /3 a,} du ax, / (V ' \\Ji / j and (X/a) dct/adt are of the first order of small quantities, the mean charac- teristics N, a, u", &c., obtained by integrating over a unit of volume, taking account of the motion in all directions, are taken as the mean characteristics at the centre P of the unit element. Then it follows that if PQ represents a distance r of the order X + cr, having a direction defined by I, m, n, the characteristics at Q will, to the first order of small quantities, be, putting / for any one of the characteristics, 1 4. A + n }l (2m 7 T^ III' 7 TlfT^JP V**- 1 -'/' dx dy dzj If, then, r is the range of /, whether it is X, \/2o"/(r-)/3 or trf[- )/3, \X// \X// 138 ON THE SUB-MECHANICS OF THE UNIVERSE. [169 as the case may be, and it be assumed that the group of grains arriving at P, from the direction of Q, arrive as from a uniform medium having charac- teristics which are the mean characteristics at Q, the inequalities in the mean rates of flux at p would be obtained by substituting T r fid d d\ T /rtir,\ I -I = r (l + m +n )I (218) V dx dy dz] for 9 (7) and integrating 1 1 3 (/) sin 9 dd d = to = 2?r, the equation for the mean flux is obtained to a first approximation. For the flux of mass. From equations (176), the equation for the flux of mass in direction of x is : p U " = pU-l-(a d / + p^),& C .,&c ............. (220). 3 /vV V dx r dx] Equation (220) has reference to fixed axes, for moving axes the equations become ,., ........ (221). 3 \fir \ dx dx) These equations define the values U, V, W in terms of the characteristics (u", a, p or N), the mean characteristics at the point. For the rates of flux of momentum to a first approximation. From the first of equations (213) the rates of direct flux of momentum 140 ON THE SUB-MECHANICS OF THE UNIVERSE. [170 become, to a first approximation, assuming X to be the same in all directions, For lateral flux. From the second of (213) the equations become A, }. ...(222). d - O .For Ae rates of flux of vis viva to a first approximation. From equations (215) the equations for the rate of flux of direct vis viva become For lateral flux. From equation (216) <7 2 dp da? dx " dx ...(223). The values of U u", &c., as defined in equations (221), are small quantities of the first order. Hence as these quantities, and their space variations, enter into the rates of momentum as factors of the small distances X, and a- only, the terms into which they enter are all of the second order of small quantities, as compared with p, and may therefore be neglected as being within the limits of approximation. Omitting these terms from equations (222), the rates of flux of momentum to the first order of small quantities are by convection : P" (uu'y = p" -= , &c., &c., p" (u'v'}" = 0, &c., &c., and by conduction, equation (159), P"XX = -s- ?/(? ) P" | > &c., &c., O A. \A/ .(224). 171] LINEAR DISPERSION OF MASS, MOMENTUM AND ENERGY, ETC. 141 The total rates of flux of momentum being o Ai \ A*/ \ jj >** y / p" xy + p" (u'v'y = o, &c., &c. ) Substituting in equations (223) the values of Uu", &c., as obtained from equations (221), the rates of flux of the vis viva of the component motions become by transformation : , , , , X a f dp 21 cfa 2 \ (p'u'u'u')' = 7^-7- 3a 2 -^- PT- lo V/TT V dx 2 r dx/ },&c.,&c. ...(226). , . ... ox by equation (223) * * ' And for the rate of flux of the total vis viva -*-^p dx 2 r , , , \ + pU VV 1 a S -7 9V<7r &c., &c .......... (227). The equations (221) to (227) as they stand are perfectly general. So far however these equations satisfy the conditions of steady density and steady vis viva, only, on the supposition that the conditions of mean -mass are satisfied. And these conditions explicitly involve the space variations of X; as is at once seen from equations (225). 171. The conditions of equilibrium of mass referred to axes moving with the mean motion of the medium. Differentiating equations (225) with respect to x, y, z, respectively, and transforming, the general conditions of the equilibrium of mass may be expressed as / \ r - J W _ - - &c " &c - - (228) ' (229). and from equation (146), differentiating and transforming, _dp_ 3X + XV2a 2 6 2 e": /9 vz > &c -> 142 ON THE SUB-MECHANICS OF THE UNIVERSE. [171 Adding the equations (228) and (229) the condition of equilibrium is &c " &0 .................... < 230) - The rates of flux of vis viva when the medium is in equilibrium. Substituting in the first and second of equations (226), (227) respectively from equation (228) the respective rates of direct flux by convection and conduction are expressed as: /// ' ' 'v o p (u uu ) = - s-s-H 6X 2 15AM } 1 da? + 21X \p 5 -=- , &c., & c ....... (233). The equations from (221) to (233) are perfectly general to a first approximation of the inequalities, the axes moving with the mean motion of the medium, the medium being in steady condition, and the arrangement such that a" and 6 2 are constant. 173] LINEAR DISPERSION OF MASS, MOMENTUM AND ENERGY, ETC. 143 172. The coefficients of the component rates of flux of (o- 3 . a 2 /2 \/2) the mean component vis viva of the grain. By equation (129 B) Section VIII. Substituting this in equations (233), dividing by N and putting (C 2 2 + D 2 2 ) for the product of the first two factors of the member on the right, these members take the form : \/2 dx as expressing the relative rates of flux of the vis viva of the grains across surfaces moving with the mean motion of the medium. These rates expressed by the space rates of variation of the vis viva of the grains multiplied by the coefficient ((7 2 2 + Z) 2 2 ) express the rate of flux under the condition of steady motion. But as long as the scales of the variation of a 2 are sufficiently large, as compared with the squares of the scale of the relative mass and the mean paths, to come within the limits of approximation for the maintenance of mean and relative systems, the rates at each point will be approximately the same as under the conditions of equilibrium. Then if the inequalities of mean motion are so small that the inequalities instituted in N, \ and a may be neglected as compared with N, \ a, i.e. if the scales of mean motion are sufficiently large and the inequalities sufficiently small, the coefficients (7 2 2 and Z) 2 2 , which are respectively the coefficients for convection and conduction, may be taken as constants within the limit of approximation. 173. The rate of dispersion of linear inequalities in the vis viva of the grains. Putting 1 9 3 ^ _ 1 d r . ' f \ ' ( ' '\ ' ( =\~, 1 xx == "AT ' j~ L\ PXX ~i" PU U ) U ( p m/ + pU V )V \p a rl + l\l rim *- * -* ' ' * N d 3 t dx we have Thus although not vectors the component rates of redistribution depend 144 ON THE SUB-MECHANICS OF THE UNIVERSE. [174 severally on the component inequalities, and admit of separate expressions which when added together give the expression And multiplying by N 174. The expressions for the coefficients C 2 2 and D 2 2 involve the arbitrary constant 6 2 , so that the general expression cannot be completely interpreted until 6 2 is defined. But the terms which depend upon b are very small except for states of the medium in which X is greater than cr/10 or less than 100-; so that outside these limits the coefficients are independent of 6 2 within the limits of approximation. Then, outside these limits, the expressions for 2 2 and D 2 2 , as appears from equation (233), when F 2 /3, as it would be if the volumes of the grains were zero. Neither would this stress vary with p but with p {1 + (p)} where (p) represents virtual contraction of the space free to the motions of the centres of the grains. Thus the variation of the kinetic energy caused by a mean volumetric strain in the medium is increased by the proportion of the volume occupied by the grains to the exclusion of other grains. It is thus seen that it is this excess of work in any mean strain, resulting from the virtual space from which the grains shut each other out, that is measured by the conductions. These effects have been fully expressed in equations (158) and (159), Section X., and are easily realized in the case of volumetric strain. But it is quite a different matter to realize how a purely distortional strain, which neither affects the volume of the space nor the volume of the grains, can produce a virtual alteration of freedom open to the grains or inequalities in rates of conduction ; and hence the importance of the evidence derived 178] THE EXCHANGES BETWEEN THE MEAN- AND RELATIVE-SYSTEMS. 149 from the consideration of the volumetric strain in the interpretation of the results of distortional strains as expressed in the three last terms within the brackets. From these it appears at once that the action which determines the character of any effect there may be is rate of distortion, which also determines the rate of action, while the subject acted upon is the component of conduc- tion induced by the distortional strain. In the first of these distortional terms, for instance, we see that all actions on the mean rates of conduction, expressed by p", equal in all directions, are expressly excluded. The recognition of this is important as it shows the independence of the actions, in so far that if the distortional strain does not induce any change in the rate of conduction there is no effect. This raises the question : what is it that determines whether or not these distortional strains shall have any effect ? And the answer to this is furnished from the experience derived from the volumetric strain. If the mean distortional strain, by altering the relative positions of the grains from what they would have been without the distortional strains, so alters the mean extent of freedom in the directions of the principal axes of the rates of strain, there will be effects, otherwise not. " Limiting the freedoms " is only an expression for altering the probable mean paths, and as a distortional strain consists essentially of strains in directions at right angles, such that one of these strains is of opposite sign and equal to the sum of the others, the action of a distortional strain is not to alter the mean density, nor if cr/A, is small the mean paths of the grains, taken in all directions, but to institute inequalities, increasing the mean paths in the directions in which the strain is positive, and decreasing them in those directions in which it is negative. It becomes plain, therefore, (1) that no matter what the mass or number of grains may be, if the volumes are such that the space they occupy is negligible compared with the space through which they are dispersed, the effect of distortional strains on the conductions must also be negligible. And (2) that any effect the distortional strains may produce on account of the size of the grains depends on the change in the angular arrangement of the grains, as measured by the angular inequalities in the mean paths, that may be instituted. And from these two conclusions it appears definitely that the abstract exchanges of vis viva, from the mean system to the relative system, in con- sequence of distortional strain in the former, and the space occupied by the grains in the latter, depend solely on the angular arrangements, as they are here called, of the grains. This general and definite conclusion brings into view, for the first time, 150 ON THE SUB-MECHANICS OF THE UNIVERSE. [179 the fundamental place which the conditions to be satisfied by the relative mass, as set forth in Section V., as resulting from first principles, occupy in the exchanges between the two systems. It also calls our attention to the fact, pointed out in the preamble to Section IX., that the tacit assumption in the kinetic theory of gases, that the redistribution of vis viva entailed the redistribution of mass, has limited the application of this theory to circumstances in which the conductions are negligibly small, and reveals the necessity, for the general theory, of a study of the law of redistribution of mass resulting from the dispersion of mass as a subsequent effect of encounters, and as being in some respects inde- pendent of, and of equal importance with, Maxwell's law of redistribution of vis viva. Although in such studies of the kinetic theory as I have seen I have not found any reference to the existence of such a law or the necessity of its study, in a recent reference to the celebrated paper by Sir George G. Stokes, " On the Equilibrium of Elastic Solids," I was much relieved to find that, in his discussion of Poisson's theory of elasticity, he expresses the opinion that it is important to take into account the possible effects of new relative positions which the molecules may take up, in which I recognise a reference to what I have called the angular distribution of the grains. 179. The probable rates of institution of inequalities in the mean angular distribution of mass. When the condition of the granular medium is such that the probable mean path of a grain is the same in all directions that is, when the mean of the paths of all the grains moving approximately in one direction is the same, whatever direction this may be there are no angular inequalities in the arrangement of the grains. And when the means of the paths of grains moving approximately in the same directions are different for different directions, these differences serve to measure the inequalities in the angular arrangement of the grains. And in exactly the same way the angular inequalities in the number of encounters between pairs of grains having relative-mean paths approximately in the same direction serve (and are rather -more convenient) to measure the angular inequalities in the mass. Such relative angular inequalities are instituted solely by distortional motion in the mean system. And the rate of distortion is one of the factors of the product which represents the rate of institution of the relative inequality ; the other factor being the ratio of the average normal conduction of momentum at an average encounter of a pair of grains, divided by twice the average convection by a grain in the direction of its path. 179] THE EXCHANGES BETWEEN THE MEAN- AND RELATIVE-SYSTEMS. 151 By equation (147) the normal conduction at a mean collision is and by equations (155) and (156), there are two mean paths traversed for each collision, and the mean displacement of momentum, by the convection of a grain between encounters, is X V. Therefore the ratio of the corresponding normal conductions and normal convections is 2 3 ~ ( 39) ' And the rates of institution of relative angular inequalities in the arrange- ment of the mass are represented by " - 2 - + + - .. 3 X 7 w ( dx 3\dx dy dz )} This is, only, when u", v", w" are referred to the principal axes of the rates of distortion. And da'/dt, db'/dt, dc'/dt, represent the relative rates of increase of the mean paths of pairs of grams having relative motion in the directions of x, y, and z respectively. The rates of relative increase of pairs of grains, having directions of motion other than the directions of the principal axes, are obtained from those in the directions of the principal axes as in the ellipsoid of strain. Besides expressing the inequalities in the angular distribution of mass and in the mean relative paths, da', &c., express the rates of increase of the inequalities in the numbers of encounters between pairs of grains having relative velocities in the directions of the principal axes. But they do not, without further resolution, properly represent the rates of increase of the inequalities in the rates of conduction in the directions of the principal axes ; since the directions of encounter, that is, the normals at encounter, may depart by anything short of a right angle from the direction of the relative motion of a pair. Before proceeding to consider the relative-inequalities in the rates of conduction, however, it seems desirable to call attention to the distinction between rates of strain and strains. It will be noticed, after what has already been said as to the difference between the effects of volumetric strains and distortional strains, that in what follows, the expressions da'/dt, &c. are used to express the rates of increase of relative-inequalities resulting from rates of distortion, while * N.B. The a', b', c', in this article have no relation to (a, 6, c) as used in equations (181) &c. for inequalities of vis viva. 152 ON THE SUB-MECHANICS OF THE UNIVERSE. [179 these expressions are equally applicable to the rates of volumetric strain. Thus the expressions, \/2 a l du dv dw A/2 a , f/ T \Z-r --; K-+-T- + ^H&, &c., &c. ...(241), 3 X/ \X/ { dx 3\dx dy dz /) or since X is not affected by the distortional strains we may put for the actual rates u" 2 /du" dv" d ......... (242), which express the increase in the mean paths of pairs of grains having relative velocities in the directions of the principal axes. 180] THE EXCHANGES BETWEEN THE MEAN- AND RELATIVE-SYSTEMS. 153 Then since the numbers of encounters between such pairs are inversely as the increase of the paths, we have, equating the reciprocals of both members, l_ '-l- g- + + ...(243). 3 X y \\J (da; 3\dx dy dz )} From which we have for the rate of relative increase of encounters the numbers of pairs with relative motion in the directions x, y, z, V2 , n = sin cos , -"ft Jo ( d cos 4 6 1 id cos 2 Id cos 4 6 , ,. a + - - + - - (6 + c) - ' v 4?r 27T V 2 4 4 a 1 (248). 181. The mean relative inequalities in normal conduction are obtained after the manner in which equation (148) is obtained, by resolving the com- ponents of mean normal conduction in the directions of x, y, z respectively, and multiplying them by the expressions for a, 6, c, &c. equations (247). Then, since a + b + c = 0, we have for the probable inequalities respec- tively a/4, 6/4, c/4. Our object however is not to obtain the inequalities in the probable number of encounters, but the inequalities in the mean normal conduction in the directions of the principal axes. The mean relative inequality of normal conduction is obtained by the same method as in Art. 104. This requires that for the direction of x, l^ must be multiplied by -^- *J^f\ -} V v l, and then integrated. Thus o \\J Tt - /'2V-f(-\ [*[ dcos5 1 (dcos 3 dcos 5 0\ ] (249), reduce to (250). These are the inequalities in the probable normal conductions in the directions of the axes of x, y, z respectively, and it remains to find the inequalities in the probable conductions in the directions of the principal axes. 181] THE EXCHANGES BETWEEN THE MEAN- AND RELATIVE-SYSTEMS. 155 The probable inequalities in the conductions resulting from an encounter, having the normals in the direction of x, are obtained by substituting the expressions for a, b, c in the preceding equations, then resolving the normal components of V v and ,* = " 3 \ f W { dx 3 (dx + dy + dz )} ' dw" .(254). To convert these into rates of institution of inequalities in the probable rates of conduction they must be multiplied by the constant coefficient of the d 1 (a')/d 1 t in equations (252) which by equations (159) may be expressed as: 0'32//'; the coefficients of the right members of equations (254) may also be expressed by 2,p"/pa. 2 . Therefore 0-32X' 2 L duT _ 2 /<&/' ^_ ^\ dw"^ = a 2 I <& 3\ '2 " dw" , (255) express the initial rates of increase of probable angular inequalities in the rates of conduction, resulting from distortional rates of strain in the mean-system, which are expressed in the last term but one of equations (117 A). The rates of increase of conduction resulting from rates of change of density. By equations (239) the relative rates of increase of p" are the products of the relative rates of change of density multiplied by the ratio of the rate of conduction to the rate of convection ; the last factor is 3 \\\ . ' Thus for the relative rate of increase of p" 1 a 1 (j? // ) = _(du^ dif d*/V p" 8^ ^ o 2 V cte dy rf^ / ^ o the actual rate of increase being d 1 (p") = _ p" 2 (duT dv^ dw"\ dj a? \ dx dy + dz ) .(256). 182] THE EXCHANGES BETWEEN THE MEAN- AND RELATIVE-SYSTEMS. 157 182. The transformation of vis viva or kinetic stress. This as expressed in the last term of equations (117 A) and multiplied by 2 so as to express the rate of increase of vis viva (not energy), is MY (W'U'Y I &C &C If the axes are principal axes of rates of distortion and the medium is in uniform condition the last two terms within the brackets are zero. Then taking a', b', c' for the relative inequalities, which are initially zero, we have for the rates of increase 2 9,a' , , , w , ( n du" 2 /du" dv" dw"\} D P -^ 4r =P (uuj \2-j 5 ( j- + -j- +T- H , &c -> &c - ..-(257). z oj { dx 6\dx dy dz /J Putting ^Ttt^nj, ^m 2 n 2 , Izm^n^ for the direction cosines of the principal axes referred to any system of rectangular axes and taking a, b, c, f, g, h as expressing the inequalities when referred to other fixed axes, by the well-known theorem V + m 3 2 c' f= m 1 n 1 a' + m^ntf + m 3 n 3 c f &c. &c. where a + b + c = a' + b' + c' dit dt dt dt and substituting for the values of da'jdt, &c., from (257) .(258). .(259), P r a 2 d l a . (da" 1 fdu" dv" -- =3 a ----- + -- o 2 dw" -j -j- dy dz a*/dv" dw"\ , &C..&G (260), Then putting aV 2 for (263). These equations express the initial rates of increase of angular inequalities in the rates of convection resulting from distortional rates of strain in the mean system, which are expressed in the last terms of equations (117 A). 158 ON THE SUB-MECHANICS OF THE UNIVERSE. [183 183. The institution of linear inequalities in the rates of flux of vis viva of relative motion by convection and conduction. Thus far the analysis for the rates of institution of inequalities in the vis viva and rates of conduction has been confined_to the effects of uniform rates of strain in the mean-motion extending throughout the medium, whether distortional, rotational, or volumetric. When however the rates of mean volumetric strain are other than uniform, as long as the parameters of such motion are large as compared with the parameters which define the spaces over which the means of the relative mass and relative-momentum are approximately zero, the analysis of the effects resulting from small variations in the rates of strain in the mean-motions, in instituting linear dispersive inequalities in the mean vis viva, p(a?)"/2, of relative-motion, follows as a second approximation on that which has preceded. In Section V. equation (93), it is shown that provided the relative motion and relative mass are subjected to such redistribution as to maintain the scales, over which they must be integrated, small compared with the corre- sponding scales of the mean-motion, the conditions for mean- and relative- systems will be approximately satisfied. The expressions for the rates of institution of linear dispersive inequalities by convection and by conduction are given by equations (261) and the last of equations (256) 2 \2 dx -?. 1 / "\_ _ ^P f^L 4.^ ^ w/ ^' (264). dit^ 3 a? \dx dy 184. The institution of inequalities in the mean motion. In the case of a space within which there are no inequalities, in either system, the institution of inequalities in the mean system within the space must be the result of some mean inequalities in the mean state of the medium outside the space of some action across the boundaries; since in an infinite medium, including all the mass, all actions must be between one portion of the medium and another. For the sake of analysis however it is legitimate to consider the mean actions on the boundaries of any space, as determined by the scale of mean- motions, as arbitrary. And it is important to notice that such mean actions on the mean motion are the only actions that it is legitimate to treat as arbitrary ; since, as has been shown in the last article, the institution of inequalities in the relative motion results solely from the action of the mean motion. 185] THE EXCHANGES BETWEEN THE MEAN- AND RELATIVE- SYSTEMS. 159 Arbitrary accelerations may be finite or infinite and by assuming the accelerations infinite we are enabled to institute finite inequalities in the mean motion in an indefinitely short time, and this without instituting any inequalities in the relative motion, as the instantaneous result of the institution of the inequalities in the mean motion ; whence, it appears, that we may, for the purpose of analysis, start with a medium without any inequalities in the mean mass, relative mass, or relative motion, but with arbitrary inequalities in the mean-motion. With such an initial start we have, from equations (120) Section VI., p-L = 0, &c., &c (265). 185. The redistribution of inequalities in the mean-motion. The effect of the instantaneous institution of inequalities in the mean motion is an instantaneous finite acceleration to the institution of inequalities in the relative motion as expressed in equations (255) to (263) as the result of transformation ; the action including both the convections and conductions. This acceleration of the inequalities, in vis viva of relative motion, including conduction, is also an acceleration to the institution of the space-rates of variation of these inequalities, and these space-rates of variation of the inequalities of relative motion are transformed back as accelerations of the mean motion. Thus, although diu'/dj = 0, the institution of du"/dx, say, has instituted an acceleration to the institution of inequalities, the space variations of which react as accelerations on the mean-motion. That these reactions are dis- persive, of inequalities in the mean motion, follows definitely from the sequence of the rates of action already defined. To prove this we may consider the acceleration of any one of the inequalities, instituted by the mean motion, as to its rate of reaction, on the inequalities of position of the mean-momentum, by itself independently of other inequalities. Considering the effect of acceleration of the inequality on the acceleration of the rate of increase of mean-momentum, it appears, at once, from the equations (120) that the reaction resulting from this inequality affects both u" and v". These effects may be considered separately. But from equations (255) to (263) it appears that the rate of institution of the inequality p" (u'v')" + p" xy depends on the mean inequalities du^ dif. dy dx ' so that if du"jdy is zero there will still be reaction unless dv"/dx is also zero. 160 ON THE SUB-MECHANICS OF THE UNIVERSE. [186 From equations (255) to (263) the rate of institution of the inequality is *" dv " Then changing the sign and differentiating with respect to y we have for the rate of increase of reaction from this inequality, Differentiating this last equation with respect to y the acceleration of the rate of increase of the inequality in the mean motion is M } _ ( 2 , 0-64 N d* fdvT dv" x - This equation expresses the partial effect of the inequality p" (u'v')" + p" X y on du"jdy. And proceeding in a similar manner we have for the other partial effect on dv"/dx ......... <-> Then adding, the total effect becomes 0-64 N / d &\ du" + ~ + dx " It is at once seen that this equation represents a positive acceleration to dispersion of the inequality in the mean motion, du"/dy -f dv" jdx, as the result of the rate of institution of the inequality p"(uv')" +p" X y In a similar manner it may be shown that the effects of the five distortional inequalities, in the rates of convection and conduction, are accelerations to the dispersion of the five remaining inequalities in the rates of increase of mean motion. These, together with rates of dispersion of the volumetric inequalities, admit of expression in a general form. 186. The inequalities in the component of mean motion. du" dv" (du" _ 1 fdu" dv" dw"\\ dy 1 J~. o I / ~r 3jT7 j_ ) ( > """"" da; efce 3 V dx ' dy T ^ y j ' 2 /rfw" dw"\ \lte + ~dx~) I /du" dv" dw"\ 3 o -j- + T~ + -j > &c -> &c -> 2 3\dx dy dz J admit of expression after the manner of expression of component stresses by simply substituting I" xx for p"^, &c., &c., and we may further simplify the expressions by putting /" for (I" m + I" yy + /")/3. 187] THE EXCHANGES BETWEEN THE MEAN- AND RELATIVE-SYSTEMS. 161 In the same way we may take /^ for {p" (u'u')" + p" xx }. In this way we have for the three typical expressions of accelerations to rates of increase in inequalities of mean motion ^ (T" T"\ ( *r * + *** P 5T (* a "~ * ) = P a + /-.- J3 , vi a 2 . 0^4 \ (d* d*_\, j,, , j,, yx) "''\ p 2 + 2p"a* P )(dy* + da?) (2 xy + 1 * rft At \ ( A* ft* " a , _ _*rt I tt / T" \ , a / r// 3 T 3p /x a 2 D fL //" N - f " a , _ n *} a (T" \ + *L Pm^ *) \P O+Q-TT^P M53V "/^A4 Each of these types, it will be observed, expresses acceleration to the dispersion of the inequality of the mean motion. Whence it appears that the instantaneous institution of inequalities in mean-motion is also an instantaneous institution of accelerations to the dispersion of the inequalities in the mean motion. Q. E. D. It will be observed that since by definition the mean relative components taken over the scale of relative motion are all zero, there can be no change in the mean momenta as the result of exchanges between the two systems. And hence the action of dispersion can be, only, changes of the position of the momentum from one place to another. 187. In the consideration of the equations for momentum the question of dissipation of energy of mean-motion to that of relative-motion does not arise. But, as an acceleration to dispersion of inequalities of the mean- motion is an acceleration to decrease the component momentum where it is greater and increase it where it is less, so that there is no change in the integral momentum of mean motion, it follows, as a necessary consequence, the acceleration to dispersion of momentum entails an acceleration to dis- sipation of energy of mean-motion to that of relative-motion. The expression for these initial accelerations to dissipation of energy may be obtained in various ways, one of which is involved in the proof of the following theorem : The initial rates of institution of inequalities as expressed in equations (255) to (263), for convections and conductions, are essentially accelerations to mean rates of increase of the vis viva of relative-motion as well as to the redistribution of inequalities in the mean system. The terms which express exchanges of energy by transformation from the mean system to the relative system, which are the only exchanges between the systems, are the last of the terms in each of the equations (116 A). Then putting p"&(t*V)/3j($]i &c., &c., as the initial effects of the instantaneous R. 11 162 ON THE SUB-MECHANICS OF THE UNIVERSE. [188 institutions of inequalities in the mean motion on the relative motion, we have ( 1 T u'u + v'v' + w'w' 1 " fdu" dv" 1 ' [ . I ,...(272), + +P\ ( : j-+ j- + j- V dx dy dz "du" , , , v ,/du" dw"\\ [p wu-p zx \ ( 4- ,-J \ \Aj4t \JU*As / \ and two corresponding expressions for the other components. By equation (265) di^'/dj, &c., &c. as well as all inequalities of relative motion are initially zero ; so that, initially, both members are zero. Then performing the operation 3 1 /9 1 < on both members and observing that by equation (265) this operation has no effect on the mean inequalities, dv" dw"^ ^ dx dy dz dx dv" (273), and two corresponding equations for the other components. These three equations taken together express in terms of the differential coefficients the rates of institution of inequalities of the relative motion, expressions for which in terms of the mean motion are given in equations (255) to (263): and substituting these expressions for the differential coefficients in each of the three equations, and adding the corresponding members, we have for the total initial rate of acceleration of the rate of increase of relative energy c.(-^c(~} = (V64 ,, 2 \((du^\ z /^"V ( dw " p"o?P J \\dxj \dy ) \dz dv" dw"\ 2 fdw" dz dy J 2 \\dy dx J \dz dy j \ dx dz J)" The member on the right is essentially positive while the left member expresses the acceleration of the mean rate of the vis viva. Q. E. D. 188. The first term on the right, equation (274), expresses the accelera- tion of the rate of mean-energy of relative motion resulting from the inequalities of the direct space variations of the mean motion, including 189] THE EXCHANGES BETWEEN THE MEAN- AND RELATIVE- SYSTEMS. 163 both volumetric and distortional effects, while the second term expresses the acceleration of the rate of mean-energy in consequence of the tangential space variations of mean-motion. These accelerations are all positive, tending to produce a dispersive con- dition of relative-motion. The tendency, thus proved, of the effect of transformation from energy of mean-velocity to energy of relative-velocity, at each point, so to direct the signs of inequalities in relative vis viva as to cause dispersion of both energy of mean and energy of relative-velocity, and to render the effect of transformation, of mean-motion to energy of relative-motion, positive, is quite independent of all other actions or effects ; and, although not hitherto analytically separated in the theory of mechanics, is now seen to be one of the most general kinematical principles the prime principle which underlies those effects which have long been recognised from ex- perience and generalised as the law of universal dissipation of energy. The analytical separation of this principle does not immediately explain universal dissipation. It accounts for the initial acceleration to the dispersive condition, but it does not, alone, account for irreversibility of the dissipation. The proof of this at once follows from equations (271), the general solution of which is which expresses two reciprocal inequalities of mean motion proceeding in opposite directions uniformly at velocities V 0-64 P"*+:^ If then u" be everywhere reversed, the direction and the rate of propaga- tion of the reversed inequality remaining the same, will bring the state of the relative motion back to the initial condition. And this applies to all inequalities, so that if there were no other action than that of transformation including its effects on the mean and relative inequalities, these effects would be perfectly reversible. 189. The conservation of the dispersive condition depends on the rates of redistribution of the relative motion. By equations (271) and (274) it appears that as long as the inequalities of relative-motion are zero while the inequalities in the mean motion are finite the signs of the acceleration to the dispersive condition are always positive. Therefore if these inequalities remain small as compared with the energy of relative motion, while the signs of the inequalities of the mean- motion are not changed, a dispersive condition is secured. From which it 112 164 ON THE SUB-MECHANICS OF THE UNIVERSE. [190 follows that any cause which maintains these inequalities small, compared with the relative energy, will render the dispersion irreversible by reversing the mean motion, no matter how great the acceleration to the dispersive condition arising from the prime tendency to the dispersive condition. Such actions exist in the angular and the linear dispersions, of the angular and linear inequalities of vis viva of relative motion, and rates of conduction through the grains, equations (195) and (205), Section XI., and (236), Section XII. From equation (266) it appears that the instantaneous reversal of the mean motion has no effect (instantaneous) on the relative motion ; so that this is not simultaneously reversed. And thus it is not the resultant motion that is subject to reversal, but only the abstract mean motion, while the abstract relative motion continues as before to redistribute the reversed mean motion. This explanation of irreversibility of the mean motion and the irreversible dissipation of energy could not have been obtained until the analytical separation of the abstract mean motion from the relative motion had been accomplished. And this fact fully explains the obscurity which has hitherto surrounded dissipation of energy. The general reasoning in this article, although sufficient to afford a general explanation, is, of necessity, supplemented by the definite analysis by which the inequalities in the vis viva of relative motion are determined in the next article. 190. The determination, in terms of the quantities which define the con- dition of the medium, of the inequalities maintained in the vis viva of relative motion, and in the rates of conduction, by the combined actions of institution by transformation, and redistribution by relative relative-motion. In entering upon this undertaking it is in the first place necessary, in order to render the course of procedure intelligible, to point out that as far as mechanical analysis has as yet been developed, including the present research, it has not included such analysis as is necessary to express the means of the instantaneous transmission of accelerations, and thus we are unable to deal definitely with continuous initiation from rest of continuous inequalities. This inability, which is generally recognised, was discussed in a paper read before Section A of the British Association at Southport, though not further published. In this paper it was suggested that such inability was evidence of some property in the constitution of the medium necessary for the instantaneous transmission of acceleration, and showed that if the medium consisted of rigid particles as in Maxwell's Kinetic Theory (1860), then since any acceleration at a point would, necessarily, extend through the thickness of the grain, it would therefore afford instantaneous 191] THE EXCHANGES BETWEEN THE MEAN- AND RELATIVE-SYSTEMS. 165 linear transmission of acceleration, and so render the necessary analysis for dealing with initiation possible. As we are here dealing with a granular medium, this analysis, if fully developed, would remove the disability. But, having assurance of this, we may avoid the development of the analysis by following the method of Stokes considering only such inequalities as are steady or periodic when referred to moving axes. Under such conditions the determination of the inequalities maintained is practicable, and indicates the general form of the equations for the general inequalities. The incompleteness of the analysis for the expression of the linear instantaneous transmission of accelerations is not the only reason for con- fining the application of the analysis to steady or periodic inequalities. Putting aside uniform continuous strains and rotations in the case of a granular medium, of which the mean condition is uniform and indefinitely continuous, it is the properties of such a medium, of transmitting undulations, that first claim our attention. And as such undulations are the only motions, in such a medium, that can extend to infinity throughout an infinite space, they must be considered as the principal form of mean motion. However, before proceeding to consider the undulations, it may be well to point out the several classes of mean motion which may be recognised at this stage of the analysis. Other than undulations, the only possible mean motions, including mean strains, are such as involve some local disarrangement of the medium, together with displacement of portions of the medium from their previous neighbourhood as in the vortex ring which may have a temporary existence when C.?, and A 2 , 2 A 2 for A, 2 , A. 2 . Q. E. D. The equation for the rate of increase of the mean vis viva (a 2 / 2). Multiplying the expression for /, equation (284), by the corresponding expression for I", it at once appears that / consists of two parts, the one being continuously positive and the other periodic. Thus: //"= - [ ri A*qA 2 sin (mt - ax) in- j- Afqm cos(mt ax) (285), + -0.2 170 ON THE SUB-MECHANICS OF THE UNIVERSE. [194 from which it appears that the dispersive inequality in equation (284) is expressed by sn * ~ the remaining part of /, o - ax), representing that part of the inequality the effect of which is purely periodic, or non-dispersive. Therefore the equation for the rate of increase of the mean vis viva is which is a general form for all rates of dispersion of mean vis viva. Q. E. D. 194. Having, in Art. 193, obtained the general expression for total inequalities maintained by relative-motion as the result of institution by transformation and redistribution, as well as the general expressions for the dispersive and periodic components of the inequalities, it appears that the analytical distinction between the corresponding inequalities in vis viva, and rates of conduction, may be expressed by substitution for A* and A.?, &c., the values of these constants as expressed : , .... (convection, in equation (277), for angular inequalities in \ . . [conduction, (278), f ,. ,.,. . (convection, ., (279), tor linear inequalities in J (conduction, (280). They are, for angular inequalities in convection : q \-T a sin (mt ax) m cos (mt ax) [ . . .(287) ; J for angular inequalities in conduction : i _! -/( g )p sin (m j _ ^ w cos (mt ax) . . .(288) ; m + u 4 195] THE EXCHANGES BETWEEN THE MEAN- AND RELATIVE-SYSTEMS. 171 for linear inequalities in a 2 /2 in convection : 5 a 2 v ' 3^2 fa 2 3Xa ) Ivx = m N2 q \ . sin (mt - ax) - m cos (mt -ax)\ . . .(289); /ft oAOt\ I A/7T I m 2 + v V VT / for linear inequalities in oc 2 /2 in conduction : , . , T a 2 sin (ra - OMJ) - m cos (m - ax) * I O > a T a V 77 " X 4 ...... (290). The equations for angular inequalities are general for all states of the medium. But the expressions for the linear inequalities are those to which linear inequalities approximate according as \/ &c - \dx dy dz J Expressions for the resultant institutions of inequalities of mean motion when the direction of propagation is perpendicular to the direction of motion. If *'o> 2/0) ar> e measured in the directions of propagation and mean motion respectively, the resultant rate of shear strain is expressed by Then taking x l} y l} z l for the principal axes, 1 1} m l} n^ for the direction- cosines of the principal axes referred to x , y , z , we have, resolving for the principal strains, du^' _ j dv dvi' j dv 1 dw l j litfli j , - = froWto ^ -J = 0. 196] THE EXCHANGES BETWEEN THE MEAN- AND RELATIVE-SYSTEMS. 173 And since ni = n . 2 = n 3 = Q, Zj 2 = 4 2 = mi* = m/ = % ^m, = - L 2 m 2 = , du l /dx l = dvjdx-t = ^dv /dx n , and referring to any rectangular axes x, y, z, the partial inequalities are (du" /du" dv" dw"\\ 1 , dv" du"\ / ~ \ dx ~ \dx dy ~dz~ ~~ 2 dx dy dw" du" o -J +-j 2 V dx dz \ )> &c -> &c ....... (290 B). ) 196. The equations of motion of the mean system in terms of the quantities defining the state of the medium. Having obtained the four general expressions for: The total angular inequality in convection: equation (287) linear (289) angular conduction (288) linear (290) Adding the two first together we have the total inequality in vis viva. And in the same way adding the last two together we have the total inequality in conduction. Then again adding we have the total inequality. Thus reverting to the forms A^, Bf, &c., for the respective constants, arid introducing the actual expressions for the general expressions I" , or the harmonic expressions p (u'u), &c., for the inequalities, we have, for angular and linear inequalities in vis viva, A-\ f A 91 (du" 1 /du" dv" dw"\] MM) = ri 4ii! i ~T H j + ~j ) r 31 (du" dv" dw" 77^- 2 -K- i- -,- - m 2 + (a6 Y 2 ) 4 _ dt dx dy dz A i 2 r A 91 \ (dv" dll"} o /ono\ p(v'u') = -- ^I'-S olT- + T~f >&c.,&c (292), m? + AJ L 9^J 2 ( dx dy } . , ,. Af [ . 311 (dw" du"\ s s /OOQ\ p(wu)= \A 2 Z -^- \a\-j-+ j-h& c v&c (293). w 2 + A j~, j- i & c -> & c -> & c - dx dy dz But otherwise the inequalities of mean motion as expressed in equation (291) are partial. (3) The coefficients of these partial equations must be such as will, within the limits of approximation, resolve into the resultant equations for the resultant inequalities. (4) The coefficients in the partial equations which express component angular inequalities satisfy the condition of resolution stated in (3) as a matter of form. (5) The coefficients in the partial equations which express component linear inequalities do not obviously, as a matter of form, satisfy the condition of resolution to a first approximation unless a*C//m* is small. But treating 197] THE EXCHANGES BETWEEN THE MEAN- AND RELATIVE-SYSTEMS. 175 this quantity as small, it can be shown that they do satisfy the condition even to a second approximation. Thus omitting the square of d 2 C.?/m? as a first approximation, and putting (a 2 + 6 2 + c 2 ) 2 2 4 /4m 2 , the mean value of a 4 C 2 2 , in the second approximation, the terms expressing component linear inequalities take the form , & ,, &c .. .. (297) , ra 2 \ dtJ \ 4ra 2 / and these obviously satisfy the conditions of resolution for inequalities in both vis viva and conduction : C* ( d\ (du" dv" dw"\ ( (a? + 6 2 + c 2 ) 2 G, 4 ) a 2 G 2 - 5i) I ~j -- H-j- + j II I , &c., &c....(298), m 2 V dtJ \dx dy dz J { a 2 which satisfy the conditions of resolution, and the second approximation may be neglected. (6) The proof that these 2 (7 2 2 /m 2 are small, is not possible as long as m 2 and a? are considered as arbitrary, and subject only to the conditions of being small as compared with tr/\ and 1/X, since the proof depends on dynamical analysis which is effected in a subsequent article, in which it is shown that for any disturbance propagated through the medium these constants are extremely small. (7) Although small the second approximation is finite as long as the first approximation to the inequalities is finite. Beyond reminding us of this fact there is no object in retaining this second approximation. 197. The equations of motion to a first approximation. Substituting in the equation of mean-motion (119) from equations (291) to (296) for the inequalities in the relative vis viva and rate of conduction, these take the form : du" A? r a ' dt 6 dx \ dx dy dz &c. &c. d 2 ^7 I H : (300), with two similar partial equations for the rates of increase of dv" /dt and dw''/dt, and the conditions dw dv - _ dy dz 176 ON THE SUB-MECHANICS OF THE UNIVERSE. [198 As explained in (7) in the last article the last factor in the second term on the right, which adds the second approximation, may be omitted within limits of a first approximation. Substituting for the coefficients Af, A, &c. their values in terms of the quantities which define the state of the medium, as given in equations (277) to (280) and (287) to (290), we have, to a first approximation, the equations of motion in the mean system in terms of the quantities, referred to axes moving with the mean-motion of the medium, the general ex- pressions for which are stated in equations (119). Q. E. F. From these partial equations (300), we get the partial equations for the component vis viva of mean-motion, in terms of the quantities which define the state of the medium, by multiplying the partial equations of motion by u", v" , w" respectively, as in equation (122), and these added together resolve into the several equations of vis viva in terms of the quantities the general expression for which is given in equations (125). 198. The equations of the components of energy of the relative system in steady or periodic motion. It has already been shown, equation (285), that the rate at which the component of energy of relative motion is increasing, at a point moving with the mean-motion of the medium, is the product of the total partial component of the inequality in relative motion multiplied by the inequality of mean-motion in the general form : A? a*AS-^\r*. oil Therefore, proceeding as in the last article to take account of all the inequalities angular and linear, since the constants are the same, and the linear inequalities a, b, c are the parameters of the variations, the equations for the partial rates of increase of the energy of relative motion by trans- formation from the mean-motion become i a ia?\ ( A, 2 r, i a "-z ^ ~. t) r%+ 1 9 / ,2 i A 4 2 9 7\t z ot\/ ( m T -0.2 I Gi dv dw\~] 2 Ifdv duY Ifdw du^ dx 3\dx dy dzj] 4>\dx dy) 4<\dx dz + ~^n J" 2 A 2 -^-W' + ^ + ^7 (301), ra 2 + (aiA,) [ 2 dJ \_dx dy dz J with two corresponding equations for the directions y and z. 198] THE EXCHANGES BETWEEN THE MEAN- AND RELATIVE-SYSTEMS. 177 Then substituting for the coefficients from equations (287) to (290) we have, to a first approximation, the partial equations for the vis viva of relative motion in terms of the quantities which define the state of the medium, terms for which the general expressions are given in equations (123). Then considering the partial equations (298) we have for the resultant equation of relative vis viva, the general expression for which is given by equation (126), P2 " _ _ _ 3\dx dy dz l((d/uT_ dvy / 2 [(dy + dx) + \dz dy dx dz /dv'^ duf\ 2 fdw" <^V \dz And putting for the right-hand member its equivalent \ | [p" (*' + * + w'*)] - i | [>" (u' + .' 2 + i^)], we have the expression which would constitute the first member of equation (126). Therefore we have, in the second member of equation (302), the ex- pression, to a first approximation, for the rate of variation of the energy of the relative system in terms of the quantities which define the state of the medium. Thus equations (300), (301) and (302) are, to a first approximation, respectively the partial equation of momentum of mean-motion, the partial equation of energy of relative motion, and the resultant equation of energy of the relative system. And it may be noticed that the equation of energy of mean-motion corresponding to equation (125) Section VI. is at once obtained by multi- plying equations (300) by u", v", w" respectively. And thus the dynamical theory of a purely mechanical medium is established and defined for periodic inequalities to a first approximation. Q. E. D. B. 12 178 ON THE SUB-MECHANICS OF THE UNIVERSE. [199 It is to be noticed here that the three equations (300) of momentum in the mean system, to a first approximation, . when multiplied by the respective components of mean motion, become the component equations of energy of mean motion, and on being reduced and added together form the resultant equation of mean energy. And since, in a conservative system, such as that under consideration, the only exchanges between the two systems are between the energy of mean motion and the energy of relative motion, we should have as the sum if the approximation is complete ; and this is the case. That is to say, the approximate expressions for energy of mean motion obtained from equation (128) become, on changing the sign, the equations for energy of relative motion. It thus appears that there is only one equation of energy although there may be two systems of partial equations for the energy of the components of mean and relative motion. There are, however, two systems of equations for momentum, one for momentum of mean motion, and the other for the mean momentum of relative motion, the second of which is expressed by (O" = o, 00" = o, (o" = o, while the first is the system expressed by equations (300). This affords a check on the method of approximation which only becomes apparent at this stage. 199. The equations of motion to a second approximation. In proceeding to a second approximation, it is to be noticed that the rates of increase of a or a 2 , Af, Bf, CV 2 , and D^, the coefficients in the first approximation, are the result of the irreversible dissipation from vis viva of mean motion in consequence of the inequalities in mean motion, as considered in the first approximation, tending to increase the value of a, and to institute linear inequalities in the value of a or a 2 ; such secondary inequalities are instituted both by angular and linear inequalities in the first approximation. But it is not in taking account of these secondary inequalities that the second approximation consists, for, as will appear as we proceed, such secondary inequalities are of no account as compared with the first. The second approximation consists in taking account of the rate of irreversible dissipation of energy resulting from each of the several actions, 200] THE EXCHANGES BETWEEN THE MEAN- AND RELATIVE-SYSTEMS. 179 as expressed in the first approximation, as cause logarithmic rates of diminution in the linear inequalities of mean motion. In this portion of the analysis, since the general expression for the equations to a first approximation has been effected, attention may be confined to the two primary undulations, approximately simple harmonic, referred to axes in the direction of mean strain ; taking the axis of ac for that of propagation and the axis of y for that of shear, so that the inequalities (/") in mean motion are expressed by du" , dv" -f and - 7 . dx dx The equations for the undulations are obtained to a first approximation by taking all the rates of variation of the mean motion zero, except those which enter into the two expressions respectively in the equations (300), (301) and (302). 200. The determination of the mean approximate rates of logarithmic decrement. To do this it is necessary to know two quantities : (1) The ratio which the mean of the total undulatory energy bears to the mean of the energy of mean motion, including resilience, per unit volume. (2) The rate of irreversible dissipation per unit volume in terms of the energy of mean motion to which it is proportional. Let R be the ratio of the total energy of undulation to the total, including resilience, per unit volume ; T the coefficient by which mean energy of mean motion must be multiplied to express the rate of dissipation. Then, the bar indicating the mean, -r, 8 fu"' 2 -f v" 2 + w" 2 \ A /u" 2 + v" 2 + w" 2l \ The logarithmic rate of decrement is r * -T *Ju"* + v"*lw"* = e* R The values of T are all to be obtained from equation (302) omitting the d/dt. The values of R are a little more complex. But as in the first * No connection with T (tau) the rate of propagation of light. 122 180 ON THE SUB-MECHANICS OF THE UNIVERSE. [201 approximation the motions are a simple harmonic function of t and x or x and y, R = 2 for normal waves, R = 2 for transverse waves when there is no diffusion, R = 1 for transverse waves when diffusion becomes easy. This last case, whatever other interest it may have, is of great interest in affording a check on the correctness of the approximation, since Stokes has obtained a complete solution of this case for a gas as well as any viscous fluid, and as y omitting the d/dt in the coefficients of both the terms of equation (302) and dividing by p. For convenience putting A for the sum of the coefficients for the angular inequalities, and L for the sum of the coefficients for the linear inequalities, resolving in direction x, we have for the respective rates of dissipation And we have for the mean square of the inequality, mean energy of motion, and total energy, q*/2a 2 , and q z /a 2 respectively. Thus R = 2 and = = QA + L) a\ 5 ........................... < 306 >- And the equation for the normal wave is (306). 201] THE EXCHANGES BETWEEN THE MEAN- AND RELATIVE-SYSTEMS. 181 In a similar manner for the transverse wave .(307). The mean values of (dv"/dxy*, (v") 2 , and total energy are, when 1 > v v =---e ^3 VT ' cos ( mt ax) (olo). a From equation (310) the coefficients A, B, L, are ^ = 3V^' and for - small L=^. /^A\ X m 2 2 \/7r f \^^)- , . , and for - large X 5 j9 2 4 o- 2 a G 9 = - - v-~~ ~ a " , 3m 2 2 3 We have thus obtained the complete equations for indefinitely small steady continuous undulations, including rates of decrement for normal and 182 ON THE SUB-MECHANICS OF THE UNIVERSE. [201 transverse waves, in terms of the quantities a, X, = OAC, L B = AB, L c = AC. Then a = L H cos 9 = L c cos (f> b = L B sin 6=0 = L c sin and cot 6 diminishes as 6 increases, we have for the maximum coefficient cot 2 + cot 2 <-l = l, and this is when the axes of no contraction are inclined to the axes of dis- tortion at 45. Further, it appears that as 6 increases from 45, cot 2 diminishes until dilatation is zero, when the condition of the medium is unstable. This may be demonstrated graphically. In Figs. 3 and 4 A A, BB and CC are the three axes of symmetrical distortion, and the full-line circles represent the spherical grains in contact. (See also Fig. 1, page 83.) Fig. 3. Fig. 4. Fig. 3 shows a loss 2 A A' in height. Fig. 4 shows a gain 4AA' - (AA) 2 . 2AA' = (AA)* . 2AA'. Whence we have the dilatation dV_(AAY.'2AA' ~V = (AA) 3 And dividing by the strain 2AA'/AA and changing the sign, we have for the coefficient of dilatation AA (AA) 2 . 2AA' 2AA' ' (AA) 3 = 1. 207 A. Then as regards the inequalities of pressure p r = 2p t = \p' ', resulting from such symmetrical distortional strains in the principal axes of strain, since there is no work done on the grains it follows directly, putting p" for the mean pressure, p r for the normal in the direction of the strain, and p t for either one of the tangential since these are principal stresses Pr+2pt = 3p" (320), and since there is no work done on the grains, Pr = tyt (321), whence by (320) Pr = ip", Pt = lp" (322). 208. It is to be noticed that contraction strains, such as that discussed in the last article, the strain being in the direction of one of the axes of distortion, are the only symmetrical strains when a = 0, and it does not follow that the coefficient of dilatation for small unsym metrical strains is unity. But it does follow from virtual velocities that if p" is the mean pressure in a kinematical medium without limit, that the normal pressure resulting from a local disturbance cannot be greater than 2p" and must be greater than zero if p" is finite. From this we have the proof of the important theorem : That whatever the coefficient of dilatation may be, a disturbance such as might be caused by the removal of any number of grains from a space in an otherwise uniform medium, without relative motion, would be attended with inward radial displacement of the grains from infinity throughout the entire medium. For, as has just been shown, p r must be greater than zero ; so that there can be no cavity greater than the space from which the grains can exclude 190 ON THE SUB-MECHANICS OF THE UNIVERSE. [209 other grains, and there can be no dilatation without the displacement of grains, so that as the ideal excavations proceeded the grains would follow inwards, and as there is no elasticity and the grains are all under pressure, each grain as it disappears must cause inward movement from infinity; for as the coefficient of dilatation cannot be infinite, the grains being smooth spheres without friction (so that any binding or jamming would be impos- sible) every grain would be under pressure. Q. E. D. Thus the relation between the tangential and normal pressures would depend upon nothing but the coefficients of dilatation, and if these were constant the normal and tangential pressures would be constant. But such constancy would depend on there being angular similarity in the arrange- ment of the grains about every axis through the centre of disturbance, which similarity does not exist in the normal piling. It is therefore certain that the inward strains, although having six axes of similar arrangement symmetrically placed, would be influenced by the crystalline formation of the uniform piling; particularly at great distances from the centre of disturb- ance. For when the distances from the centre are large the strains would be so small that the crystalline characteristics of the uniform medium would have undergone very slight modification, whereas near the centre where the displacements are greatly larger the unsymmetrical characteristics would be greatly modified. On these grounds it appears certain that the coefficients of dilatation would be greatest at an infinite distance from the centre and would gradually diminish ; in which case the tangential pressure would fall and the normal pressure rise gradually as they neared the centre, satisfying the conditions of virtual velocities and the condition for equilibrium, which latter requires that at any distance r from the centre p r + 2p t = p". What the mean of such coefficients might be is doubtful, but it seems probable that they would not differ greatly from the coefficient unity, which is the smallest coefficient for symmetrical distortion. Whatever these coefficients may be it follows from the paragraph last but one, that the dilatation resulting from the inward strain must occupy the space from which the grains were absent, so that the sum of the normal and tangential stresses would be equal to the mean pressure of the medium, or p r + 2p t = 3p". 209. From the conditions of geometrical similarity in the case of uniform continuous media it appears : (i) The size of the uniform grains has no effect on the dilatation or mean pressures resulting from continuous uniform distortions. Therefore similar and equal continuous finite distortional strains will produce similar 209] CONSERVATION OF MEAN INEQUALITIES AND THEIR MOTIONS. 191 and equal dilatations whether the grains are indefinitely small or of any finite size. (ii) The size of the uniform grains in a continuous medium does affect the dilatations resulting from strains other, than continuous uniform distor- tional strains. To prove these theorems. If we consider two finite media of which the parts are exactly similar in shape, number, and relative position, but in one of which the scale is A and the other B, these media will be geometrically similar except as to scale. Thus whatever strains in proportion to the constant parameters, A and B respectively, these media may undergo, the proportional similarity will hold, and this extends to the dilatations, the coefficients of which will be equal. Q. E. D. If however instead of considering these similar actions within spaces proportional to the scales A and B, we consider these proportional actions within equal spaces, the principle of similarity disappears unless the positions and strains are such that there is perfect uniformity throughout the medium. This proves the first theorem. Perfect uniformity exists in the case of grains in uniform piling subject to equal distortional strains whatever the values of A and B, provided the spaces are such that there is no sensible effect from the boundaries. Q. E. D. It is thus proved that for other than equal uniform strain there cannot be similarity in the effects in equal spaces in media of which the scales of similarity A and B differ. Thus if the strains in the medium in which the scale is A are subject to variation on that scale, while those on the scale B are subject to similar strains on that of B, then the effects of these variations taken over equal spaces will of necessity differ. Q. E. D. Then since the dilatations resulting from parallel continuous strains are in no way dependent on the size of the grains, even if these are infinitely small or have any finite size, the question arises as to what would be the difference in the dilatations resulting from finite similar local disturbances about negative centres in two media in one of which the grains are infinitely small and in the other finite. In the first place it appears that as far as regards the dilatations resulting from uniform parallel distortional strain these would be independent of the size a. And it can be shown that these are the only dilatations if r 6 whence it appears that X--^ (340). Also dividing the last term in equation (338) by r 2 we have for the radial displacement at a distance r which is the same expression for the radial displacement as that assumed. So that both conditions are completely satisfied. 216] CONSERVATION OF MEAN INEQUALITIES AND THEIR MOTIONS. 201 216. In this section it is assumed that there is no diffusion. Having in the previous articles in this section effected the analysis of the inward strains and the consequent dilatations for only negative spherical disturbances resulting from the absence of grains, before proceeding to consider the corre- sponding analysis for the other inequalities in the density of mean matter, it seems convenient to proceed with the analysis necessary to determine the effects such negative disturbances may have on each other when existing within finite distances of each other. Any such action must depend on the interference of the strains outside the respective singular surfaces, and any attraction of the centres resulting from such interference must be a function of the distance between the centres. From Arts. 209 and 212 we have perfect similarity in the strain resulting from uniform distortions, from which it follows that such strains from different negative centres superimpose without affecting their respective dilatations, and hence can in no way interfere or attract one another. In the case of the strains resulting from finite values of a- owing to the curvature resulting from distortions, the strains from different negative centres at any finite distance must interfere. This appears in Arts. 209 and 212, in which it is shown that for other than equal uniform strains there cannot be geometrical similarity in the effects in equal spaces, in media of which the scales are different. For, applying this to the case in hand, since the diameter of the grains, ff 1 say, is common to all the grains, while the number of grains absent as well as the radii of the singular surfaces may differ in almost any degree, the dissimilarity at once appears. For the sake of clearness we may consider in the first place two cases in both of which the a- has the value o- a , and the singular surfaces both of radii r 1} but in one of which the volume of the grains absent is r a 3 , and in the 3 ., 4?r other r b 3 . Then by equation (331) we have for the dilatation at a distance r for the centre a !zr r 3^i/' 1 _ M 3 a r* ( ZrJ ' and for the centre 6 477 477 3 (Tj / 0-!\ T Tb ?A ~2rJ' 202 ON THE SUB-MECHANICS OF THE UNIVERSE. [216 and neglecting 0^/2^ for the present, as small, multiplying by r 2 dr and integrating from r, to r = oo we have for the dilatation, taking &> to express it, From the expressions in the preceding paragraph for the total dilatation resulting respectively from the two centres considered as if each were the only centre within an infinite distance, it appears in the first place that the dilatation resulting from the product to express the total dilatation from r to r = oo resulting from a single negative centre, then as has just been shown (342). Then the number of such singular surfaces which would occupy an empty spherical shell of radius r B when arranged in closest order would be approximately And by equation (341) the total dilatation of each of the N' surfaces outside the surface 4?rr 2 is 216] CONSERVATION OF MEAN INEQUALITIES AND THEIR MOTIONS. 203 Multiplying <>! and a> B by N' we have for the respective total dilatations N'u^N'-- *^ (345) and JX-W B =J\- ~~-- (346). 3 r B Subtracting these equations as they stand we have (347). Then from equation (347) it follows that the dilatation resulting from any number of negative similar disturbances (if the singular surfaces are at an infinite distance from each other) will be , 47rr 3 a- "T"-* 1 while if these surfaces are arranged in closest order the dilatation will be 477T 3 ff ~3~ ^- Whence since r B is greater than r^ it is shown that, no matter how accomplished, the dilatation resulting from negative centres diminishes in the ratio n r*' as the centres of the singular surfaces approach until they are arranged in closest order. This proves the diminution of the dilatation owing to the diminution of the variations of strain as the centres approach or the diminution of the dilatation owing to the diminution of the curvature of the normal piling in the medium due to dissimilarity. Q. E. D. From the proof of the foregoing theorem it also appears how it is that the dilatations resulting from distortion do not interfere however much they superimpose, for since the dilatations resulting from distortion in no way depend on the curvature in the medium, as curvature, they depend only on the strain, whereas the diminution is in the variations of the strain. In order to prove the attraction of the negative centres it is necessary to consider the effects of the pressures in the medium. These have already been discussed in Art. 213, equations (332) to (334), in which it is shown that the dilatations resulting from curvature are subject to the mean pressure p" and satisfy the condition of virtual velocities. In dealing with attraction it might seem necessary first to prove or assume that the singular 204 ON THE SUB-MECHANICS OF THE UNIVERSE. [217 surfaces are also surfaces of freedom which can be propagated in any direction through the medium, for as the medium is elastic in consequence of the finite relative motion, if we can find the variation of the work done by the external media on the singular surfaces owing to variation of their distances, it becomes possible to separate the active effort from the passive resistance. Multiplying the member on the right of equation (347) by p" we have as the expression for the difference of the energies in the media when the N' singular surfaces of radius r^ are at an infinite distance from each other, and when the N' singular surfaces of radius ^ are arranged in closest order within the surface r B . This difference in the energy proves the existence of attractions what- ever may be the passive resistance owing to want of mobility of the singular surfaces. These attractions as obtained by neglecting a- 2 are the only attractions between negative centres of disturbance which are small compared with their distances apart, as follows from the fact already proved that the aggregate dilatation resulting from distortional strains depends only on the volume of the absent grains. 217. The law of the attraction of negative centres appears at once from the analysis. If instead of taking the total dilatation from r B to r = , as in equation (346), we take the dilatation from r R to r, where r is greater than r B , the dilatation from the JV' singular surfaces in closest order is 7rr 3 [ r a- , the second member of which expresses the potential of attraction between the two equal negative centres. This multiplied by a second negative inequality and differentiated with respect to the distance between the centres expresses the effort of attraction of the centres as 3 (352). And again, although not previously noticed, it appears at once from equation (351) that, if instead of the limits of integration being from 1\ to r, they are taken from r to r = oo , we have 47rr 3 f*r .,, 4nrr 3 a- p v> -. -r 2 dr = p ' ^ . - (363). 3 J r ?' 4 3 r This integral must have some significance as a potential. And it appears on multiplying equation (353) by 4nrr 3 /3, which is an expression for a positive 222] CONSERVATION OF MEAN INEQUALITIES AND THEIR MOTIONS. 209 inequality equal to the negative inequality, and differentiating with respect to the distance between the centres, when the equation becomes : The second member expresses an attraction between the positive and negative centres. 221. The significance of the two integrals. In Art. 216 from equation (346) it is shown that negative centres attract, therefore if there were a choice of two general integrals of the dilata- tion from a negative centre, from one of which in the case of negative centres there would result a repulsion, while the other would result in attraction, it is certain that the integration which would result in the attraction is the only one between negative centres whatever might be the significance of the other integration. And this is what actually occurs. If instead of the limits from r x to r as in equation (351) the limits are taken from r to oo as in equation (353), then taking account of a second negative singular surface we should have for the complete potential : which differentiated with respect to r is: f r which expresses a repulsion. Hence this cannot be the integral for the attraction of one negative centre for another. As already remarked this form of integral of the dilatation from a negative centre must have a significance, and significance appears when we substitute a positive inequality 4>7rr 3 {3 in place of the negative inequality 4<7rr 3 /3 in the last expression for the attraction, which becomes /4arr * Thus we have the expression for the attraction of equal positive and negative centres resulting from the finite size of the grains. 222. The intensity of the attractions of equal positive and negative inequalities. In the first place it is to be noticed that the intensity of the attraction between equal positive and negative inequalities as in the last expression R. H 210 ON THE SUB-MECHANICS OF THE UNIVERSE. [223 (Art. 221) is as cr to r^ of the total intensity of attraction between positive and negative surfaces. Indeed the expressions last but one and last (Art. 221) only indicate the significance of the two integral potentials. And such intensity as they express in no way depends on the curvature. This becomes clear if we recognise that in the case of a displacement of n grains the strains from the negative centres are negative and extend to infinity, while the strains resulting from the positive centres are positive and extend to infinity. The components of the negative strains cancel with the components of the positive strains with which they are parallel ; hence the diminution of the dilatation as the displacement diminishes in no way depends on the curvature but wholly on the cancelling of the distortional strains. It thus appears that in order to express the effort to restore the normal piling in the medium, we have only to substitute the radius of the singular surface in the place of cr in the last expression (Art. 221). Thus for the total effort, in the complex inequality resulting from the displacement of a volume of grains 4?rr 3 /3 through a distance r, to restore the normal piling we have R - - n" / 47rr 3 y T 1 mTl P ^ 3 ) r z ^ooo> Q. E. F. 223. It may be noticed that in obtaining equation (355) no use has been made of the potential of attraction. This is because the inequality caused by a displacement of a volume of grains under the pressure p", which has the dimensions ML 3 T 2 , is essentially one displacement, not two equal and opposite displacements as in the case of two equal negative centres, in which the relative displacements of energy have no effect on the mean position of energy in the medium. This may be shown by subjecting the expressions for the effort of attraction between negative centres, and the effort to reverse the displace- ment in the case of complex inequality, respectively, to further analysis. Taking the effort of attraction of two equal negative centres, as in equation (354), to be : ,/ 4-7rr 3 o- P 3 >2' and the effort to reverse the displacement in the complex inequality, as in equation (355), to be : \~2TJ f-' and then integrating each of these expressions from ^ to oo , we have as 224] CONSERVATION OF MEAN INEQUALITIES AND THEIR MOTIONS. 211 the energies resulting from the dilatation from outside the singular surfaces of radius r lt /47rr 3 Y 1 "nrK' and Then to obtain the expressions for the potential of attraction for either of these respective energies, the factor 1/r must be separated into two factors proportional to two inequalities of the same or opposite sign in accordance with the sign of the product of the inequalities. Then multiplying the factor which has the positive sign by 1/r we have the potential, while the other factor is numerical and represents the attraction of the centres. In the case of two negative centres, taken as equal for simplicity, as the signs of the inequalities are the same we have for the potential : and for the attraction : And in the case of the complex centre, since the product of the centres is negative, we have for the potential : , /47rr., 3 Y 1 Tl v~3/ r and for the attraction : Whence it appears that in the complex inequality both the potential and the attraction are irrational. Whence it is proved, since the effort is real, that the absolute displacement of energy is one displacement and not two. 224. The electrostatic unit of electricity is defined as the quantity of positive electricity which will attract an equal quantity of negative electricity at unit distance with unit effort. This unit as is shown in Art. 223 is irrational. An expression for the unit corresponding to the electrostatic unit is obtained from either of the last two expressions in Art. 223. Thus from the first of these, putting r\ = r and r= 1, we get: 142 212 ON THE SUB-MECHANICS OF THE UNIVERSE. [225 And from this, since all the quantities under the radical are positive, we have the condition = l ........................... (356), from which if p" is known r may be found. 225. From the analysis in Art. 223 it is easily realised that there is a fundamental difference in attractions between two negative centres, and the attraction of two equal centres one positive and one negative. It has been shown (Art. 217), that the attraction of two negative centres corresponds, in every particular, to the attraction of gravitation as derived from experience. And it now appears that the alteration from a positive to a negative inequality correspond to the statical attraction of the positive for the negative electricity. Not only then has the step at which Maxwell was arrested that of accounting by mechanical considerations for the stresses in the dielectric been achieved, and a moot point of historical interest settled, but as now appears a definite error as to the actual attractions has been revealed. This error is in the general assumption that electrified bodies repel each other. As this may not be at once obvious it will be discussed in the next article. 226. To show that positively electrified bodies do not repel. It has been shown in Art. 225, neglecting the small attractions of two positive or two negative centres, that the efforts of attraction between equal positive and negative centres, at any distance r, are equal and opposite. If then in the same line we have two equal complex inequalities arranged so that their displacements are opposite, the negative centres being outwards as -\ ---- [-, the effort of attraction of one of these complex inequalities would not in the least be affected by the other complex centre. Hence there is no attraction between two positive centres, the only effort to separation of the two positive centres being between those of the two complex inequalities, the effort in either being the same as if the other was not there. Hence the only efforts are those of attraction. Q. E. D. It should be noticed that these attractions are quite apart from the repulsions resulting from two positive centres owing to the curvature and finite size of the grains as in gravitation, and further that, other things being the same, the ratio of the attractions between positive and negative and the repulsions between positive centres is as rjo; and hence the repulsion may be neglected as compared with the attraction. 227. In the analysis for the effort of attraction of negative inequalities and that to reverse the displacement of a complex inequality the terms in 228] CONSERVATION OF MEAN INEQUALITIES AND THEIR MOTIONS. 213 the expressions for the contraction strains which involve powers of r*/r* the ratio of the volume of grains absent divided by the volume enclosed by the singular surface have been neglected (Art. 214, equation (337)) and it is this simplification only which renders the law of attraction as the inverse square the law of attraction of the singular surface at a distance. But this in no way limits the variation of the stresses over those portions of the space in and between the parts of the two singular surfaces which are within indefinitely small distance of each other. Such limits can only be de- termined by taking into account the higher terms which have been neglected. This analysis I have not attempted. But it seems to me very important to notice this omission, as it appears that the attractions or repulsions ex- pressed by the higher powers of 1/r, when the surfaces are indefinitely near, must be of great intensity, so that owing to sudden variations the work done in separating the surfaces must be extremely small. These characteristics are those of cohesion and surface tension and they promise to account by mechanical considerations for the hitherto obscure cohesion between the molecules as belonging to the attractions resulting from the finite value of the diameter of the molecules divided by the curvature resulting from distortion, or, we might say the complement of gravitation. 228. The fourth and last class of possible local disarrangements causing strain in the normal piling, with some degree of permanence, in the schedule (Art. 203), is that which does not depend on the absence, presence, or linear displacement of grains, but does depend on local rotational displacement of grains about some axis. Then since there are no resultant rotational stresses or rotational strains in the medium, or rotation of the medium, the rotational inequalities must be arranged so as to balance. Any such rotation of a portion of the medium would be attended with dilatations. But it does not follow that the dilatations would in all cases be so small that the coefficient would be unity. Then noting that the medium in virtue of relative motion of the grains is in some degree elastic, if we conceive that by two opposite couples about parallel axes at a finite distance two equal spheres of grains in normal piling having their centres on the respective axes, could be caused to turn about their axes through opposite but equal angles 6 and 6, the actions would be reciprocal, and supposing the actions to start from the medium in normal piling, when the angles were so small that at the surfaces there was no change of neighbours, the only effects would be strains attended by dilatation about the axes, which on removal of the couples would revert, restoring the 214 ON THE SUB-MECHANICS OF THE UNIVERSE. [229 unstrained medium. And in this case the coefficient of dilatation would be unity. Then if the angles were increased the strains would be such that over the equators of the spheres the grains would change neighbours, diminishing the dilatation; so that on the couples being removed the spheres would not revert and would not restore the unstrained medium, nor would the angles 6 and 6 be zero. Those portions of the surfaces of the spheres nearer the axes, where the strains had not been sufficient to cause a change of neighbouring grains, would be subject to stress tending to diminish the angles 6 and 6, while in those portions where the grains had changed their neighbours the stresses would be resisting this change, so that the result would be a balance of strains and stresses, leaving the system in equilibrium under the relative rotational strains and stresses and dilatations extending outwards from the surfaces of each till they vanish at an indefinite distance. The strains and stresses extending from the sphere of which the residual angle was 6, since the axes are at a finite distance, could not in any way affect strains of shear having the angle 6. But if the shears were in a plane perpendicular to the axes arid at a finite distance from each other, the strains and stresses being opposite would cancel, and the dilatations would diminish in such manner and proportions that there would be efforts to approach proportional to the inverse square of the distance. Or, if, other things being the same, the spheres were at finite distances on the same axes, they would still be under efforts to approach, owing to the cancelling of the strains and diminution of the dilatation. And in either case, other things being the same, if one of the poles at the axis of either one of the spheres were reversed the result would be an effort of repulsion. Q.E.F. Thus efforts of attraction correspond exactly with those of fixed magnets, and thus we have been able to account by mechanical considerations for the magnetism which has any degree of permanence. 229. Having in the foregoing articles of this section accomplished the analysis necessary for the determination of the attraction of negative centres of disturbance, the efforts to reverse the displacement in the complex inequalities, discussed the probability of cohesion as the result of the terms neglected in the analysis for the efforts of the negative centres, and effected the analysis for the efforts of attraction resulting from opposite rotational strains about parallel axes at a distance ; it remains to complete the section by effecting the analysis for determining the mobility of the singular surfaces. 230. From Theorems 1 and 2, Art. 204, and more particularly in Art. 214, we have defined the effects of local inequalities in the mean mass, when a/\ is large, on the arrangement of the grains and the distribution of the strains or THE UNIVERSITY 232] CONSERVATION OF MEAN INEQUALITIES AND THEIR MOTIONS. 215 in the medium about both negative and positive centres. Thus it has been shown in the case of a negative centre that the inward strains would be such that the resulting dilatation would pass the point of stability and reform, causing a nucleus of grains in normal piling which might increase until it was stopped by meeting the inward strained, and consequently dilated, normal piling. This meeting of the two closed surfaces, the outer surface of the nucleus in normal piling with the inner surface of the inwardly strained normal piling, affords the first clue to the possibility of a surface of freedom. For, since the grains are uniform equal spheres, there can be no fit between the grains in normal piling at the one surface and the grains in strained normal piling at the other. To use a mechanical expression the grains cannot pitch, and consequently there is a spherical shell of grains in abnormal piling which constitutes the singular surface a surface of weakness if not a surface of freedom. Then by Theorem 1 it follows, whatever may be the arrangement of the grains and whatever the exchange, there can be no change in the arrangement or number of the grains. Therefore these surfaces of misfit are fundamental to all inequalities in the mean mass. 231. Since there is no regular fit in the shell of abnormal piling at the singular surface, say of a negative centre, and each of the grains is in a state of relative motion, each of the grains is in a state of mean elastic equilibrium such that half the grains are on the verge of instability one way and half in another. If, as by the existence of another negative centre at finite distance there is an effort of attraction, however small, it would, since there is no finite stability, in the first instance cause change of neighbours, and if sufficiently strong it would entirely break down the stability and cause one or both the centres to approach at rates increasing according to the inverse square of the distance, since as by Theorem 1 there would be no change in the mean arrangement of the grains and the viscosity may be neglected. 232. This brings us face to face with questions as to the mode of dis- placement of the singular surfaces, as well as the manner of motion of the inequalities in the mean mass which constitutes the centre, which have not as yet been discussed. In the first place it appears at once, however strange it may seem, that in the case of a negative inequality, to secure similarity in the arrangement of the infinite medium the mass must move in the opposite direction to the inequality, otherwise there would be no displacement. And further the opposite displacements of the positive and negative masses must be equal, subject to the condition that for every indefinitely small displacement of the negative inequality there should be an equal and opposite and exactly similar and similarly placed displacement of positive mass. 216 ON THE SUB-MECHANICS OF THE UNIVERSE. [233 233. Then, apart from vortex rings which cannot exist in a medium in which tr/\ is so large that there is no diffusion of the grains, it appears that the only way in which the conditions in the last paragraph are realised is by propagation. This admits of definite proof. If we conceive a singular surface about a negative centre to be moving upwards through the medium, as it rises the upper surface will be con- tinuously meeting fresh grains. Then if the motion continues one of two things must happen. The grains must be shoved out of the way, in which case all similarity of the arrangement would be destroyed, or the grains must cross into the singular surface. If this were all we should again have the similarity upset, as the singular surface must increase to accommodate grains coming in. But if at the same time as the grains enter the singular surface from above grains cross out of the singular surface in exactly the same numbers and vertically under the grains which enter from above, the motion of the singular surface would not disturb the similarity of the arrangement beyond such limits as the elasticity of the medium admits. This manner of progress of a singular surface is that which has several times been referred to as propagation. It is strictly propagation. For if there is no general uniform mean motion the grains within the singular surface are at rest, while if the medium has such mean motion it would not affect the motion of the singular surface though it would affect the rate of propagation since that would include the propagation through the moving medium. This then is the only mode of displacement of a singular surface the propagation. N.B. This law of propagation would not prevent strains in the singular surfaces such as might be caused by undulations in the medium corresponding to those of light. 234. It may seem that displacement by propagation does not of necessity entail displacement of mass; nor would it if there could be propagation without local inequalities in the mean density of the medium. But in a uniform medium, without inequalities, there can be no propagation as there is nothing to propagate. Thus it is that the inequality in density, the integral of which is the volume of the grains, the replacement of which would restore the uniformity of the medium, obliterating the inequality, constitutes the mass propagated. And as this, for a negative centre, is negative, its propagation requires the displacement of an equivalent positive mass in the opposite direction to that of propagation of the negative inequality. 235. It thus appears that the distribution of the density of the positive moving mass is at all points the same as the distribution of the density of 237] CONSERVATION OF MEAN INEQUALITIES AND THEIR MOTIONS. 217 the negative inequality, and as this on changing the sign is the same as the dilatation at all points, the density of the positive moving mass is equal to the dilatation. The dilatation at any point in the medium resulting from a negative centre is expressed by : 4 Trr^o 3 3 r 4 in which r is greater than r l9 while r /r is small. It thus appears that, since the density of the medium is unity, the motions of the medium of unit density necessary to equal the displacements of the positive mass at density 47rr 1 r 3 /3r 4 , which can under no circumstances be greater than 4 < 7rr 3 /3r 1 3 are almost indefinitely small. 236. Taking U a as the velocity of the singular surface and u" as the velocity of the medium at any point outside the singular surface, since there is no mean motion of the grains within the singular surface, u' is everywhere small compared with U s , Of course this does not affect the integral displacement of mass integrated over the medium from r t to oo . But it does affect the displacement of the apparent energy of the motion of the inequality which is taken to be 4?rr 3 /3. For if we integrate w" 2 over the medium it is small compared with u s \^ This apparent paradox, however, is explained on recognising that the grains being uniform, since <7/\ is very large, the conduction of energy is nearly perfect ; so that the rate of displacement of momentum does not depend only on the convections of the order u" 2 p but depends also on the conductions tt> since these actions are the direct result of the propagation of the singular surface through the medium, so that there is no change in the strains, dilatations, or the mean arrangement within or about the singular surface for an infinite distance. It is easy to realise the way in which the strains at . any fixed point contract and expand as the singular surface moves away from or approaches the point. 237. In the foregoing reasoning in this section no account has been taken of the possibility or impossibility of any lateral motions of the grains which might be necessary to maintain the arrangement. That such lateral motions of the individual grains would be necessary is certain ; but it does not follow as a matter of course that they would be possible without creating temporary strains which would in the first instance require a certain 218 ON THE SUB-MECHANICS OF THE UNIVERSE. [238 acceleration to start them. But once started the action, since it involves a certain definite rate of displacement of mass, would proceed at a uniform rate, supposing no viscosity, and the medium unstrained by other centres. That the necessary acceleration to effect the start must depend on the particular arrangements inside and outside the singular surfaces, is clear. And from this it may be definitely inferred that the number of definite primary arrangements in which the stability to be overcome by acceleration is within finite limits, is finite. Whence it follows that the number of singular surfaces having different numbers of grains absent, in which the limits of stability are within finite limits, is finite ; and these would be the only surfaces of freedom. Q.E.D. . It should be noticed that the expression " primary arrangements " is here used to distinguish those singular surfaces which do not admit of separation into two or more singular surfaces of freedom. It is thus shown that singular surfaces about negative inequalities admit of motion in all directions, by a process of propagation, without any mean motion of the grains within the singular surfaces, while the motion of the mass outside the singular surfaces, when there is no other inequality within finite distance, is such as to maintain the similarity in the arrangement about the centre entailing the displacement of the mass (47rr 3 /3) in the direction opposite to that in which the singular surface is displaced by propagation. 238. We have thus effected the analysis for the determination of the mobility of solitary negative centres. And it may be taken that the analysis for positive centres would follow on the same lines with the exception of the sign of the inequalities. There still remains to consider the possibility of the combination of primary singular surfaces, forming singular surfaces with limited stability in which the grains absent or present are the sum of the grains, the absence or presence of which constitutes the inequalities of the primary singular surfaces combined. It has been shown by neglecting certain terms (equation 337) that negative inequalities attract according to the inverse square of the distance and in Art. 227 it has been pointed out that the terms neglected are such . as would indicate cohesion or repulsion between the singular surfaces when closest ; and in such conditions there would be a connected singular surface however many were the primary singular surfaces cohering, so that mobility of the whole group would be secured. In the case of two primary negative inequalities in which the numbers of grains absent are different, although neither of these admit of separation into two or more separate inequalities, there does not appear any impossibility, 241] CONSERVATION OF MEAN INEQUALITIES AND THEIR MOTIONS. 219 except such as results from their limited stability, why they should not combine if their velocities are sufficient to break down the limited stability. In such case it seems that one or other of two results must happen ; either the breakdown would be temporary, the two centres immediately reforming as by the rebound, setting up a disturbance in the medium which would be propagated through the medium, or they would reform into a single negative centre, in which the volume inside the reformed singular surface would be less than that of the sum of the volumes within the two singular surfaces of the two primary inequalities, or in some other way manage to diminish the dilatation ; and in this case also there would be a disturbance in the medium. 239. It is certain that when negative inequalities are arranged in their closest order, there is cohesion between the adjacent singular surfaces which resists the separation of the adjacent singular surfaces but does not cause attraction between the singular surfaces when these are at a distance which is greater than some small fraction of the radius (rj) of the singular surface (Art. 227). It is also certain that, when under the conditions stated, the singular surfaces would still attract one another at a distance as in equation (348) : And thus if we consider N the number of such negative centres within a distance r 3 to be indefinitely large as compared with r lt since they are in closest order the centres would be in stable equilibrium under normal and tangential pressure, as in the case of gravitation. 240. If the number of grains absent about each of the centres which constitute the total negative inequality is the same, and by some shearing stress the inequality is subject to a shearing strain, there would result dilatation, doing work on the medium outside, which would be maintained as long as the shearing stress ; but since all the centres are equal, whatever arrangements of the grains under the stress take place between the centres, there would be no absolute displacement of mass. And the result would be the same whatever might be the number of grains absent in the primary inequality. 241. Thus we may consider what the action would be if we had two such total inequalities A and B differing in respect to the number of grains absent in their primary inequalities say that the number of grains absent is greatest in A. If these total inequalities are brought together by their attractions the grains in abnormal piling which separate the two total inequalities A and B 220 ON THE SUB-MECHANICS OF THE UNIVERSE. [241 may be, for simplicity, taken parallel to a plane which is a plane of weakness in the medium. If, then, there are shearing strains parallel to this plane such as cause grains from the inequality A to pass to the inequality B in the abnormal piling in the plane of weakness, so that in this piling the arrange- ment, instead of the two primary inequalities in which the numbers of grains absent are A and B, is two equal negative inequalities in each of which the number of grains absent is : A+B A+B 2 '2 s and one complex inequality in which the numbers of grains absent in the positive and negative centres are : A-B B-A 2 ' 2 ' in this case it at once appears that besides the attraction correspond- ing to gravitation and cohesion, the effect of the rotational strain would be to cause absolute displacements of mass, which, by Art. 225, would cause efforts of reinstitution between the strained aggregate inequalities, correspond- ing to electric attractions. But as the attraction would be normal to the surface of weakness, while for reinstitution the action must be tangential, the rotational strain might be stable, and the attraction might hold when the strained aggregate inequalities were forced apart. If the rotational strains were sufficient the normal attractions might overcome the normal stability of the complex inequalities, and in that case there would be a sudden tangential reversion, which, as there is absolute displacement of mass, would in the recoil reverse the complex inequality and so on, oscillating until the energy was exhausted in setting up undulations in the medium which would be propagated through the medium at the velocities of the normal or transverse waves as in light. If we have two aggregate inequalities in one of which the primary inequalities are not combined, while in the other the different primary inequalities are combined, we should have three total inequalities A, 5/2, C/2 in the arrangement : B ' C B C 2 + 2 + 2 + 2 2 "' 2 "' and two complex inequalities : B C 2 2 Then if the strains were sufficient the normal attraction might overcome the normal stability of the complex inequalities, causing a reversion. In this case however it does not follow that the reversion would be complete and so 241 A] CONSERVATION OF MEAN INEQUALITIES AND THEIR MOTIONS. 221 reinstitute A, B/2, 0/2 ; for since the work done by the strains might be sufficient to overcome the resistance to combination of B/2 and (7/2, the recoil from the breakdown would cause a total or partial combination of 5/2, (7/2, instituting B the aggregate inequality and so diminishing the energy available for undulations, thus affording an explanation by mechanical considerations of the part electricity plays in instituting the combination of molecules into compound molecules with limited stability. It is to be noticed that the effects of rotational strain between the aggregate negative inequalities which differ as to the number of grains in the primary inequalities, correspond to the effects produced when resin is rubbed by silk or frictional electricity and thus the so-called separation of the two electricities by friction is accounted for by mechanical considerations. Having shown that negative inequalities may not only attract, but may also cohere when in contact, we may return to the consideration of the significance of the fact mentioned in Art. 217, that the attractions correspond- ing to gravitation as well as cohesion depend solely on the numbers of grains absent, while the volume within the singular surfaces, which determines the volume from which one centre excludes other centres, depends on the possibility of some arrangement between the grains in abnormal piling and those in strained normal piling (Art. 214). 241 A. It is shown in Art. 217 that for any displacement of a negative inequality there must be a corresponding displacement of positive mass in the same plane and in the opposite direction. From this it follows that as two negative centres approach under their mutual attractions the mass in the medium recedes, which is an inversion of the preconceived ideas. Such action however is not outside experience, since every bubble which ascends from the bottom in a glass of soda-water involves the same action. The matter in the bubble having the density of the air requires the descent of an equal volume of water at a density 800 times greater than that of the air. It is the negative inequality in the density of matter which under the varying pressure of the water causes the negative or downward displace- ment of the material medium water and the positive or upward displace- ment of the negative inequality in the density within the singular surface. In order to recognise the significance of the parallel drawn in the last para- graph it must be noticed that in this research we have adopted a definition of mass, which, although satisfying the laws of motion and the conservation of energy, is independent of any other definition of matter. Hence it is open to us to suppose that what we call matter may be such, that if expressed in the notation so far used in this research, would represent local negative inequalities in the mean density of the medium. Then since, as has already been shown, and will be confirmed in what is to follow, the definition of matter as representing negative local inequalities 222 ON THE SUB-MECHANICS OF THE UNIVERSE. [241 A in the mean density of the granular medium completes the inversion and removes all paradox, this definition of matter is adopted as the only possible definition. We then have for the negative inequality : where p" = 1. And for the volume from which one negative inequality excludes other similar inequalities, when in closest order, we have by equation (343): 4 4 3'3 ir ' n ' Then dividing the negative inequality by the volume from which other centres are excluded we have as the expression for the mean density of the negative inequalities when in closest order : Then again dividing p" the density of the uniform medium by IT, the mean density of the inequality, we have in the ratio of the two densities a number without dimensions as expressed by < 358 >- In equations (357) and (358) II is used to express the mean density of the negative centres when in closest order. Thus II is the maximum mean density of the negative centres for any particular negative centres. It does not however follow that U expresses the maximum mean density of negative inequalities for all negative inequalities when in closest order. For as pointed out there is no proportional relation between the number of grains absent and the volume within the singular surfaces for inequalities which differ. But it does follow, from the fact that the number of centres which have surfaces of freedom is finite, that there must be some negative inequality of which the mean density is a maximum. And from this it again follows that p"/II must have a minimum value. Then taking fl to express the minimum value which, whatever it may be, is constant and without dimensions, we may express the densities of all the other negative inequalities in terms of ft, making use of any system of units. Then if, as before, the density of the medium is unity, the maximum density of negative inequalities is : 241 A] CONSERVATION OF MEAN INEQUALITIES AND THEIR MOTIONS. 223 and if the mean density of an inequality is n times less than the maximum inequality it is expressed by: J^ nil' And again, if, changing the unit of density, the density of the medium becomes nfl, the maximum density of negative inequalities is expressed by n. The proof that the quotient O of the density of the uniform medium divided by the maximum mean density of the negative inequalities is a numerical constant, independent of units, giving us, as it were, the gauge by which we can compare the quantities, as obtained, in this and the previous sections, with the evidence derived from actual experience, completes the consideration of the possible strains other than the undulatory strains (con- sidered in Section XIII.) resulting from the conservation of inequalities in the mean mass, which formed the subject of this section. SECTION XV. THE DETERMINATION OF THE RELATIVE QUANTITIES a", A", o-, G, WHICH DEFINE THE CONDITION OF THE GRANULAR MEDIUM BY THE RESULTS OF EXPERIENCE. THE GENERAL INTEGRA- TION OF THE EQUATIONS. 242. IN the last paragraph of Section XIII. it was noticed that, up to that stage, it was not possible, for want of evidence as to the actual rates of degradation of light, to complete the determination of the values of a", cr, \". And further, that as the equations (310 313) have been obtained by neglect- ing all secondary inequalities, they afford no evidence as to the limits imposed by dilatation on the shearing and normal strains. These disabilities have not as yet been altogether removed. But we have, in the last section, obtained expressions, in terms of p", a", a, \", for the attraction of negative centres, which correspond to those of gravitation. Also in the last article it is shown that what is known as "matter" corresponds with the inequality in the medium resulting from absence of grains. Also it is proved that there must be a finite maximum mean density for negative inequalities when in close order, which corresponds to the mean of the heaviest matter. And further, it is shown that the mean density of the uniform granular medium, divided by the maximum density of negative inequalities, is a number without dimensions expressed by H whence we are enabled to measure the density of any inequalities in closest order, in any system of units. We are thus in a very different position, as regards evidence, from what we were at the end of Section XIII. 243. By the last article of Section XIV., taking 22 as expressing in c.G.S. units the density of the matter platinum, which is approximately the densest form of matter, we have unity for the density of the matter water in C.G.S. units. Then for the density of the granular medium in C.G.S. units we have 220, where the constant number H has still to be determined. 245] THE VALUES OF Ot", X", " ............................... (359). Also the mean density of the medium p" or unity becomes p = 220,0" ............................... (360). And, if in C.G.S. units of matter, p expresses the mean density of any negative inequalities in closest order, however complex, such as the mean density of the earth 5'67, the corresponding expression, when p" is taken as unity, is 245. From equation (359) we may now proceed to find an expression for the mean pressure in terms of the rate of degradation in the transverse undulations when (f_\i I and r = r B ; rj substituting, the expression for the attraction of unit mass becomes, if the .. r 3 4 5-67 Then, supposing that rfjr? is a maximum, we have from equation (358) < 380) - And as the density of the mean negative inequality is 5'67/22 of the maximum inequality, we have for the attraction which becomes, on substituting from equation (380) and reducing, 4 4 5-67 Then transforming so that the density of the medium is 22O, since r B is 6-37 x 10", we have for g A K.ay 981 = 22%/Vl TT . 6-37 x 10 8 ................ (381). 248] THE VALUES OF a", X", 0- AND G BY EXPERIENCE. 229 Then substituting the value of 22Hp" in equation (377) we have 4 fn \* g7r5-67 x 6-37 x 10 8 x 1-8574 x 10" x (-)