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VMW. OF
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THE SUB-MECHANICS OF
THE UNIVEESE
UonDon: U. J. CLAY AND SONS,
CAMBRIDGE UNIVERSITY PRESS WAREHOUSE,
AVE MARIA LANE.
©laagoto: 50, WELLINGTON STREET.
ILeipjifl: F. A. BROCKHAUS.
£eto gorfc: THE MACMILLAN COMPANY.
Bombao anU Calcutta : MACMILLAN AND CO., Ltd.
[All rights reserved.]
THE SUB-MECHANICS OF
THE UNIVERSE
BY
OSBORNE 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.
PUBLISHED FOR THE ROYAL SOCIETY OF LONDON.
CAMBRIDGE:
AT THE UNIVERSITY PRESS.
1903
dambritigr :
PRINTED BY J. AND C. F. CLAY,
AT THE UNIVERSITY PRESS.
^42
PBEFACE.
rr^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.
ERRATUM.
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.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
The actions necessary to satisfy the condition that action and reaction
are equal ......
The conservation of the position of momentum
Conservation of moments of momentum
Bounding surfaces
Energy
The general equation of energy in a medium in which there are no
physical properties
Simplification of the expressions in the equations of energy
Possible conditions of the mass in a purely mechanical medium
The transformations of the directions of the energy and angular redis
tribution .......•••••
The continuity of the position of energy . . . .
Discontinuity in the medium
PAGE
22
23
24
ib.
25
27
ib.
28
ib.
30
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. Kineniatical definition of mean motion and relative motion
43. The independence of the mean and relative motions ....
44. Component systems of mean and relative motion are not a geometrical
necessity of resultant motion .
45. Theorem A .
46. Theorem B .
47. General conditions to be satisfied by relative velocity and relative
density
48 — 49. Continuous states of mean and relative motion ....
50. The instruments for analysis of mean and relative motion
51. Approximate systems of mean and relative motion ....
52. Kelation between the scales of mean and relative motion
42
44
45
ib.
46
48
50
51
ib.
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 nnit 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 defined
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.
PAGE
ART.
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 grains, 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. xi
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 y2 V{ 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 \J'2\ 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 s]2V( and \l2(V^+dVJ) 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 x having velocities V{ 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.
115.
116.
117,
118.
119.
120.
121.
122.
123.
124.
125.
126.
The number of collisions which occur between pairs of grains having
mean velocities between V1 \]2 and ( V{ + d V{) \]2
The mean velocity of pairs having relative velocities s]%V{ and V{\\j2
All directions of mean velocity of a pair are equally probable, what-
ever the direction of the mean velocity
The probable component of mean velocity of a pair having relative
velocity r2 = s/2 Vx
The probable mean transmission of vis viva at an encounter in the
direction of the normal
The mean distance through which the actual vis viva of a pair having rela-
tive velocity \]2 \\' is V{\ si 2 and the actual vis viva of such a pair
is 2(rS + -f) = 4(V1/s/2)* . . .
The probable mean component displacement of vis viva at a mean
collision by conduction ..........
The probable mean component displacement of vis viva by convection
between encounters by a grain having velocity between ]\' and
VJ+dV{
The mean component flux of vis viva .....
The mean component flux of component vis viva
The component of flux of mass in a uniform medium
Summary and conclusions ........
PAGE
105
ib.
ib.
106
ib.
ib.
107
ib.
ib.
108
ib.
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 .........
128. The logarithmic rates of angular redistribution by conduction through
the grains as well as by convection by the grains — Rankine's method
129. There are always masses engaged in each encounter
130. When twTo hard spheres encounter, &c. .....
131. The fundamental effects neglected in the kinetic theory hitherto
132. The concrete effects of encounters between grains .
133. Variations in the complex accident ......
134. The effects which follow from the three instantaneous effects.
135. The instantaneous and after effects of encounters before the
encounter of either of the grains ......
136. Theorem
137. The theorem, Art. 136, includes the redistribution of the actual vis viva
138. The redistribution of inequalities of the angular distribution of normals
139. The redistribution of the rates of limited conduction
140. The analytical definition of the rates of angular redistribution of in
equalities in the direction of the vis viva of relative motion
141. The energy of component motion in any direction cannot by its owi
effort increase the energy of the component motion in this directior
next
110
111
ib.
112
113
ib.
ib.
114
115
ib.
116
ib.
ib.
117
ib.
CONTENTS. Xlll
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.
PAGE
ART.
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 ......•••••• w.
174. The expressions for the coefficients G and D 144
175. Summary and conclusions as to the rates of redistribution by relative
motion ........••••• to-
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 170 — 1
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 decrement 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
205 — 207. (1) The coefficients of dilatation. (2) The normal pressures when
a" = 0 186—7
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 a- multiplied
by the curvature on the normal piling of the medium . . . 190 — 2
211. Granular media with relative motion 193
212 — 213. The relation between the mean pressure and the constant mean
tangential and normal principal stresses resulting from a negative
spherical disturbance about an only centre on the supposition that
the coefficients of dilatation are unity ....... 194—5
214. The arrangement of the grains about the centre ..... 197
215. The expression for the contraction strains 199
216. The effects negative disturbances may have on each other when within
finite distances 201
217. The law of attraction of negative centres ....... 204
218. The mechanical interpretation of the " potential " 205
219. The analysis for the effects of positive centres ...... 207
220—223. The first of the class of complex local inequalities — electricity . 207 — 11
224. The mechanical interpretation of the electricity unit . . . . 211
ib.
xvl CONTENTS.
PAGE
ART- 919
225—226. Positively electrified bodies do not repel *l*
227. Cohesion and surface tension
228. The mechanical interpretation of magnetism 213
229. The mobility of the medium • • • 214
230—231. Misfit of the grains where the nucleus, in normal piling, meets
the grains in strained normal piling, causing singular surfaces of weak-
ness or of freedom • • 214 5
232. The mass moves in the opposite direction to the negative inequalities . 215
233. Motion of the singular surfaces by external propagation .... 216
234—235. The density of the moving mass is equal to the dilatation at all
points ; l°'
236—237. The completion of the analysis for the mobility of rotating centres
positive or negative 217
238. Combinations of primary inequalities to form singular surfaces with limited
stability 218
239. The cohesion between adjacent singular surfaces 219
240. If the number of grains absent about each of the centres which con-
stitutes the total inequality were the same whatever the strain there
would be no mean displacement of mass
241. Absolute displacement of mass resulting from two total inequalities
differing in respect of the number of grains absent in their primary
inequalities when subject to shearing strains w.
241 a. An inversion of preconceived ideas 221
SECTION XV.
The Determination (1) of the Relative Quantities a", \", a
and g which define the condition of the medium by
the Results of Experience : (2) The General Integration
of the Equations.
242. The advance upon the position as regards evidence from that at the
end of Section XIII 224
243. The change of the units of density from that in which the density of
the medium was taken as unity to the density as measured in units
of matter io.
244. In c.G.S. units of matter the mean pressure p = 22Qp" and the mean
density is p=22Qp" 225
245. The expression for the mean pressure in terms of the rate of degra-
dation - in the transverse undulations, when a/\" is large, 22S2//'
= 22Q x 1-8574 xlO11 ( — ) , where n.2 is the wave-length measured by
the diameter of a grain, and tt is the time to reduce the initial energy
from 1 to 1/e2 ib.
246. The value of Q 227
247. From the evidence afforded by the known law of gravitation, the value
of n2tt is obtained which satisfies the law of gravitation g = 98l . . 228
CONTENTS. XV11
ART. PAGE
248. From the evidence afforded by the limits of the intensity of light and
1'52
heat, the value of v('2 is tentatively obtained as v1"2=- "'xlO-1
in c.G.s. units 229
249. From the value of i\" the genei'al equation
22i2 x 1-8574 x 1011 f^Y =\/22Q x 1-172 x lO*2
is obtained, and from this the value of X" is obtained as 8-612 x 10-28
and the values of a and a" are obtained in terms of *J-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 . 231— 2
252. The X-rays 232
253. The rate of decrement of the normal wave in terms of \/22Q . . 233
254—255. The density of the medium in c.G.s. units is 22Q = 10,000 . . 234
256. The inferior limit obtained from the evidence of Rontgen 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 compared 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
6-777 x 10 = a".
The mean path of the grains is
8-012 x 10-28 = \.
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
io4 = 22a
The mean pressure in the medium, equal in all directions, is
1-172 xl014=p.
The coefficient of the transverse elasticity resulting from the gearing of
the grains, where the piling is normal, is
9-03 x 1024 = n.
The rate of propagation of the transverse wave is
3-004 x 1010=t or \Tnfp.
R. 1
2 ON THE SUB-MECHANICS OF THE UNIVERSE. [2
The rate of propagation of the normal wave is
7-161 x 1010= 2-387 x r.
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*3, 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 10_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 ;
Pt=§P>
and a normal pressure pr = § 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
(-1),
the reciprocal of the density of the medium being TO-4,
the mean pressure of the medium 1172 x 1014,
a the diameter of the grain 5'534 x 10-18,
the mean radius of the earth (v3709 x 108;
1—2
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 47r/3, or
pa x 10-4 x f 7T x 567 x 6"3709 x 108 = 981 x 102.
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 (ncssure 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 r0. 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
ndt~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 — ( — e2)* - and (— e2)*, which are both irrational. Their product
being the rational quantity
e3
r '
which, differentiated with respect to the distance, is
p2
and the mechanical explanation of these is,
p[-a) ro\z- -p[-» r°7\=-2>
2 U
and for the effort to revert, we have
Then for the electrostatic unit we have, since r = 1, ami 22 = — 1,
'fir)**-1-
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 1045,
and r0 = (y493x 10-3, 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 UN 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 1015,
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. 1874 — 5, p. 7.
t Royal Soc. Phil. Trans. 1879.
t Royal Soc. Phil. Trans. 1883.
8 ON THE SUB-MECHANICS OV 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.
+ Royal Soc. Phil. Trans. 18D5, 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,
SS, 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 :
2 {QSS)= 1 1 \Qdxdydz is put for the quantity within a space S enclosed
by the surface s at the instant considered,
2 (oQ&M) is the quantity enclosed at a previous instant.
X(pQS8) is the quantity which has been produced within 5 during the
interval, and
2 (CQSS) is the quantity which has crossed the surface inwards during
the interval.
Then 2 (QBS) = 2 (0QSS) + 2 (PQBS) + 2 (eQBS)
is a complete expression for the Axiom.
Using 8 [ I to express any change effected in the time St this may be
written
S[2(QS5)] = S[S(i,QS^)] + S[2(cQ8/S)] (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 Q1} Q2, &c. respectively refer to the several
particular forms or modes of the entity.
Then since
8[%(pQ8Sy\ = 0,
8[X(pQ18S)] = -8[X{pQ28S + &c.)] (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 8t may be and whatever may be the motion of the surface s enclosing
the space $ ; for the 8 covers the X ( ).
If however the surface s be steady or fixed in space the 8 may be covered
by the X( ) and the equations written
X[8(Q8S)] = X[8(PQ8S)] + X[8(CQ8S)] (3),
S[S(3,Q1^)] = -S[a(i,Q28^f + r)] (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 : —
X(Q8S) = Qdxdydz (5),
and X[8(Q8S)]=~(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.
VL. 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 (f> has positive values on one side of the surface and negative
values on the other, then putting Qi for the continuously varying value of
Q where (/> is negative and Q., for Q where <f> is positive, Q is at all times
expressed by the limiting value of the function
F_Qi + Q3en*
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 Qf.
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 dxdydz 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 j% (CQ) dxdydz = - dt W ( >Q) + j- (vQ) + ^- (wQ)\ dxdydz '
or at a point the rate of change by convection is
t n = - \d (uQ) + d M) + d (vQ)\
dtc \ dx dy dz \
•V),
* Electricity and Magnetism, Maxwell, § 8.
+ Electricity and Magnetism, Maxwell § 8.
dQi , ^Qa .mi,
dF _ dt dt, _n(Ql- Q2) <** d<f>
dt ~ l + c^ (l + e11^)2 dt '
„ , . . . , . .-.-xi • L. dF (JO, . . ... dF dQn
h rom which, taking n mfanite, when <f> is negative — = -y- , when <b is positive — - — —^
dt dt dt dt
and when <£ = 0
dt (l + e"*)a
which is infinite, but which, integrated, from <j> negative to <p positive over an interval dt, indefi-
nitely small, gives
/
JE1
dF * = «.-«!■
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 j! (PQ) - ^ (uQ) - ± (vQ) - ~ (wQ)\ dxdydz (8),
dt -j- (jjQi) dxdydz = — dt \ -r (PQ2 -\- &c.)[ dxdydz (9),
or at a point the rate of change is
d
dt
^(/«=-l(pft+&=-) (II).
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.
In the equations (5) to (11) the surfaces of the element of space (SS 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 5.
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 a, 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
i 2 «w> = -jjj{i K- - u) <a+ 1 K. - v) Q]
d )
+ [(w - w) Q] J dxdydz (12).
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
- am k« - *> « + 1 k- *) « + a k- - ») «i • • -<is>'
and integrating equation (10) over the surface, the rate of change in the space
instantaneously enclosed as by a fixed surface is
susf
uiSM
at
- l\l{i «> + 1 <•« + a (w«>} *•** (14) ;
whence substituting in equation (13) for
from equation (14),
+///{s(''(3)+<|(5<2)+a(rae)}&rfy^ (15)-
or as it may be written
|p(«as)]-s{«(|+s^ + sJ+w^)o
+«U+^+&)[ 0<i)-
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 P&S
for the capacity for momentum or mass in the indefinitely small space 8S
and substituting p for Q in equation (2) the equation for conservation of
mass becomes
S[Z(pP83)] = 0 (17);
and by equations (1) and (17) the equation of motion of mass becomes
S[Z(P88)] = S[Z(ePSS)] (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 + djm + dpv + dpw = Q ^.
dt dx dy dz
which is the general equation for density of mass or medium at a point.
19. Position of mass.
Taking x, y, z~&s defining the position of the indefinitely small steady
space 8s, 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
$[8{p(px)8s}] = 0, 2[8Uf>y)<fe}] = 0, t[S{p(pz)ds}] = 0...(20).
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)
8[S(par&)] = 8[2{c(/M08s}], &c., &c (21).
Since x, y, z are not functions of the time, it follows by equation (19),
if .'•, y, z define the position of the centre of gravity of the mass in the
steady space over which the summation is taken, that
dx \\\{X ~ *> ( Ifc + ^ + ddz) dxdydZ
°±J21 V ,,• , , , , &c.,&c (22).
dt (Up dxdydz v '
For in a fixed space,
<
Also ^2 (pds) = - ///(d£l +&o.) dxdydz.
20] GENERAL EQUATIONS OF MOTION IN A PURELY-MECHANICAL-MEDIUM. 17
For a space moving with the mass by (15)
~ 2 [(p») &] = 2
dt
+ X [{dx {pXU) + Ty <*"»> + Jz (PXW)} 8S
.(22a).
whence since x is not a function of t,
( p -tt 8s j = 2 (puSs), &c, &c.
20. Before proceeding to the consideration of momentum and energy
it will be found convenient to express certain general mathematical relations
between 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,
d(pQ) _„d(pp) t ^d(pq)
dt
'"A +'
dt
.(23).
And since by equation (17)
d(pp)
dt
= 0,
d(pQ) __ d (pq)
= P
dt
.(24).
Also
and
dt
dQ _ dp dq
dt'~qdt+pdt '
d (eQ) (d , d . s d
~dT=~ \dx{pUq) + dy <™> + dz {m)
•(25);
= -q{^ + &C]-p{u^C] j
whence subtracting and having regard to equation (19)
dQ d(cQ) jdq dqM„
therefore by equation (8)
d(pQ) = n jdq
dt
.(26).
dq „
=p\dt + udx+&c
Again, if Q = pq = pq1q2 and Qx = pqu Q1=pq2, by equations (26),
+ &c.l )
d(pQ) _ frfgiga , dq,q,
dt ~P\ dt + U~dx~
= P
(h
<%* , .. d& , ^L« idy±±o,d2i + ,^c V
dt
4 v ^2 + foe \ 4- a Pl + M^+ &c <-
+ U dx + j +q" {dt + dx +^C-j
•(27),
R.
18
ON THE SUB-MECHANICS OF THE UNIVERSE.
[21
and putting q1 and q2 respectively for q in equations (26) and substituting
in the right member of the equations (27),
dQ_d(M)_ (dQ2 d(cQM (dQi_d(M\\
dt' dt ~qi[dt ' dt )^ h\dt dt I
d(PQ)_ d(M , ^d(M
dt
dt +q'2 dt
J
.(28).
21. In the equations (25) to (28) p is subject to the condition of
conservation of mass, equations (17) and (19). If instead of p we take p"
as an abstraction of the density we obtain a corresponding but more general
theorem, by putting
dp"
W
dp"u dp"v dp'iu) d ( vp")
dx dy dz ) dt
.(23a),
where the last term on the right expresses an arbitrary density ; then
(24a),
d(vQ) „d(pq) d(pP")
— p ^ f q
dt r dt
dQ dp
dt
dt
d(cQ)
dt
dt
+ P
dq
dt
-^'^)-?i"t+4
Equating by (23a, 24a, 25a),
r \dt dx
dt
dt
dt
d(pQ) _ „ d(pP") , // (d1 , „ dq „
~dr-q^r +p [Tt + udx+&c-
And putting q = q,q, and Q, = p"Qlf Q2 = p"Qi} we have
ldQ2 d(cQ2)\ d(vp") „{dq2i dqa
<U U ST) = ** "it + ™ ft + U dx + &C'
(dQ, _ d{cQ,)
dt
d(pp")
q* [ dt ~¥" ) = q^~^~ + W" ( Tn + u± + &c-
dQ _ dJcQ) _
dt ' dt ~Ma-
From which it appears
dt
d(pP")
dt
dq1 + udqi
dt dx
dq,q2 dq,q2
'p -dT-udx+s*c'
dQ d (CQ)
dt
dt
</i
dQ2 d(.Qt)
dt
dt
+ <1>
d(PQ) _„d (pQs) , n d (PQJ
dt
dt
+ q*
dQi _ d(A)
dt dt
d (Pp")
M*
d{pp"\
dt
dt
7.rA'
dt
.(25 a).
.(26a).
..(27a).
.(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 as before u, v, w 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
Mx = pu, My = pv, Mz = pw.
Substituting Mx for Q in equation (1) it becomes
B[Z(MX8S)] = B[Z(PMXBS)] + Z[(CMX8S)], fa., &c (29).
By equation (2) substituting PMX for PQX,
B[X(pMxB8)] = -B[X(pQ2BS + &dc.)], &c, &c (30),
where — B [S ( PQ.2BS + 8zc.)] expresses the rate of destruction of momentum
in direction x, in all other modes than that represented by MXBS within the
space of S.
23. Conduction of momentum by the mechanical medium.
As % (MXBS) represents the sum of all the momentum in direction x
within the space S, there is difficulty in realising how momentum in direction
x can be produced or destroyed in any other mode. If, as in this research,
pBS is defined as including the total capacity for momentum within the in-
definitely small space, BS, the production or destruction of momentum in
direction x in any other mode than MXBS, at a point within the space BS,
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,
o o
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 the 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-
mediuin, 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 Ji, the intensities of the rates of exchange of
momentum, in direction h, across the surfaces yz, zx, xy are respectively
pi, pin, 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", pirn,
pin,
zx
pinl, pin",
plan,
xy
pnl, ptun,
pn2.
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
I
.(30 A),
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, pxy> Pyy, Pzy> Pxz, 2V> P™>
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),
_ {dphcx + d]>y* + <%**} dxdydz = Fxdxdydz \
{ ax dy dz )
%+% + %}***-*.***
d$>xz. + dpyz + d}^\ dxd dz = Fzdxdydz
dx dy dz )
Fx, Fy, Fz 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 Fx, &c,
respectively represent the rates at which the densities of the respective
components of momentum, in other mode than that of Mx, &c, are destroyed
within the element, and as these are the only rates at which momentum
within the element is destroyed — Fx, &c. define the values of (PQ2 + &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)
d-±J^ = Fx, &c, &c (31),
and substituting in equations (29) we have by equation (10)
d-^ = Fx + ^(eMx), &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 Fx equations (30 A) and d (cMx)/dt from (7) we have
the full detailed expressions for the equations of the densities of the com-
ponents of momentum at a point
dMx_
(it
[dec (pxx + pmL) + dy (pvx + pUV^ + dz ^Pzx + Patvy &c>' &c-' • ^33^
The equations (32) and (33) are the equations of conservation of mo-
mentum in a purely-mechanical-medium, at a point, in 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 S.
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 pxx, pyx, &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
dMx_
at
\ dx ^xx + pU^ + &C' I ' &C'' &° ^^'
.(35),
or transforming, since x, yy z are not functions of t,
7 (A
21 (®PU) - Pxx ~ PUU =-\jxX( Pxx + PUU) + &C-
^ (ypu) - pyx - pw = -\([xy (p*x + puu) + &c-
jj: (zpu) - pzx - puw = - |^ Z (pxx + puu) + &C.
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 pxx, &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,
dt
{ZpV - tJpW) - (pZy - PyZ)
d
•(36);
dx ^ (J>,7/ + UV^ ~ V (Pxz + UW^ + &C'
whence in order thai 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),
Pzt, = Pyz > and similarly, that pxz = pzx and pyx = p3m (36 a).
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 lr, mr, nr, ls, ms, nS) lt, mt, nt for the direction cosines of
r, s, t respectively, then since pxy = Pyx> &c-> &c->
Prr = Vxxh? + pyymr2 + pzzn? + 2pyzmrnr + 2pzxnrlr + 2pxylrmr
Pss =Pxxls1 +Pyy'ms2 + pzzns2 + 2pyzmsns + 2pzxnslfi + 2pxylsms
Ptt =Pxxk"Ji-pyym,ti +p2z7it2 + 2pyzmtnt + 2pzxntlt + 2pxyltmt
Put =PxxUt +pyymsmt + pzzn8nt + pyz (msnt + nsmt)
+pzx Ohk + h>h) + pxy {lsmt + mdt)
ptr =PxxhK + pyyintmr +pzzntnr +pyz (mtnr + ntmr)
+ Pzx (ntlr + hnr) +Pxy {kmr + Wr)
Prs=Pxx^rk + Pyyinrms + j)^,.)^ +pyz (mrns -f nrm,)
+ Pzx (nrls + h-ns) + Px!i(l'r»h+ mrk),
V...(37).
28] GENERAL EQUATIONS OF MOTION IN A PURKLY-MECHANICAL-MED1UM. 25
Where the surface separates the medium from unoccupied space the
stresses prr &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„, prs, prt 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 uf 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 2)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 + Pyx™ + pzxn, &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
Uipaj+Pyxm+Pzxll), &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 8s, the element of a closed surface, the
integral over the surface is expressed by
I j[iuPxx + VPxy + WPxz) I + (upyx + VPyy + Wpyz) ™> + {"Pzx + VPzy + Wzz) n] 8$>
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, which 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,
Id
2dt
[p (u2 + v2 + w2)} dxdydz
du dti du\
da
dy
Pxxdx +Pyxdv +Pzx dz
{ +Px
dv
dv
dv
+ P
"ydx
dw
t-Pwfy+Pv dz
> dxdydz
dw
dw
* dx + Pyz dy + Pzz dz )
(upxx + Vpxv + Wpxz) I
= [[|+ ( uVyx + vpvv + wPyz) m
(+ (uPzx + vpzy + Wpzz) n ■
BS. . (38).
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 than 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~ -i- v- + w-) and transforming equation (38) by equation (15), the
equation of energy for a fixed space becomes
[(itPxx + VPxy + IVPxZ + UE) I
dt
SIS
+ (upyx + vpyy + wPvz + vE) m + (*fP» + vPzy + WP& + w^) 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 , v d . > d .
di = ~dx ^Upxx + VPxy + WPxz> ~ dy ^UPyx + VPyy WPyz' ~ dz ^Pzx vPzv+wP^
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 Nx, Ny, Nz 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
N = NX + Ny + Nz.
dW = - [ix (UPxx) + aij ^Pyx) + Jz (^24 ' &°-' &°-
Then
(41),
and
dN dN„ dN, dN
+
+
dt dt dt dt
s
dN
dt
8S
V...(42).
= I I {{uPxx + VPxy + U>Pxz) I + (UPyx+VPyy+Wpyz) 7)1 +(upzx+Vpzy+Wpzz)n} dS
Whence substituting in equation (40) it becomes
dE dN d , „.
li = + -dt+dt(cE)
(43) ;
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 fdv dw\ (dw du\
- \P** in. +Pw JT+Pzz T„ +Py* ^ + jt, + P*° [jz + ~- ~ '
f~ dx T *** dy T ** dz T *» \dz T djj) " Izx \dx ^ dz)
(da dv\\ ....
+**\fi + <£•)}= ° ^
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 pxx, &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, arc measured in the directions
of the principal axes either of the rates of distortion or of the stresses at a
point p and a, v, w are the components of the velocity in these directions,
respectively, transforming to these axes we have by equation (44) ; since
either ; —
dv dw _ 0 . „
j- 4- -j- = 0, &c, &c. ; or pjk = 0, &c, Ace (45),
du dv dw
Piid7+^Tj+Pkk^ = 0 <46>
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
Pa = Pa — Pkk = 0 ; which is the condition of empty space (46a),
, du dv dw n „ , . ,
Pa =Pjj =Pkk ; and -r + -=- + - - = 0 ; perfect fluid.
"'i a* clfc
du dv dw dw dv du dw dv du . . . . , .
32. The transformations of the directions of the energy, and angular
redistribution.
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
\ dtSSI(pu2) dxdydz - SSSipxx £ + *- % + pzx s) dxdydz
= - jjj te (#«tt) + ^ (Py*u) + Jz (P**u)\ dxdydz . . . (47),
&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 Rx, &c, &c. for the
densities of the rates of angular dispersions at a point, from the directions
x, y,.z respectively, these are
dRx l du du duy
( du du du\ 0 .
= - [P»*fa + Pyx d~ + Pzx j-zj , &c, &c.
dt \r™dx ' iyxdy
It is to be noticed that in a medium such that u, v, w do not represent
the velocities of points in mass, Rx does not represent angular dispersion
only, unless equations (44) are satisfied ; and if not so satisfied dRx/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 . x / _ du dv dw\
1 j /du dv\ /du d ii'
+ 2 fyx [dy " dx) + pzx [dz ~ dx
1 . v (du dv dw\
-^P**+Pyy+P**){^ + d;+dz)
Pxx +Pyy 4" Pzz\ du f 1 (^ (du ^ dv\
rdu dw^
- 1 ft. f-^) ffi + 2 Jn. [Ty + dxl
(du dw\\ /<K >
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)
8 [2 [EasBS}] = 8 [X{c(Ex) 88}] + 8 [2 \v(Ex) 8S}], &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,
x"Z{E8S} = Z{Ex8S], &c, &c (49),
whence, differentiating with respect to time,
~ % {E8S} =~xjt [2 {E8S\} + jt [X {Ex8S\l &c, &c (50).
Then, by equation (15), these equations become
(^%{E8S\=-x%
dE dEu dEu dEw] g '
dt dx dy dz )
^n dE d(Exu) d(Ex'o) d(Eayw)\ ^"
\ dt dx dy dz J
(x _ m (*i + *m + *m + dJm\ BS\
V dt dx dy dz J j
_ v
+ S (Eu8S), &c, &c (51).
Whence, for a fixed surface, since u = v — w = 0,
dx
di
s|(,-S)fas}
, &c, &c (52).
% (E8S)
For a surface moving everywhere with the mass so that u=u, &c,
equation (51) becomes
2 \(x - x)% (PE) 8S) + S {Eu8S}, &c, &c.
Jl _-4__ (53),
dx
di
[E8S\
or,
~[l{(Ex)8S}] = %i[x^t(pE) 88^ + ^(Eu8S) (54),
where, as in equation (42), differentiating with respect to the limits
dJv, ^ = ~ \dx ^PxxU + PxyV + P**0) + &c- + &c- • • • | (55X
or
dN
dt
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 vel - 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.
> before, u, r. u\ to express the components of the actual
vel- - of the mass the point a and time f. and p for the dens
the mass, and as express g the components, with respect to the
same axes, of the velocil i component system, there exist at each point
the residual componc
= - . = - . . - V
sums of these components & - fcisfj the equations
S tion III., and the following equation, for the resultant system, and if one
- -.-- ... any definition, actual or conditional, the
equation for the s tant system becomes the equation for the residual
- - m.
Ej neral method in mechanical analysis to separate the motion
he ma-- .eh point iuto two component systems, whenever the condi-
fche independence of these systems is obvious. A-. for
the motion of the m ss each point at any instant is considered
sisl _ of the motion of the centre of gravity of the whole mass; at
the ins _ her componei t sys m which is the motion at
the point i the motion of the centre of gravity. But such instances
been considered as depending on special theorems, and do not
app - __ - d the study of the method which they involve as a
- - ' sis apart from the existence of conditions which
systems ■mpletely 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
jt (,Q) = J{ (e"Q) + J( (,'Q), &o (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", piv" 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" 0 0
-j- + u -= — h v -5 — h w —j— , &c. &c.
dt dx ay dz
are subject to arbitrary definition independent of the actual accelerations of
the mass. And
R. 3
34 OX THE SUB-MECHANICS OF THE UNIVERSE. [35
W "
Taking -± , &c., &c.
P
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.
*£ - i c*o+ ,|<^***> w.
and again by the equations for the resultant system
jt(pi'-'' + pu') = jt(cPu' + cpu')+ Fx, &c., &c... (59).
Subtracting equation (58) we have for the other system
^.' = (|(^') + ^-p^(X)> &*. &c (60).
It thus appears that
p jt (Pu"), &c, fee.,
express rates of transformation of density of momentum from the component
system pu' to the system pu", &c, &c, consequent od 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 q1} &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
pT(E"), &c, respectively, since there is no transformation of mass,
(ci).
35] THE EQCATI« ? IS 7 : SRUUfTY ? SYSTEMS
From these eq s if will be seen, at once, that the sum of the trar.--
formati - mpor. - - t necessarily rea hat the
transformation is not whollv I . . B .
m
rhe equal - g compone: - si as *Ter in form
from th - - sum of
the dei- s . aent syst - > not
equal to the de: - gy of the resultant sysl
or tha:
=±p - - *+w*+i -- --: - - •:
whence p B—E -E') = p - - . «/)
Whence it appears the transformation of energy is -
between th systems E and S\ but also between each of thes
- fee - oat besides the Equations rf ei :_ : : he component
stems there fe equation of energy :: the : -idual system to be
considered.
The den;i:v ; 1 ::rmation to the resiJual s; - by
definition equal in valoe and op -:~ in s the sum of the rates
- . 1 v nergies :" the componrL" sysl
p(%(* *%<?>).
Another expression for the transformation to the residual -
obtained by multiplying each of the rates o: jf comp: l
of momentum tc the compone nt ae corresponding componer
: the »thei - on and adding, as in equations (28).
The de^ : rhe rate of production into residual energy may be
obtained in the same way by equati . :. 28 thenhyequ
exr ressi ds i i
p^ B-B -E and j: .
In the equation of motion for the resultai.: - :em of motion in a
purely-mechanieal-medium, d B the he rate at which
is iuced in i E, is lefim I as :
expression for this production disappears from the equation of energy. It
does not i r follow a .- g metrical consequence th
for tanddiL-: dt, obtained from the equ ns of mom
equ::: b 28), ar. ro, But it d follow th
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{TR')jdt.
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.
dTp" _ dfS dp"u dp"v dp"w ( ,
dt dt dx dy dz
as the arbitrary expression for this defined rate of increase, we have the
equation of continuity for the component density
dp" dc{9") _dT{p")
dt dt dt
(64).
37] THE EQUATIONS OF CONTINUITY FOR COMPONENT SYSTEMS OF MOTION. 37
And by the equation for the resultant system
d(p" + p') dc(P" + p) \
dt dt
dp' dcp dTp"
dt dt
dt
(65).
Then, since by equation (24), dp(pu)/dt = pdpu/dt, substituting in equation
(32), the equation at a point for the resultant system is
(66).
du du du du _ dpu
dt dx dy dz dt
rJ " rl "
Then multiplying by p" and adding u -~ — u -~- to the left member
and the equivalent udpp"/dt to the right member, we have for the equation
of momentum of the defined density :
dp'u dc(p"u) „ dpu dTp"\
~dt dt~ = P ~dt +U~aT
dp (p"u)
dt
(67),
and in precisely the same manner
dp'u dcp'u _ ,dp(u) dTp
dt dt
= P
dt
— u
dt
_ dp (p'u)
dt
(68).
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" dc (p"u") _ dp {p"u") „ dTu" „ dcp" , r „ „-, \
~dt dt d^ = p ~W~U -df~d[Tpu}
'I J . /'
de'P
= p ^r + w dt
dp'u" dc (p'u") _ dp (p'u") , dTu
dt dt dt
dp"u dc(p"u) ^dp(p"u') „ druT _ , dc-p" [ p"Fx - pFx"
dt dt dt P dt ~U dt p"
} (69),
dp'u' de (p'u') dp (p'u,') , dTu" , d^p p'Fx - pFx"
~dt dT ~dT ~ ~ p ~df + U 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") _ dc(pu-p"u") = dp(pu-p"u")^
dt " dt
dt
7 H II
dTp u
=F*-~iir- i
(70),
38 ON THE SUB- MECHANICS OF THE UNIVERSE. [38
whence putting Mx" for p"u", Mx for pu — p"u", &c, &c., we have
dMx" dcMx" ^dpMx" _ „dTu" „dTp" \
dt dt dt P dt dt
dMx _ dcMx _ dpMx' F _ „ dTu" _ „ dTp"
dt dt dt x P dt ~U dt
(71).
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,
////|s UM"")] + Tt (**")} dxd'Jdzdt
where so far qxx, qyx, &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
j J J Why ~ (hz) dxdydzdt = 0, &c, &c (73),
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 (6.9) by u" and transforming by
equations (28 a), Section III., and putting p"Ex" for p" (u")2/2, we have
d (p"Ex") dc {p"Ex") _ dp {p"Ex") _ „ „ dTu" u"> dTp"
dt dt ~ dt ' ~ P dt + 2 dt +<*°-
Also multiplying the third of equations (69) by v! and trans-
forming (28 a) we have
dP"Ex de(p"Ex) dp(p"Ex)
dt
dt
dt
"F
= « [E£ - p» d-^>) + u" d~W + u> d-^}
dt
dt
dt
Then multiplying the first by u and the third by u" and
adding, &c.
d(p"Ex) dc(P"Ex) dp(p"Ex) dTp"v!u" r,„p"F,n
'■ — — = ^ = ; -+- p U ■+" OiO.
dt dt dt dt ^ p
= u u
,.„dT(p")
dTu" o"u"Fr.
dt ~df
p (u — u )
//\ "Tc
+
+ &C.
Again, multiplying the second by u" , &c.
dp' Ex dc (P'EX") = dp (P'EX") = u„ ,dTu" t u"*dTp | fca
dt dt dt
Multiplying the fourth by u', &c.
dp'Ex dc {p'E£) _ dp (p'Ex)
dt
2 dt
dt
dt dt
, , dTu u 2 dTp , , „ o
dt ' 2 dt
Then multiplying the second by u' and the fourth by u" and
adding, &c.
dp'Ex dc(p'Ex) = dp (P'EX) = dT(p'uu") _ PWF + &q
dt
dt
dt
dt
= u>u» clTf + P' d-i^pn +pj^+ &,
dt r dt p
.(74).
40 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 (PE - P"E") dc (pE- p"E") _ dp (PE - P"E")
dt dt dt
for the energy of the system of momentum pu - p"u"
hSfSgtVD-vr.+vr^vr.-t&iP (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 -^— ' = '°^~ + -~ -J are mechanically
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
motiou after the abstraction of the residual motion, we take
u" + u>» + &c., v" + v"' + &c, w" + w" + &c.
and for CQ put &Q + d>Q + C-Q + &c., for TM' put PM" + pM'" + &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 defined 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 with
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 after the 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) and (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 (1, 2, 3), and the
condition of relative energy (4), but not, of necessity, that of position of relative
energy (7); together with another system ivliich 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)Ax + {y-y)Ay + {z-z)Az + {t-t)Au &c, &c....(77),
where the barred symbols refer to the mean-position of the mass within the
limits, whether time or space, thus
JJJfxpdasdydzdt „ m
X= ffffpdxdydedt ' &C (78)'
the limits being assumed ; the conditions to be satisfied by the component
velocity u" are :
46 ON THE SUB-MECHANICS OF THE UNIVERSE. [4G
(1),(2),(5); that
I jp (u — u") dxdydzdt = 0,
(3), (6)
(4)
Iff Up (u - u") dydxdzdt = 0, &c, &c, &c.\ (79)
p (u — n") 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
JJJKpu) dxdydzdt
JjjJ(p) dxdydzdt ^
then integrating the four equations of position and obtaining the values of
Ax, Ay, Az, At by elimination from the resulting equations. These values
must be real since the Ax, &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
| fffxp (u - u") u" dxdydzdt - 0 (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 x (y — y) pAy, &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 tlie conditions of energy of
relative inass, (15) and (10).
Taking the mean-density of mass at x, y, z and t to be
p" = D + (x-x)Dx + (y-y)Dy + (z-z)Dz + (t-i)Dt (82),
where, as before, the barred symbols refer to the mean position of mass
between limits of time and space. And putting ^, x1, yx, &c, as referring to
4G] 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)
(10), (12) [...(83),
\\\\x(p — p") dxdydz dt = 0, &c, &c, &c.
which five conditions determine D, Dx, Dy, Dz and Dt whatever may be
the distribution of mass, so that putting p = p — p" the conditions (11)
and (12),
\p dxdydz ■-- 0
re , <8*>
I jxp dxdydz = 0, &c, &c, &c.
are satisfied.
Again, since the constants A and D in the equations (77 and S3) for u"
and p" are respectively the values of u", p", at the mean position of mass
respectively, and the constants Ax, &c. and Dx, &c, are the differential
coefficients of u" and p", respectively, the equations may be written
u"=u" + u, &c, &c.) /0_x
\ (bo).
p" = p" + p\ &C, &C.J
Then multiplying the corresponding members,
pu = p"u" + p'u" + pit, &c, &c (86),
whence it appears, since the integrals of the last three terms on the right
are by definition of necessity zero, that
jpudxdydzdt= 1 1 1 ipii'dxdydzdt (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 x2p' are not
of necessity zero, and the expression of xpu, &c. includes the terms
ft'}/
x-p -j— , &c, it appears that the product p"u" does not of necessity satisfy
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
jffLpu"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 which 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 pit , pv', piv 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,
I \xp"ii' dxdydzdt —III jxpudxdydzdt, &c, &c.
requires that the products x*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 — pu =pu + pu (k"),
47] THE MEAN AND RELATIVE MOTIONS OF A MEDIUM. 49
and since u" is a linear function of the variables
(Pa - p"u" ) u" = p'u"* + pu'u" (90),
whence the integrals of both the terms on the right vanish by the previous
conditions.
And further, the conditions
ljlfcc(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"2 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 defined
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. 123—164.
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, Ffp 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
fffj'Q dxdydzdt = -„
ffffdxdydzdt
and
ffff(Q-Q") dxdydzdt =
fffjdxdydzdt
and putting h and k for any two variables, r W^/-
(h (Q - Q") dxdydzdt = 0,
hk (Q - Q") dxdydzdt = 0V
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
4—2
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 x0y0z0t0 as referring to a particular point and time, then at
any other point x, y, z, t,
■ (• - "0 ffi + &c' + 5 (« - *»)' S) + *»
f
•(93),
i
where the differential coefficients are all finite. Therefore as (x — x0), &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 — x0), &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
Po"uo" + &c. within the limits denned 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, to 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"'2 — 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, 8y, 8z, 8t,
= ffffQdxdijdzdt _Bxd<y + &c )
JJfjdxdydzdt 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 -j*- is a derivative of any quantity
dx ) dx
.(95).
and Q W = d (Q,"&) _ QfdQt
dx 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"
-j- + u" -j + v" -=- + w" -=-
dt dx dy dz
+ U Hx +V ~dy+W dz=dt {*U }> &C-' &C (96)-
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'Jdt 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
%^' = M'*': + &c. + f%i-T (97).
dt dx \ dt J v '
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),
dpu" de(pu) „ dvu" 0 . /noN
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.
i Clpll
¥)"=ffM' (99)'
or dividing by p" it appears that the transformation from relative-velocity to
mean-velocity, at a point, is expressed by
1^ jde (pit)
P
dt
+ FX[ , &c, &c;
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 dpu" '/dt in equations (58)
and (60), Section IV.
m
, du" , du" , du" 1 de (pu)" Fx" . . \
+ V -7 + w
dy dz
dt
.(100).
dv (u) . du" , du" , da" 1 de (pu) Fx" 0 0
). =~U -j V ~j W -j j, f. - + —tt , &C, &C.
dt dx dy dz p dt p
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 v! , &c, &c. respectively ; thus
2 dt 2U dx +&c- + p»\ dt + *.J , «&, *c
1 dT (u'Y { . . du" , . du" . , du")
2T- = i""¥+"T,+"*j
u' (dc {pu') „ '
&c, &c.
dT (u"u)
dt
v.. .(101).
1 \dT{u'J , dT(u'f\
-z\—d^+^r)> &c-' &c-
i d_^r_f (u-- u') (dc{Pu)
~ 2U dt ~ p" \ dt + a
■ + It'll -%- + &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 and 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 (u"u) 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 {u'f _ 1 d[c(puj + u'Fx]
2
(de (pu)
dt
dt
, , da"
pu u
dx
pu
P
dt
+ Fa
.(102).
Then putting
(uj = ((u)r + ((.ujy (103),
where {(u'Y)" is obtained after the same manner as u" ; putting d(T((u')'2)")/dt
for the total rate of transformation, we have as in equations (97) and (98),
substituting {(u'J)" for u" and the three last terms in equations (102) for Fx
in equations (100), since the last term has no mean values,
ldp((uj)"_l( d((uj)"
2"dT'~2\U dx +&C'
+
d [dcP {{uj)']
1^
V
, , du
pu u
dx
dt
+ &c
+ u'Fx
.(104)
and
i4MJf iKpr.a,, {104A).
2 dt 2 dt
dTu
=ar"°-
Then since
dp (w2) _ dT {u'j dT({ujy
dt -~~dT~+ dt ~+UJi*>
\dp\y?-(u*)"] 1 ,d((u'y + (uY)
dt
2U
u
+ —
P
dx
+ &c. - u'Fx
[1 dcpu'
+ Fa
+
1
2 dt
1 dcp((u')T
2 dt
dx
\-u'Fx-[ pwV^ + &c.
.(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
( , , du" , , da" , ,du"\ 0 0
— p \u u -? \- v u , +w u -7— > , &c, &c.
r [ dec dy dz )
59. The rate of transformation from mean to relative energy.
From equation (64), at a point,
dTp" _dp" dp"u" dp"v" dp"w" dp'ti dp"v dp"w
dt dt dec 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
dTp" dcp" (dc>p"\" (dTp
dt dt \ dt J \ dt
Also from the last of equations (65) \- (107).
dTp" dp dp'u" dp'v" dp'w" dQp')
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
drp" _ (dcp)'
dt dt
.(108).
Then adding the mean and relative parts ; since
(dC'(P'))"= (4- (/>"))
dt dt
and (pu + &c.)" = 0,
7 // 1 n
Orl_ = _a^_
dt dt v '
60 ON THE SUB-MECHANICS OF THE UNIVERSE.
60. The transformations for mean and relative momentum.
[60
We have
dT(p"u") _ „ dTu" „ dc>p
dt
= P
dt
— u
dt
.(110).
Then substituting from the first of equations (101) and (109), and trans-
forming,
dT{p"u")^ dc-{p"u") , \d0{p"u)
dt
dt
+
dt
VF%\ +&c (Ill),
and we have
dT{pu-p"u")_d,{p"u") \dc(p"u') J
df -—it i"rfT~"+^r&c (lllA)-
61. The rates of transformation of mean- energy of the components of
mean- and relative-velocity.
From equations (74), (100) and (109) we have
id[T(P"(uy)]_ id{c.P"(u"y\ \ AdcPu)" , „„}
2" dt ~~ 2 dt + r * r
1 d[>'' ((^)T] = _ 1 d[c,(p"(uj)"]
2 dt 2
+
1
"<k (P ((ujy
dt
dt
+u'Fa
pu U —j V &c. -
.(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
dc (pu')"
its equivalent
or
dt
dpu'u' dpv'u' dpiv'u'y
dx dy dz
{dp" (u'u)" dp" (v'u)" dp" (w'u)'
\ dx dy dz
and we have
„ (dcPu)" (d(p"(u'u)"u")
u
di
dx
+ &c.J L"(kV)"^ + &c. l...(l 13),
also
dx dy dz '
61] COMPONENT SYSTEMS OF MEAN- AND RELATIVE-MOTION.
so that by equation (95), Fx" may be expressed by
\dp"xx
61
dx
+ &c.
Then we have
u
r r = r h <KC. — V Tr. -, &C.
.(114);
dx dx
also (u'Fx)" = (uFx-u"Fx)" (115),
and this may be expressed as
\d{wpxx)''
+
UPXX)'' . O 1 . (d(ll"p") g
-f- 1- &c.\ + { \j r V &(
dx ) I dx
Then substituting in the first of equations (112) we have for the rates
of transformation to the energy of mean motion
\d[T (p"(u"Y)] _ \d[e (p" {u'J)-\ id [u" (p" (uu)")]
dt
dt
dx
+ &c.
- ^'£m) + &cj + K (mV)" + ^-j % + &c j -(u6)j
and again substituting in the second of equations (112) we have for the rates
of transformation to the energy of relative motion
id[T(p"({ujm=_\d[,P"{{U'w]
2
dt
dt
i d [C-P ((uyyr [d (upxxy &j
2 dt 1 dx
U*> ^Y
+ \i
>xx
dx)
+ &C,
J
+jl^k)+&cl
du'
{p- (uUy + p"*x}"g + &c.
•(117).
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 TEE SUB-MECHANICS OF THE UNIVERSE. [62
the energy of residual motion, or of relative energy, these become
1 dT [pu2 - p" (un _ 1 d [cp" (u?)]"
dt 2 dt
id[c-(p(uyyr | \d[u"{p"{uu)")] fcj
2 dt \ dx j
+ f¥- + &4 - {*- s + &c-}" ■ &°- &°-
.(118);
and these are the exact forms in which the rate of transformation to relative-
energy, obtained by substituting u2, (u2)", (u2)', u^ wr u> u"> u> ^ 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')" -, - + &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
K-rf|r+&cj' K£+&cj
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
(p"xx -^ + &c.j , &c, &c,
and
P*
dx
du
dx
+ &c. , &c, &c,
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=t—t — ~t — — , the first two terms in these equations (47a)
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
df_
dx
„ (du" dw"\
\dz
&c, &c,
du
dy
dv
dx
Pv*\j7.-jz)+P**
du
dz
dw\ |
dx))
&c, &c.
The corresponding expressions for the rate of increase of the resilience
are
&c, &c,
dx dy dz
) + (Pxx - P)
du dv
dy dx
+ Pz
du
dz
dw\)
dx))
&c. &c.
64
ON THE SUB-MECHANICS OF THE UNIVERSE.
da" „ \ , / dl
[62
Substituting these for \p"xx -~j— + &c- ) and (pXx -r- + &c. J as they enter
into equations (116) and (117), these equations become
1 dT [p" {u'J] ldd[p"(u'J] [de„[p(u'u)"-\ , s ) \du"V"xx , gT |
2 ' ~5T "2 d£ ( dm ~ + &C-j " \ ~dx + 65C'j
c?m" dv" dw'
+ {I p" (^dx-dy-ih)*^ Y'y*Kdy ~^)+p'
+
doc dy dz
1 „(du" dv^_ dw'
3 V dec dy
du" _ df\
'du" dw'
[~dz~~llx~
„ /du" dv" dw"\ , „ „. d
1 ( „ (du" dv"\ „ (da," dw"\\
if «*{ly+7te) + p"{w + lte)\
a
+
, &c, &c (116a).
ld[Tp"(u'u')"}_ 1 d yP" (uu')"] 1 d Up (u'u)']" \d (u'pzx)" I
2 " dt 2~ dt 2 dt ' \ dx ')
V
p
du
dx
dv' dw'V
dv'
dy
du dv'
dx dy dz
dz J
dw"
+
P>
+ \(pxx-p)
du'
-dy
dtf\"
dx I
~Pzx[
(du
\dz
dw"
dx
dx
1 ( „ (du dv'\" , ,
2\Py^ + Tx)+P
p"(u'uyit\ + 8zc. + 8zc.
du dw'V
dz dx J
, &c, &c (117a).
In these equations the first three terms in the members on the right
express rates of linear redistribution of the energy of components of motion
of the respective systems, while the fourth terms express, respectively, rates
of energy received from the other components of the same system by angular
dispersion, and the fifth and the last terms express the direct exchanges
between the two systems, of mean density of energy, by transformation.
This last statement however is only true when, as in the case of the
resultant system, in a purely mechanical medium, there is no resilience in the
resultant system, for the fifth term in the last equation expresses rates of
decrease of the resilience in the resultant system less that of the abstract
resilience in the mean-system ; so that, if the former is not zero, this term,
besides the exchange by transformation, expresses the total rate of increase
of the resilience of the resultant system.
In a granular medium when u, v, w are the component velocities at points
in mass, and there is no resilience in the resultant system, the sum of the
65] COMPONENT SYSTEMS OF MEAN- AND RELATIVE-MOTION. 65
resilience of the mean and relative systems is zero, and the fourth term in
equation (117) has the identical value, under opposite sign, as the fourth
term in equation (116), which expresses rate of decrease of abstract resilience
in the mean system.
The first term in the brackets represents the angular dispersion by
distortion under mean strains, equal iu all directions, and the second re-
presents the rates of angular dispersion by rotational motion of the mass.
63. The equations for the rates of change of density of mean- and
relative-mass.
By equations (64) and (109) we have for mean density
dp" _dc»P" . ,
It ~~dT { W)'
and by equations (65) and (109) we have for the equation of relative mass
MdW) dQp) (n9A)
dt dt dt v
64. The equation for mean momentum.
By equation (58) and the first of equations (100) we have for the equation
of mean momentum
d£^ d,,p''u" _ (d (Pxx + p" (u'u')") |
dt dt \ dx J '
and by equations (60) and the second of equations (100) we have the equation
of relative momentum
d{pu-p"u") = de (pu) _ de» jp"u") _ , . ,y, _
dt dt dt KcP }
dx
...(121).
65. The equations for the rate 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.
Substituting for the transformation function in the first of equations (74)
from equation (116), the equation for mean density of energy of mean motion
becomes
i_ djp^jurn _ i jMp: (u'7)] \d[u" (p"(uu'r + P„")] | ^ + krn
2 " dt 2 dt { dx
+ |(p''(«V)'' + ^)^, + &c. + &c.|, &c, &c....(122),
R.
66
ON THE SUB-MECHANICS OF THE UNIVERSE.
[66
and the equations for the mean density of energies of relative-velocity
become
i d [P" {{uj)"] _ i d[c>-(p"((uj)")] i d[,(p((ujy)r \d(uP„Y
dt
+
dt
+
dt
dx
+ &c
+ |^) + &cl
- f[/)"(uV)" + pxx"] ^+&4 , &c., &c (123).
66. The equation for density of relative-energy.
Proceeding in the same manner as in equations (74) and substituting the
rate of transformation to relative-energy equation (118), the equation for
relative-energy of component velocities becomes
1 d [pu* - p" (M»)"] _ 1 d [> (pu* - p" (a*)")] Idy (p (u*)")\
dt
dt
+
■2
dt
1 d{c,(p(u>)')] 1 d [.p ((uj)'}"
2 dt 2 dt
+pjg«n+&,j
&c, &c (124).
67. Complete equations.
1 d[p"{(u'y+(vJ+{w'J)\ 1 d [> [p" ({u'J + (v'J + (w'J)}-]
2 dt
2 dt
( \d [u" (p" (u'u')" + pxx")]
+
(/,
&c.
X
-< +
+
id[v"(p"(vu)"+pXy")][tc)
d \w" jp" (w'u)" + pxz")] &e
die"
+ ](p"(u'uy+pxx")^+&c.
+ \ip"(tv'u)"+pxz")d^ + ^
.(125).
67]
COMPONENT SYSTEMS OF MEAN- AND RELATIVE-MOTION.
67
i d[p\{uy+(vy+{wyy] _ i a [Ap"(W+ mw)]
dt
dt
id[AP(w+(vy+(Wyy)Y
+ 2 dt
( + [d [(upxx)" - n"pj'] &)
\ dx ')
d \ivpyx)" - v"Pyx"] ] &c } } +
-■> +
+ \d [(wpzx)" - v/'pm"] + &c I
\ dx ') j
+ \(P"(WuT+pJ')d^+^c
( + {(*- s) +&a
^
~j +{(p"(^/r+^/)^,+&c.|
^ +{0»'V»T+ ft.") ^ +&<?.}
1 d [p (a2 + i>2 + W>) - p" {{a?)" + if)" + (w2)")]
2 rZi
.(126).
+
i d [c» (p (u> + v>+ w>)) - p" ((Uy + (vr + <y p]
2 d«
1 d [, (p (u2 -M2 + <Q]
2 d£
1 <Z [c, (p Q2 + ^2 + w*)') - [e, (p (m3 + i;2 + w2))}"]
+ 2 cZi
f {d\u"p"(u'ii)"} .
+ \ L 'y — + &c.
{ dx
\
K
(
(d[v"p"{v'u)"] 9 )
+ < +{ Pdl ;J+&4i
rfa;
;
♦f^Ml r+WW
— < +
+
f(Z [t^Vr]' . fc
eZ#
+ &C. )-+< +\[Pyxj-) +&C.
<*[«?«]' &(J
cZ#
cZ«V
dw\'
) \
+ ?■
[V^^
+ &c.
.(127).
5—2
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, pxx,
&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
r'djg) tHjg) r^ w
J o at J o at J o at J o 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
n£r dt Has ^V) + J ("oV) + s ^v>} rf<
/ . du" ,dv" ,dw"\ 7,
-plu' -j-+V -j- + w -j- (ft
r V ow? a# a.z /
z{s^v>+|^v> + sW*
-^Fx'Bt + F'^&cSzc (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
d(pu')
dt
, &c, &c,
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 actious 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
w^uld 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 defined 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
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 Fx', &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 Fx', &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
Fx', &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.
(b) 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 p'urely 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.
(f) 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 0 to ir 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 con firming
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. Tn 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 :
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
ay
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 cc2, or by the product of two variables, as yz, and taking the mean over
all four variables ; as
„ iMl = _,= {<L(g'^+&cl -* HM>+&cl (128 a).
dt [ dx } { dx J
Then if x2p' integrated over all four variables satisfies the conditions to
any degree of approximation, the maintenance of the same degree of approxi-
mation requires that
X~ 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 ter
depends on, and is proportional to, the inequality instituted.
m.
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
wC^]^ /
/ i^-0^0"^ yv. s
Fig. 1.
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
6—2
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 pxx — 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 pxx 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, and 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 versa 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 cr/2 with the same internal constitution — anything or
nothing — and the interstices between them are unoccupied ; these surfaces
having the property of maintaining their motions, uniform in direction and
magnitude, across the intervals, and that of instantly reversing the com-
ponents of their relative velocities in the directions to the surfaces at
contact on encounter without having changed their shapes ; such a medium,
however far it might go to satisfy the kinematical conditions necessary for
the physical properties of matter, would of necessity entail the laws of
motion and the conservation of energy ; and would thus constitute a purely
mechanical medium in which the results would be the same whatever might
be the constitution of the space within the surfaces.
The mean density in such a medium would be measured by the number
(N) of closed surfaces divided by the space occupied. And the density
within the surfaces would be the reciprocal of the volume enclosed (7rer3/6).
Since each of the grains represents the same mass, this mass becomes the
standard of mass ; and being common to all the grains, is of no analytical
importance.
In the same way <r, the diameter of the grains, becomes the standard of
scale in the medium ; and being the same for all the grains has no analytical
importance.
It is, therefore, important and convenient, as adapting the notation to
any arbitrary system of units, to define the mass of a grain in terms of
the dimensions of the grains in the arbitrary units.
The most definite and convenient definition appears to be that which
makes the mean density of the medium, when the grains are piled in their
closest order, a maximum, that is when each grain has contact with twelve
neighbours at the same time. In this way the mass of a grain is expressed by
f
Vl'
where a is the diameter of a grain expressed in arbitrary units.
Then if p" expresses the mean density of the medium
>"=f (*>
And thus p" becomes unity when the grains are in closest order.
SECTION IX.
THE PROBABLE ULTIMATE DISTRIBUTION OF VELOCITIES OF
THE MEMBERS OF GRANULAR MEDIA AS THE RESULT OF
ENCOUNTERS, WHEN THERE IS NO MEAN MOTION.
95. Maxwell's Theory.
Since the only action between elastic hard particles, as considered by
Maxwell, is that of exchanging each other's relative motion in the direction
of contact at the instant of contact, and the action of the grains, as defined
in Section VIII., is identically the same, notwithstanding that it is not
ascribed to elasticity, Maxwell's* proof of the law of probable distribution
of velocities to which the action between the particles tends, applies equally
to the grains. This law of Maxwell's is perfectly general and independent
of all circumstances as to shape and size of the particles, and the extent of
their freedoms, as long as there is freedom in all directions, and there is
no distortional mean motion.
According to this law the mean of the energy, taken over limits of space,
such as define the scale of the relative velocity of the motion in each degree
of freedom, is the same for each and every degree of freedom, and is
constant when equilibrium has been established. From this it follows that
the time-mean of the energy of motion in each degree of freedom is the
same, and is equal to the space-mean.
In the case of all the grains being similar and equal the mean component
velocities positive or negative are the same, whether taken with respect to
time, or to space. And when the grains differ the mean component
velocities are inversely as the square roots of the masses.
This law of distribution, to which the relative- velocities, in any granular
medium, tend when the mean motion ceases, being general requires no
further exposition here.
* Phil. Mag. 1860, Part I., pp. 20—23, Props. I, II, III, IV.
90 ON THE SUB-MECHANICS OF THE UNIVERSE. [96
In following up the consequences of the law, to which the mean com-
ponent vis viva tends, on the mean distribution of the spheres, Maxwell,
it appears, has tacitly introduced an assumption which, although legitimate
in cases in which the diameters of the spheres are negligible as compared
with the mean-paths of the spheres between encounter, has completely
obscured the fact that the mean arrangement of the grains does not depend
solely on fulfilment of the law of distribution of the vis viva; but also
depends on the hindrance which the surrounding grains may offer to the
enclosed grain in changing its neighbours.
When the grains are small compared with spaces separating them this
hindrance becomes negligibly small. And, further, whatever effect it might
have is entirely dependent on the conduction through the grains ; so that
the neglect of the displacement of momentum by conduction renders any
account of such mutual constraints which the grains may impose on each
other futile.
It now appears, however, that taking account of the conditions, we have
in these a class of actions which, however insignificant they may be when
the density is small, entirely dominate all other actions when the density
approaches maximum density. And it thus becomes evident that the
failure of the kinetic theory, as applied to gases, to apply to the liquid and
solid states of matter is owing to this tacit assumption that the distribution
of the mass depends only on the action which secures that the distribution
of vis viva shall approach that of uniform angular dispersion as the medium
approaches a state of equilibrium.
It will thus be seen, that accepting Maxwell's law of probable distri-
bution of vis viva, it still remains necessary for the purpose of definite
analysis, to define the limits of its consequences on the probable arrange-
ment of the grains, i.e. of mass.
96. Maxwell's law of probable distribution, of vis viva is independent
of equality in the lengths of the mean paths.
This is founded on the demonstration (1) that when two elastic spheres,
having relative-velocities in any particular direction, undergo chance en-
counter, all directions of subsequent relative-motion are equally probable,
and (2) the demonstration that whatever may be the shape of the elastic
bodies the same law holds, as to the linear velocity, and is further extended
to their rotational motions. As consideration here is confined to the case
of smooth spheres it is sufficient to take into account the first case only.
The most general expression of this law for uniform grains is, taking
x, y, z to represent the component velocities of grains in the directions x, y, z
respectively, and N for the number of grains in unit space, the numbers
97] DISTRIBUTION OF VELOCITIES OF MEMBERS OF GRANULAR MEDIA. 91
of grains which have component velocities which, respectively, lie between
x + 8x, y + 8y, z + 8z, are
XT {X2 + ,f. + zn
SN= — 7--,e Z* 8x8y8z (130).
a* (7r)5 a
From 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 pairs 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, y1, z as the relative components of the same pair, and xly yl7 z1}
#2> 2/2. #2 as the components of the individual motions, we have
X1 = X + X', y1 = y + y'l Z1=Z + Z ,
x2 = x- x', y, = y- y', z, = z- z' .
Then for the numbers of grains for which x\ is between ar, and x1 + 8x1,
yx between y1 and yx + 8yx, zx between zx and zx + 8z1} and x2 is between x2 and
x2 + 8x„, &c, &c.
N't /(g+s')'. (y+y'Y2 ,(5+g'Fl
— e l o? + a? + a? i dxdydz
.(131).
a3(7r)^
No Hx-x'V2. (y-y'y.g-gTl
n2= . , x»e «■ a8 a2 a2 sdx'dydz'
2 a3 (tt)5 y
The first of these equations expresses the probable number of grains
having mean-velocities between x and x + 8x, &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' + 8x', &c, &e, for any
value of x, &c, &c. Whence the probability of the double event is expressed
by the product
a6irs
.(132).
92 ON THE SUB-MECHANICS OF THE UNIVERSE. [97
Then if r = x2 + y2 + z2 and r = x'2 + y'2 + z'2, the number of pairs having
mean-velocities between f and r+Br and relative velocities between r' and
r+Br is
nin2=^lK2e-^-+r^ 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 = — go to # = co we find
TV" TV r'2
,1 x 2 e-wdxdy'dz' (134)
(\/2a)3 (tt)* * 7
for the whole number of pairs whose components of relative velocities are
between x' and x + Sx', y and y' + 8y', z' and z' + Bz'. And integrating for
r instead of r we find
N N J^L
jL — - — e~ a2 dxdydz (135)
(a/\/2)3(7r)S J V '
for the number of pairs whose mean components of velocity are between
x and x + Bx, &c, &c.
These may be expressed in a more convenient form by substituting
— r^cos 0d<fi 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 \f%r and Vz {r + Br),
of pairs having mean velocities between ?-/V2 and (?*+ Sr)/v/2,
since N is the number of grains in unit volume and N (N — 1) is the
number of pairs of grains,
7V4 (r\2 r"
)J^e~^hr = n1 (136),
a3 V7T
{A«rk a-g-Vg^-^-D^, (137),
(iV-l)^(;VV2)%_(^gr/V- = ^_1)Mi
(a/v 2)3 V7r
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
ot 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 iWrs2
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
l
1 =
iW27TS2
In this result there are two things to be noticed.
In the first place the 7rs'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 A, 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 with 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 pxx, pyx, &c., and p"(uu')", p"(v'u')", &c,
admit of expression as
P+Pxx-P* pyx, &c; %p" (v'v')" + p" {u'u)" -y (v'v)"t p"(v'u')", &c,
where p = $ (pxx +pw + pzz), p (v'v)" = p" (u'ti + v'v' + w'w')"
and in this case p and ^p"(v'v')" represent the mean action, equal in all
directions, while pxx—p, p" (u'u')" — \p"(v'v')" &c, pyx, &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/cr
becomes large.
Maxwell has shown that neglecting <r/X the mean path of a grain and the
relative path of a pair of grains are expressed by
X= -— ^ and V2X= — -^ (139)
respectively, while both of these are obtained from
V2ttXo-2 = ^ (140),
96 ON THE SUB-MECHANICS OF THE UNIVERSE. [101
where JV 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/cr is large we have all positions of the projection, in the direction
of relative motion of the striking grains, over the disc 7r<r2/4, equally probable,
and then the probable mean relative striking distance in the direction of
relative motion is
2
This is a relative distance and the corresponding actual extension of their
actual paths is, by equations (136) and (137),
V2
101. The assumption that all positions of the projection, in the direction
of relative motion, of the striking grains are equally probable over the disc
area 7rcr2/4 is obviously legitimate when \ is large compared with a, and
hence these estimates of the probable mean striking distance when \/cr is
large are precisely on the same footing as Maxwell's estimate of the mean
path neglecting a/X. But there does not seem to be the same ground for
this assumption when a/X is large ; while, on the other hand, there is
evidence, as pointed out in Section VII. (Arts. 88 and 89), that, when the
grains are close, the normals at encounter fall into line (approximately) with
the direction of a finite number of axes, fixed in space, not more than six.
In this article the arrangement of the grains is assumed to be similar in
all directions ; so that, whatever may be the law of distribution of the pro-
jections of encounters on the disc-area, the probability will be equal in all
directions at equal distances from the centre of the disc.
Therefore taking 6, as before, for the angular distance from the axis of the
disc at which the normal at encounter meets the hemisphere of unit radius,
the law of radial distribution on the disc may be expressed by a function of
cos 6, which function will depend only on the ratio a/X. Thus as a general
expression for the probable mean striking distance we have
2-rra I cos 6 (1 + A1 cos 6 + &c.) sin 6d sin 6
j" -=§-/©-<141>-
2-7T I (1 + ^.jcos 6) sin 0dsin0
Jo
102] EXTENSION OF THE KINETIC THEORY. 97
in which Al &c. are functions of a/X only ; and as the law of radial distri-
bution of the striking distance is perfectly general we have in the right
member a perfectly general expression for the mean relative striking distance
of a pair of grains in the direction of their relative motion. And dividing this
by V2 we have for the mean probable actual striking distance of a grain
$•/©
Thus as a general expression for the mean path of a grain we have
= ±^-l,f(?l
V2 [ira'N
and for the volume maintained by a grain
^Ht-OK'
(142).
102. Further definition of f(a/\).
Since the foregoing expression for the volume from which a grain excludes
other grains applies to all conditions of the medium it must include the case
in which X is indefinitely small; in which case, if the medium is in uniform
condition with three perpendicular axes of similar arrangement, the unique
condition is that in which the volume maintained by each grain approximates
to as/\/2, as explained in Section IX., each grain being in contact with 12
neighbours. In this case N approximates to \/2/<x3 which is the reciprocal of
the volume maintained by the grain, which thus approximates to the volume
of the spherical grain multiplied by 6/\/27r. Substituting this for the right
member of the second equation (142) we have for the limit when a/X is large
/©-i
6
(143).
4 V2tt
Then, again, if X/a is large the value to which f(<r/X) approximates is unity.
Whence for an expression satisfying all cases in a uniform medium with three
axes of similar arrangement it appears that we may take
where a? = 1 — 6/4\/2 ir and b2 is arbitrary
.(144).
It is convenient however to render the expression for this function a little
more general, since in a granular medium although generally in uniform
condition, with three axes of similar arrangement, there may exist localities
where the arrangements vary about local centres ; the medium being still in
equilibrium and X/a being small. Under such conditions the limits of
variation are defined by the fact that equilibrium requires that each grain
shall be in approximate contact with at least four grains. And it seems that
r. 7
98 ON THE SUB-MECHANICS OF THE UNIVERSE. [103
these may be included by substituting 1 - 6r/4, where G has the value 6/V27T
when the medium is in uniform condition, and values ranging to the limit
18/4V27T when the medium is in varying condition, as about centres of
disarrangement, instead of 6/4a/2 it in a'\ Then
/©-^-SK"' (145)-
By definition (Section IX.) p = N^J^I, and by the second equation (142)
a
2ttP
.(146).
103. In order to render the expressions for the mean relative-path of a
pair of grains and the mean path of a grain, taking account of the three
dimensions of the grains, general and complete, use lias been made, equation
(139) in Art. 100, of the ratio (1/V2) of the mean path of the grain to the
mean relative-path of a pair of grains as determined by Maxwell for con-
ditions in which the third dimension is negligible.
The legitimacy of this assumption therefore remains to be proved. But
before proceeding to the proof of this proposition the proofs of two other
geometrical propositions are desirable, as they depend directly on the law of
distribution of the component-striking distance over the area of the normal
disc.
104. The first of these propositions is :
When a pair of grains having any particular relative velocity (a/2 V\), all
directions being equally probable, undergo chance encounter, the probable mean
product of the displacement of momentum, in the direction of the normal at
encounter, by conduction, multiplied by the component q/\/2 V7 in the direction
of the normal is
To prove this, let x De the acute angle between two diameters drawn
through the centre of a sphere of unit radius in the directions of the normal
at contact and that of the relative motion before contact, and let co 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
co cos x (1 — Ai cos % + &c.)
~ >
7T
105] EXTENSION OF THE KINETIC THEORY. 99
or multiplying this probability by the product of the normal component of
the relative velocity VSF/cos^, 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 co for all values
of x over the hemisphere, we have for the probable mean product of the
normal component of relative velocity multiplied by the displacement
■n
2-7T | - V2 o-TVcosx(l — Al cos ^ + &c.) sin %d sin %
-^~X =f ^</(0..(147).
2tt -cos%(1 - J.1cos^ + &c.)sin^^
Q. E. D.
105. The second of the two geometrical propositions is :
The probable mean component conduction of component momentum in any
fixed direction at a single collision is
2 <r
r3
3 V2
'^'t/G
To prove this we have to multiply the mean product of normal displace-
ment multiplied by the component of the relative velocity by (o-3/\/2) the
mass of a grain ; thus obtaining the expression for the mean displacement,
in the direction of the normal at encounter, of momentum at a single
encounter, as
V2
\
Then, taking 6 as the angle which the direction of the normal makes with
any fixed direction, say that in which ^ is measured, and resolving the normal
displacement a and the mean normal component of V in the direction of ^,
multiplying by sin Odd, integrating over the sphere and dividing by 4nr,
7T
^2nr\a^Vlf(^) fees9* sin 0d0 0 , ,9 . .
V2 3 J \\J Jo 2 a3 v2 T7,,M /1M
Q. E. D.
This expression for the probable mean-component conduction at a single
encounter is one of the factors of the rate of component conduction by pairs
of grains having particular relative velocity \/2 F/, the other factor being the
number of collisions that take place between such pairs in unit space in unit
time.
This second factor involves the discussion of the ratio of the mean path
to that of the relative path of a pair of grains.
7—2
100 ON THE SUB-MECHANICS OF THE UNIVERSE. [106
106. The number of collisions between pairs of grains, having particular
relative velocities, in unit of time, in unit space.
Taking N for the number of grains in unit space and substituting F/ for
r in the equations (136), (137), (138), Section IX., we have for the numbers
of grains having velocities between V( and F/ + SF/
(TY)«
m{y^]e -dV^nx (149),
for the number of pairs of grains having relative-velocities between \j2 F/
and V2 (7/+8P7)
.(.Dyr,-).^ ff| f,
(V 2 a) v71"
and for the number of pairs of grains having mean-velocities between F//V2
and(F/ + SF/)/V2,
7V7JV— 114(F7*/2¥ -(F'W2)"
wv2)v: e °"v2 JW=,f-1)" (151)-
107. From the equations of distribution of velocities, relative-velocities,
and mean-velocities amongst the grains and pairs of grains in unit volume,
it follows that the proportion of the N grains having velocities between F/
and Vi + SVi is the same as the proportion of the N (N — 1) pairs of grains
having relative- velocities between *J2 F/ and \/2(V1' + 8V1') as well as the
proportion of N(N — 1) pairs having mean-velocities between V-i\sl2 and
( V{ + 8 V1')/\J2, since for every one of the grains having velocities between
Vi and F/ + 8F/ there are (JV— 1) pairs of grains having relative-velocities
between \]2 F,' and »J2 (F/ + SF/) and (iV — 1) pairs having mean-velocities
between F//V2 and (F/ + SV1')/y/2.
Multiplying the equations (136), (137), (138) respectively by F/, ^2 F/,
and F//V2 respectively, and integrating from F/ = 0 to V-[ = oo , we have for
the mean velocity of grains, the mean relative-velocity of pairs of grains, and
the mean mean-velocity of pairs of grains,
(F/)"^, 4l(J{fjy** and (F/)7V2 = ^...(152).
And as the grains are of equal mass the relative velocity of each grain in a
pair is half the relative velocity of the pair; so that the mean relative
velocity of each grain in the pairs is
<¥-£ (153)'
108] EXTENSION OF THE KINETIC THEORY. 101
108. To find the mean path of the grains, taking V2A, for the mean path
of the pairs.
Each grain has at any instant N - 1 relative paths with the N — 1 other
grains in unit volume, and JS — 1 relative velocities, so that the N grains
have in all N (N — 1) relative paths and N (N — 1) relative velocities.
A change in the actual velocity of any one grain causes a change in the
relative velocity of each of the JST — 1 pairs of which it is a member. And
as at an encounter between the members of a pair two grains change their
actual velocities, there are 2 (N — 1) changes at each collision in the
N(N— 1) relative velocities of the pairs in unit volume. The mean
relative path of a pair of grains between changes being by definition \J2\,
the mean relative path of a grain is A./V2. And considering a particular
pair of grains, their paths and velocities relative to each other, though
continually changing, are always parallel and equal, so that the distances
relative to each other traversed by each of the grains in unit of time have
a mean value (V1')"/\/2, and the mean number of changes of relative path
and velocity in unit of time is
(H"A/2 = (7T
X/V2 ~ A, '
Whence the number of changes in all the relative paths of all the grains
is N (N — 1) (F')"/\; and since there are 2 (N— 1) changes for each collision
the number of collisions in unit volume in unit time is
N{V')"
2 \ "
Having thus found the number of collisions between the N grains in
unit volume in unit of time, since there are two grains engaged in each
collision the total number of encounters made by all the individual grains
in a unit of volume in a unit of time is twice the number of collisions :
that is
N(V')"
Therefore the mean number of paths traversed by each grain in unit
time is
Then since ( V')" is the mean distance traversed by a grain in unit time,
dividing by the number of encounters the mean path is
r/f
£U-X (154).
102 ON THE SUB-MECHANICS OF THE UNIVERSE. [109
Therefore if \/2 X is the mean relative path of pairs of grains, A is the
mean path of a grain. It also appears that the mean number of collisions
in unit of time in unit volume is
N(V')" N a .
— v — — = — . —j- (loo).
2 A A \/7r
And the mean number of grains a grain encounters in unit time is
(V)" 2a
X \Jtt . A
(156).
109. The mean path of a pair of grains.
This follows directly from the last proposition. For as the number of
mean paths of pairs of grains is identical with the number of relative paths
of pairs, and the mean velocities of pairs is one-half their relative velocities,
the mean paths of the pairs must be one-half the mean relative path of the
pairs, that is, must be equal to the mean relative path of each grain of
the pair, or
V2"
110. The number of collisions of pairs of grains having relative velocities
between V2 F/ and V2 (F/ + dV,'). '
Since the mean relative distance traversed between changes by a pair of
grains irrespective of relative velocity is \/2 A, the mean time of a pair of
grains having relative velocity V2 F/ in traversing their mean path (\/2 X)
is A/F/.
Then since the number of pairs of grains in unit volume having relative
velocities between V2 F/ and V2 (F/ + dV-l) is N {N — 1), and each of these
pairs changes F//X times in unit time, the total number of changes of these
pairs in unit of time is
F/
n(N-l) V-.
X
And since there are 2 (N — 1) changes for each collision, we have for the
numbers of collisions of the n(JSr— 1) pairs of grains in unit of time,
equation (148),
ihPi' ^F/4(F/^ Jhl
"2T = Ix7vVe 'dVl (157)'
The integral of this from F/ = 0 to F/ = oo gives the number of collisions
of the N grains in unit time.
111. The mean rate of conduction of component momentum in tlie direc-
tion of the momentum conducted. Cases 1 and 2.
3
•(158),
112] EXTENSION OF THE KINETIC THEORY. 103
Multiplying the probable mean component conduction from a mean
collision of a pair with relative velocity »J2 V-[, equation (136), by the number
of collisions in a unit of time, equation (157), and integrating V-l between
the limits V-[ = 0 to V-[ = go we have for the mean rate of conduction
*JZ<Tf(*\lWY _ffiZ wfilj, ,m V2o" f(<r\(V'V')"
P'Txf[x)-*w^-e a 'dVj06d(-cosd) = P--Jx-f{x)
whence since (V'VJ' = 3(U'U')"
P.^lf(l)(U'Uy=pxx'',&c.,&c (159).
112. The left members of equation (159) express in terms of the
quantities which define the relative motion of the medium, the mean normal
stresses, or the mean rates of conduction of momentum, in the direction
of the momentum conducted. And besides these there are the mean tan-
gential stresses, or rates of conduction in directions at right angles to the
direction of the momentum conducted.
These rates are obtained by substituting in equation (158), for cos3 0,
&c, &c, cos 0 sin 0 cos <£, &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
7T 7T
o ?/'(?) V2 Vi I I cos 0 sin 0 cos 0 sin 0d0d<j> . w,
o X' \\J Jo Jo ^ °" w^A V^ Kj
n i ¥[l) ^r -<160>'
sin 0d0d(f>
0J0
Si
J 0 J *
which multiplied by the mass and the number of collisions and taking
the mean is
H^Ms)^--**--^'-*8 (m)>
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
Pxy" = Q, &c, &c (102).
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
fn F'
2ttP X F/ -1 cos2 0 sin d dd v ,2
— " ± = ^- (163.x
2ttP sin 6d6
Jo
which becomes, taking Maxwell's expression for the mean value of v- from
0 to oo , (a2 . ■§), when multiplied by the product of the mass into the number
of grains,
I^OVPiT-pf (!64)-
And for the mean rate of momentum conveyed in the direction of the
momentum
p"^ = p'(U'Uy',&c.,&c (165).
For the lateral convections of momentum the expression is
2 n (F/FT
8 J 0 J 0
\v w cosflsin2 6d6 sin <f>d(f)
a"7-&c, +&c, -&c (166),
n " 7T
i rA r2 .
h/3 I / sm 6d6d<p
where the integration extends, as in the case of lateral conduction, over
groups of grains of which the directions are such that cos 6, sin 6, cos <£, &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'Wy = 0, &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 (105), also (162) and (167),
we obtain for the whole rates of displacement of the components of
momentum
P^+P»aruT=P»{i4^f(£)}arirr, &c. &o.) (168)
pxy" + P"(U'vy = 0, &c. &c
117] EXTENSION OF THE KINETIC THEORY. 105
115. The number of collisions which occur between pairs of grains having
mean velocities between V{\*]1 and ( F/ + d F/)/\/2.
Since the mean distance traversed between changes of a pair of grains,
irrespective of mean velocity, is X \/2, the mean time of a pair of grains
having mean velocity F//V2 in traversing their mean path is \/V. And
since the number of pairs of grains in unit volume having mean velocities
between F//V2 and ( F/ + d F1')/V2 is n(n — 1), and each of these pairs
changes V/\ times in a unit of time, the total number of changes of these
mean paths is
V
n(N-l)^-.
And since there are 2(iV — 1) changes for each collision the number of
collisions of the n (n — 1) pairs of grains in unit volume in unit time is
n v,' nv; ir,' .tza
rY = 2T^e ' -dv <169>'
which integrated gives the total number of collisions ajsjir . \.
116. The mean velocities of pairs having relative velocities V2F/ and
T7/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 I/a/2 to V2 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 \/2 F,'
and V2 (T/ + d F/) is between F//V2 and ( V{ + d Fa')/\/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
N, (No - 1) | -f — .e"*. dr\d (cos 0,) . dfr
JO ■"*
r 2 2r22
x -2 . e <*2 . dr,d (cos 02) dd>2
(— V 7T?
r, being the mean velocity, r2 the relative velocity and 0^, 0«<j>2 having
reference to the angular positions of rx and r2.
For, taking 7\S}\ and r^Sr* constant, and ascribing any particular values
to 62<f).2 and 8628(f>.,, the number of pairs, having a mean velocity Fx 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 Oid^, is to the total number which
meet the sphere as d cos 0^1^ is to 4<tt. q. e. d.
118. The probable component of mean velocity of a pair having relative
velocity r2 = V2 V1 in the direction of the normal at encounter.
Since rx = r2/2 and r2 = \J2 F/, i\ = ViJy/2. In all directions the probable
component value is
17
+
2 V2
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 <r 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 ^2 .f(a/X).Vl/S, the
probable displacement of vis viva in the direction of the normal is
41/(0 *-±$|/©W«> + rf>l? (HO).
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
± {I3 + Inr + In'} |/(£) , &c., &c.
120. The mean distance through which the actual vis viva of a pair of
grains having relative velocities between *J2 F/ and V2 (F/ + SF/) is dis-
placed at a mean collision.
Since the mean velocities of pairs of grains having relative velocity
/J2 Vi is F//V2 and the actual vis viva of such a pair is
2(n2 + !|) = 4(F1/V2)2 = 2F1,
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 Vi + BV/ 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 A. to a .f(a/X)/S, and the displacement of vis viva on encounter is
in cases (1) and (2)
X +
i/(3
It thus appears that, while, as has already been shown, the range of
mass or any mean quantity carried by mass is X, and the range of relative
velocity or momentum is
the range of vis viva is
4+&'©l-
4-is/(x
121. The probable mean component displacement of vis viva at a mean
collision by conduction.
Multiplying the mean normal conduction of vis viva at a collision of
a pair of grains having relative velocity *J2V( by cos 6 . sin 0 . dQ . 2ir and
integrating from 0 = 0 to 0 = ir/2 and dividing by 2tt we get
■n
±]J-a-f\jJ— 2-= + !sAx) :-'(m)-
122. The probable mean component displacement of vis viva by convection
between encounters by a grain having velocities between V-[ and V-[ + d V^.
Multiplying the product of the vis viva of the grain V{2 into the probable
displacement (X) by cos 6 . sin 0 . dd, dividing by 27T and integrating from
0 = 0 to 0 = 7r/2, the rate of the mean probable convection is
i * d sin2 e ja,
123. The mean component flux of vis viva.
Since there are two mean paths traversed for each collision, adding
twice the mean component displacement by convection for one path to the
mean displacement by conduction at an encounter and multiplying by
7i1F1/2X, the expression for the mean flux by grains having directions such
108
ON THE SUB-MECHANICS OF THE UNIVERSE.
[124
that cos 6 and cos <j> are positive, and for pairs of grains for the mean velocity
of which cos 6 and cos </> are positive, is
in s\
1 n
8N
124. The mean component flux of component vis viva.
The flux of the components of vis viva may be separated for direct
;ii)
2
„ ,, d sin2 6 . d sin2 6 .. , ,. ,
action by substituting cos2 6 . - — — for — in the last equation and
integrating :
N\l + 3\Jf '
TV 7T
\ r2 fid cos2 e
n
4" ■d*-^4ir
V
fl+-^
8"F
3XjJ W 4
d sin2 #
and for lateral action by substituting sin2 6 . cos2 cf>
7T 7T
"2 p rf sin4 0 /I + cos 2<f>
y (174).
M1+fx)/£)
(WO
^■*-£
/3 W /
=§-#l1 + rJ/(0T/
125. TAe component of flax of mass in a uniform medium.
Since mass is not subject to conduction, and the probability of a grain
having velocity F/ is nJN while the probable mean path is X and the
number of collisions in unit space and time between the grains having
velocities between F/ and (F/+8F/) is
the component in direction of a? of a grain of which the direction is
defined by sin 0 . d6 . d<f> 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
V
X cos On . — - sin 6 . d6 . dcf)
X
4>TT
n
4-7T
V1
,, d sin2 0.d0.d<j>
•(175).
Then integrating from 6 = 0 to 6 — tt/'2 and <f> = 0 to $ = 7r/2 and from
V; = 0 to F/ = oo
n,Fx iV 2a _
o 4 4 yV
Jo
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 0 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 uecessary, 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 defiuite 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 ; Rankine'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 defined, 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
* Royal Soc. Phil. Tram., Vol. 186 (1895) A, pp. 146—7.
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 cr, or the sum of the radii of the spheres. And
(3) the instantaneous exchange of actual component energies in the direction
of the normal.
This linear redistribution of momenta by conduction and the consequent
linear displacement of their energy, relative to fixed axes, when there is mean
motion, are the complement of the angular redistribution of energy, the
three effects being the total instantaneous effect of the encounter, which
admit of analytical separation, as long as there is no resilience.
132. The concrete effects of encounters between the grains must be
considered as belonging to the resultant system in which there is no
resilience. For when the effects come to be analytically separated by inte-
gration into effects on the mean and relative systems respectively, if there
are rates of strain in the mean system there will be, perforce, abstract
complementary resilience-effects in both systems.
It therefore appears that, if the mean effects of encounters are to be
considered as belonging to the relative system, it is necessary to assume that
the mean-motion is not undergoing strain, or that any rates of strain are
indefinitely small. Then since the relative motions are the only motions, the
following theorem requires no further demonstration.
133. If the directions, velocities and positions of the grains, constituting
a granular medium, be considered, at any instant, as a complex accident, at
the instant an encounter occurs, between any pair of grains, the three instan-
taneous effects, already discussed, will constitute an instantaneous finite
variation in the complex accident, which variation will continue the same
finite change, from the condition that would have existed, had the pairs
passed through each other without effect, no matter what other variations
might have taken place. Also, the subsequent effects resulting from the
first encounter will remain unchanged. And thus, the integral effect of an
encounter, at a time subsequent to the encounter, is its instantaneous effect
added to all effects which ensue as a consequence of the encounter. In a
granular medium, since each encounter involves two grains, the number of
r. 8
114 ON THE SUB-MECHANICS OF THE UNIVERSE. [134
changes would increase as the sum of the series in geometrical progression
with the factor 2 ; so that in a time ten times as long as the average time
between two encounters, by the same grains, the number of effects resulting
from a single encounter would be on the average 8000.
Thus taking account of the three analytically distinct instantaneous
effects, in a time ten times as long as the average life of a path, the effects
of an encounter would entail, on the average, 8000 changes in the directions
of paths of grains, 8000 linear shunts of component momenta through the
distance a in different directions, and 8000 shunts of the difference of the
vis viva of the normal velocities through a in the direction of the normals.
Assuming, then, that in these changes, or variations of the complex
accident, each has its effect in removing a portion of any mean inequality,
which portion is proportional to the mean inequality, some idea may be
gathered of the predominance of the effect of these changes in bringing
about and maintaining the mean condition of the medium to which the
changes tend.
134. In order to form definite estimates, in terms of the quantities, or
mean constants, which define the condition of the medium, of the rates of
decrement of inequalities from the condition to which the variations tend, as
well as to find expressions for the resulting condition of the medium, it
seems, in the first place, necessary to define, somewhat precisely, what are
the immediate after-effects which follow, severally, from the three instan-
taneous effects which have been analytically distinguished. For such
definition the following general theorems may be proved.
Theorem. The only effect ivhich folloivs the instantaneous effects of an
encounter, until there occurs another in which one of the grains is engaged,
is the linear change in position of mass, energy, and momentum, which results
from the instantaneous change in the direction of vis viva.
The proof of this theorem follows, at once, from the analytical definition
of the three effects and their continued existence.
For the instantaneous effect of linear displacement of the component
momenta by conduction through the distance a in the direction of the
common normal remains unaltered and hence produces no further effect
till the next encounter.
And exactly in the same way the instantaneous exchange of the energy
or vis viva of the components of the velocity of the grains, in the direction of
the normal, remains unchanged until the next encounter. Therefore it follows
that the instantaneous changes in the direction and velocity (which is obtained
for each grain by superimposing on its actual velocity, before contact, the
normal component of the relative velocity of the pair, measured in the direc-
tion opposite to the normal component of the velocity of the grain before
136] REDISTRIBUTION OF ANGULAR INEQUALITIES. 115
contact) represent the actual changes in the directions and velocities of the
respective grains, whence, as these effects are to institute rates of linear
displacement of mass, momentum and energy by convection, these are the
only changes, and they are the after-effects of the instantaneous change in
the direction of vis viva. Q. E. D.
135. From the theorem in Art. 134 it follows, as a corollary, that : —
The instantaneous, and after-effects of an encounter (before the next
encounter of either of the grains) are confined absolutely to normal displace-
ments of mass, and of normal components of momentum and energy ; so that
they have no effect whatsoever on the positions of mass, momentum or energy
as measured in directions at right angles to the normal.
Therefore whatever may be the directions and velocities of pairs of grains
before encounters, if the normals at encounter are all parallel to one axis, there
is no lateral redistribution as the result of the encounters, whatsoever may be
the extent of the normal redistributions.
136. From the principle stated in the corollary, Art. 135, that the redis-
tributions resulting from encounters are confined to the directions of the
normals at encounter, the following theorem may be proved.
Theorem. In a granular medium, in its ultimate state, without angular
inequalities in the vis viva, &c, &c, the rates of angular redistribution of the
vis viva will be equal in all directions, and equal to the rate of redistribution
in the directions of the normals, if the directions of the normals are such that
all the lines, drawn from a point, parallel to the directions of the normals,
meet the surface of a sphere, about the point, of unit radius, in points which
are symmetrically distributed over the surface of the sphere.
For in granular media, without angular inequalities, if \/a is large, all
directions are equally probable for the normals of encounters, in which the
changes in normal vis viva are equal ; so that the probable rates of redistri-
bution of inequalities are equal in all directions.
And in media in which a/\ is small, as has been shown (Section VII.
Art. 89), the directions of the normals will be arranged about n axes sym-
metrically placed ; w = 4 being the smallest number of mean normals that
admits of symmetrical arrangement; and n = 12 the largest number, and the
number in the ordinary piling. These mean normals being parallel to six
axes, so that the probable arrangement in each group, of the directions of the
normals, at encounters, in which the changes of normal vis viva are equal, will
be similar about the axes; and it has to be shown that the rates of distribution
will be the same in all directions.
This proof follows from the principle of the resolution of stresses or
component vis viva.
8—2
116 ON THE SUB-MECHANICS OF THE UNIVERSE. [137
If the angles between any line OA drawn through a point 0, and the lines
drawn through the point 0, in the directions of the normals, are respectively
6U 02,&c.,then the sum of the products of jOjCOS2^, p2cos262, &c. is the rate of
redistribution in the direction OA, and is the same for all directions if the
directions of the normals are symmetrical. Q. E. D.
137. The theorem in Art. 136 includes the redistribution of the actual
vis viva between the grains, as this results from the same exchanges in
directions of the same normals as determine the directions of vis viva ; and,
further, includes the redistribution of the limited displacement of normal
momentum by conduction. Q. E. D.
138. When the mean condition is such that there are more normals in
any one direction than in those at right angles, the rates of redistribution will
be greater in that direction in which there are most normals. But, as regards
the vis viva, as long as the distribution of the normals is such that the normal
redistribution is in no direction zero, there will be rates of redistribution which,
though not equal in all directions, all tend to bring about an equal distribu-
tion of vis viva in all directions, and also tend to bring about the normal
distribution of the actual vis viva of the grains.
As long as the inequalities in the symmetry of the directions of the
normals are small, the effect on the rates of redistribution will be very small,
that is, on the rate of redistribution of vis viva, and on the actual distribution
of velocities of the grains, whatever may be the state of the medium as regards
the ratio cr/A.
Thus for the component vis viva and actual vis viva there is a continuous
law of rate of redistribution and only one even when cr/A, becomes indefinitely
large, so that the directions of the normals approximate to steady axes which
only change their position on account of mean strain in the medium.
139. The redistribution of rates of limited conduction of momentum, or
the limited displacement of normal momentum, is primarily dependent on the
rates of redistribution of the directions of the normals. And the redistribution
of the normals is primarily dependent on the redistribution of the positions
of mass, which again has a primary dependence on diversion of the paths,- as
the after-effect of the instantaneous angular redistribution of vis viva, but this
dependence on the divergence of the path is essentially limited by the value
of <r/A,
If this is small — that is if the freedoms are great — then, after an encounter,
it is a matter of chance, like the length of the path of a grain, in what direction
the normal at the next encounter will be, all angles being equally probable,
and consequently the redistribution of the normals is determined by this
probability.
141] REDISTRIBUTION OF ANGULAR INEQUALITIES. 117
But when the condition of the medium is such that <x/\ is large the
greatest possible distance a grain can travel before the next encounter may
be much less than a, and this in any direction, in which case the possible
direction of the normal is limited by a conical surface, which may be of angle
zero, in the limit.
Then the rate of redistribution of the normals varies with the angle of this
cone. Thus, as a/\ approximates to oo , the directions of the normals approxi-
mate to fixed axes according to the arrangement of the grains ; in which case
there is a redistribution of the rates of conduction of momentum or of the
conduction of energy.
And here it may be noticed, that before the grains become virtually close,
a limit is reached at which change of neighbours, or diffusion of the grains,
ceases, and as soon as that limit is reached the mean position of the grain is
constant, except for mean strains, and then the normals group round mean
axes which only move with the mean strains of the medium.
Thus the displacements of normal momentum and energy depend on
the arrangement of the grains apart from the mean freedoms, and the
redistribution of the conduction depends on the redistribution of inequali-
ties in the symmetry of the arrangement of the grains, so that, although
both the angular redistribution of the vis viva and rearrangement of in-
equalities in the symmetry of the mean arrangement of the grains, are
included in the fourth term of equation (117 a), expressing angular redistri-
bution, they have not been analytically separated, in the terms, as depending
on angular dispersion of vis viva and rearrangement of the inequalities in the
symmetry of the mean arrangement of the grains.
The analytical separation of the abstract actions on which the two effects
of angular redistribution respectively depend, effected by the demonstration
of the foregoing theorems, renders it possible to deal with the two rates
separately and so to obtain analytical definition of the respective rates in
terms of the constants which define the state of the medium.
140. The analytical definition of the rates of angular redistribution of
inequalities in the directions of vis viva of relative motion.
As these actions do not appear to have been the subjects of previous
consideration it is necessary to demonstrate two preliminary propositions
before considering the mean effects.
141. The energy of component motion in any direction cannot by its own
effort increase the energy of component motion in this direction.
This proposition might be taken as self-evident ; but it may be definitely
proved. In the case of spherical grains the proof is simplified, and particularly
if the relative-motion is such that the only inequalities are in the energies
of motion in different directions — unequal angular dispersion.
v&c.,&c. ...(177).
118 ON THE SUB-MECHANICS OF THE UNIVERSE. [142
Taking the axes of reference fixed, I, m, n and I', m', n', and I", m", n" as the
direction cosines of the normal at the point of contact and of two other direc-
tions at right angles, also ultv1} wx, u2, v2, w2 for the antecedent velocities of the
two grains, and U^ V1} Wu U2, V2, W2, 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,
U? + U22 - mx2 - u£ = - 2 (m2 + n2) I2 {u.2 - u,)2
+ 2P [m2 (v2 - v^)2 + n2 (wa - wx)2)
+ U2mn (v2 — Vj) (w2 — ivx)
+ 2(2l—l)[lm(u2 — u1)(v2 — v1)
+ nl (w2 — Wj) (u2 — ^i)} '
Then, since for any two spheres with particular relative motion, u% — %,
v2 — vly w2 — w1, the probability of their normal, at the point of contact, having
a direction within any small area, sin 0ddd<j>, on a sphere of unit radius,
having its centre at the centre of one of the spheres, assuming all angles
of relative motion after encounter equally probable, is :
sin $d6d<f> cos ^
7T
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 7r/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 x to De measured in the direction of the relative motion, v2 — vx,
w2 — w1 are each zero, and putting
l(% + ^)8 + i(t*2-Mi)2 for uf+uj, &c, &c.
in equation (177) we have
U2 + U22 - i (Ml + u2)2 - i {u2 - Ul)2 = -2(m2 + n2) I2 (u2 - %)2 + 0 + 01
Vi2 + V-£ - i(fi + v2f - 0 =-0 + 0 + 2m2l2(Ul-u2)2 ...(178),
W1i+Wi2-^(w1+w!iy- 0 =-0 + 2n2l2(u2-Ul)2 + 0
in which the ciphers represent the values of the terms having factors (v2 - vj
and (w2 — 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
144]
REDISTRIBUTION OF ANGULAR INEQUALITIES.
119
and equal to + cos 6—±l, the equations become on transposing the last terms
in the left members
u2 + m - \ (u, + u,y = (Wl-2 %j2 - i («, - ?02 + o + o
J? 4- Vi - \ (Vl + v.2)2 =
(179),
-o + o + H^-^)2)
where, since the square of the relative motion, (u2 — i^)2, is double the sum of
the squares of differences between the actual component motions and the
mean component motions,
w2 + wA2
U2 K I +
U,
2
£^2 + ^2 - 1 («, + ^2)2 = g
J? 4- F22-|(^ + ^)2 = g
«2 S — I +
M, 4- M2\2
Ws - — ' I 4"
!!.,
2
u, 4- m2\2
+
ux
4-
w2'
2
ux
+
w2'
2
U-y
4-
zt2
.(180).
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
u2 — u1} v2 — v1} w2 — wlf respectively, in other directions than those of x, y, z
respectively. Thus the effort in the direction of the normal caused by «2 — «,
is 21 (u, - «x) and the component of the relative velocity u2 - ux in the direction
of the normal is l{%2-u^)\ so that the total result of this effort is - 2l2(u.2 — u1)2,
work spent by energy in direction of x. Of this 2l4 (u.2 - ux)2 is work returned
to the energy in direction of x ; 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 (I2 - 1) I2 (iu - u^f, and the passive accidents in the directions
of y and z are 2l2m2(u2-u1)*, 2l2n2(u2-u1)2 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,
^N= a0y (it?— tedydz* (181),
where a2, ft2, <f are double the mean of squares of the respective component
velocities.
Since the differences between a2, /32, <f and the mean (a2 + /32 + 72)/'i are
always small compared with their mean it becomes more convenient to alter
the notation and, taking a2 as expressing the mean of a2 + /32 + <f, to take
a (1 + a), a (1 + 6), a (1 + c) respectively for Maxwell's a, fi, y ; a, 6, c are then
small fractions of unity such that their squares may be neglected and for the
mean squares we have
a2(l+2a), a2 (1+26), a2(l + 2c),
and the inequalities are 2aa2, 26a2, 2ca2 ; 2a, 26, 2c being the coefficients of
inequality from the mean of the mean squares of the respective components.
It is to be noticed that in equation (136) the axes of reference are the
principal axes of the space variations of the mean motions of the medium —
the principal axes of distortional mean motions — and also of the inequalities.
146. The angular inequalities in the mean relative motions of pairs of
grains have the same coefficients of inequality as the mean actual motions.
Integrating equation (181) with respect to y and z from — oo to + oo
Ne~^il~'2a)
s/N= dx (182).
Then, after Maxwell, taking x1 as a particular component of velocity in
direction of x, the number of grains which have component velocities
between xx and x1 + S^ is
N
ol (1 +a) \Jir
Phil. Tram. Royal Soc, 1866, p. 64.
148] REDISTRIBUTION OF ANGULAR INEQUALITIES. 121
And again taking x2^=x1 + x' the number of grains between xl + x and
%! + cc' + Bx' is
^a(l + a) yV /
Then the number of pairs of grains which satisfy both these conditions is
m i^g{(.4)v?}
Then, since xx + — may have any value from — oo to + x for any value
of x', integrating for xx between these limits for any particular value of x, the
number of pairs which have component relative-velocities, in direction x,
between x' and x' + Bx is :
N* --S(i-2*w.
V2a(l +a)Vir
In exactly the same way it is shown that the numbers of component
relative-velocities between y' and y' + By' and between z' and z' + Bz' are
respectively
e 2a2 ^',
V2a(l+6)v/^
*-faa-2v.
V2a(l +c)Vtt
Multiplying these expressions by x"2, y'2, z'2 respectively and integrating
from — oo to + oo , and dividing by N2, we have for the mean-squares of the
respective components, in the directions x, y, z
2a2 (1 + 2a), 2a2 (1 + 26), 2a2 (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)
a2 (1 + 2a), a2 (1 + 26), a2 (1 + 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
Vj being the actual velocity of the grain, and \ the distance traversed before
encounter.
This distance Xj may be anything from 0 to oo . But it is proved by
Maxwell to be independent of Vx and to have a probable mean value,
neglecting a as compared with X, of
x = v^y <183>-
Taking <r into account, as will be shown, the probable value of X
becomes
v/g-/© <18*>-
V2 TT<T2N
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
Vx is independent of Vx, 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 -rjr , for all values of Vlf since this
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, z are
Xl Xm Xn
150] REDISTRIBUTION OF ANGULAR INEQUALITIES. 123
and since the chance of a collision in a unit of time is VJX the probability
of continued existence is
e A ,
and the probability of continuing for a time
t =
V,
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
t1=yj, t2 = yrj, &c (18o),
so that, multiplying V^P, V22l2, &c. respectively by tly t2, &c, and adding, the
sum will be equal to
2 {nM ( Vt + V2 + &c.)}, = | V\l,
and similarly for the other two components.
And putting Fand V2 respectively for the mean values of V and V2, the
mean time of equal probability for the continued existence of V2 is obtained
by dividing the product by V2 : -4= , and for the other components
n1\m2V thXn'F
Vhu2 ' V2n2
These mean times, it will be noticed, are independent of the directions
of the groups, being all expressed by
■ — v3-
n \ V . . ~ ~ t
t— _ - , where the probable continuance is e~n> = e KV (186).
Differentiating this expression with respect to t,
^ = S_ (187).
From equation (181) the mean values of u2, v2, w2 are found to be
|(l + 2a), | (1 + 26), f(l + 2c).
In these a2 is constant, and a + b + c = 0, and the inequalities are
|2(l + 2a)-|2=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
0 a2 a2 -[—J*
Whence if nx = 0,
a2 d (2a) a2 */ir m
2-rfr = -6ai2 2aX'&C-'&C (189)'
a" (2a)
~dT
= - 2a ( 3 ^ j , &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,
p"[(u'u')"-i(u'u+v'i/ + w'u/)"], &c, &c , (190),
and for the rates of dispersion with reference to the same axes we have,
putting d2jd2t in place of djdt to distinguish these as rates of angular
dispersion,
P z}[(uu) -h(uu+vv+ww)] = -T-ap
'. , ,.„ (u'u + v'v' + w'w'Y
(uu) —
3
&c, &c, (191),
where ^ajsjir 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 a.)
As already pointed out, Art. 146, the expressions in equations (189) and
(190) for the inequalities are with reference to their principal axes only; so
that in order to obtain expressions that shall apply for any axes it is
necessary to effect the transformation from the principal axes, at a point, to
fixed axes.
152. Rates of angular dispersion referred to axes which are not necessarily
principal axes of rates of distortion.
Taking hm^, l2m2n2, l3m3n3 to be respectively the direction cosines of the
principal axes with reference to any rectangular system of fixed axes,
a', V, c', f, g\ h'
to be the mean values of u'2, v'2, w"2, v'tv', w'u', u'v (u, &c, as before, repre-
senting the relative velocities referred to the principal axes 1, 2, 3), and let
a, b, c, f, g, h, be their corresponding mean values when referred to the fixed
axes of x, y, z.
153]
Then
REDISTRIBUTION OF ANGULAR INEQUALITIES.
125
f' = g' = h' = o
a = l*a' + l2-b' + l32c
b = m?d + m22b' + m32c'
c = n?d + n22b' + n32c
f = m^a + m2n2b' + m3n3c'
g = nf,xa' + n2l2b' + w3Z3c'
h = lxmxd 4- l2m2b' + l3m3c'
From these, adding the second, third, and fourth,
a + b + c = d + b' + c' .
.(192).
.(193).
Also since the principal axes do not change their position in consequence of
the dispersion of the inequalities
~W = h ^T +k ~U~ + h ~aj~ ' &c-' &c-
d2(f) d2(a') d2(V) 92(c') e „
- = WiWx -^- + m2W2 -^j-' + 7tt3H3 -^-y , &c, &c.
.(194).
d2t
d2t
d2t
d2t
Then substituting from equations (190) for d2a'/d2t, &c, in (194), and
remembering that lxv! + l2v' + l3iu\ 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,
P ^rA\uu) —^(uu+vv+ww)]
dot
■—-j p" — a. [(ii'u')" — \(u'u + v'v' + w'w'Y'], &c, &c.
4 A
V.. .(195).
P g^ l(v u ) J = 4 P y a (« u ) , &c. &c.
p"~[{w'u')"] =lp"^a (w'u')", &a, &c.
Ont T A.
153. TAe 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 showD, 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. 150, the probability for the continued existence for a time
k = n1y is e~n> (196).
From this it follows, as in equation (185),
ti = yhyll, U = n2yj, &c, &c (197),
in which expressions the direction cosines llt m1} n^, &c. are nearly constant
and n1; the index of probability, is constant.
Therefore taking the products (tj Vx + &c.) and dividing the mean product
by V — the mean velocity — the mean time of existence of the inequality is
found to be
l = ^\ (198),
and the mean probability of continued existence is
9-M. —
It
e "i = e a.
•(199),
155] REDISTRIBUTION OF ANGULAR INEQUALITIES. 127
which when the inequalities are small becomes
1
Q \Jtt \
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 V2.
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 X/ar is large.
But as the density becomes large, unlike the redistribution of vis viva, the
redistribution of relative mass depends on two circumstances, the inequalities
being small in both cases.
Inequalities in vis viva are not subject to any limits imposed by the
neighbouring grains and consequently all directions of motion are equally
probable, however close the grains may be, and whatever may be the arrange-
ment of the grains.
On the other hand the possibility of angular rearrangement of the grains
turns on the possibility of a grain passing through the triangular surface set
out by the centres of three of its neighbouring grains ; and this possibility
is closed at some density less than that of maximum density. The density
at which this closure is effected is that at which diffusion ceases and the
state of permanent distortional elasticity commences. Before this density is
reached the diffusion becomes slower and slower as the density increases ;
so that in a granular medium of which the mean condition is uniform, but
which is steadily contracting, the chance of a grain finding a clear way
between three of its neighbours diminishes, and each grain dwells longer
and longer in the same mean position in the medium, until all chance ceases
and its mean position is definitely defined, notwithstanding that it has still
a certain range of freedom. For the general consideration of the rate of
rearrangement of mass it is necessary to take account of the probability of
a grain returning after encounters to the formation before encounter, and
this presents great difficulties. But it will be sufficient to point out here
that owing to the instantaneous action at encounter, no more than two
grains are ever in contact at the same time, so that there is no chance of
combination of the grains, and that the mean position of two grains is not
altered at encounter while the relative motions are reversed.
128 ON THE SUB-MECHANICS OF THE UNIVERSE. [155
In the next section it will appear that the linear dispersion of vis viva of
grains is very slow as the angular dispersion is very great, so that any chance
activity of a grain of an exceptional character is immediately dispersed
amongst its neighbours and brought back to the mean.
When therefore the density is such that \fa is very small and the density
is nearly the maximum, i.e. when G is nearly Q/\Z2tt, there is no rearrange-
ment of the grains, and this will hold good as G increases provided that the
extent of the medium for which the value of G is large is very small.
Thus we have two states of the medium in which the rates of rearrange-
ment are defined, and between these a gap in which the definition is
difficult.
Fortunately this difficulty is confined to a very small portion of the total
range of density, being that between the density at which diffusion ceases
and that at which diffusion becomes easy.
This gap covers a region of which the higher limit of p is slightly less
than l/\/2, when the distribution is uniform, and is equal to 1/3 at irregular
points and surfaces ; \/tr being small in both cases.
For values of p above these limits there is no diffusion and consequently
no redistribution in the arrangement of mass, while for values of p below
these limits the change in rate of redistribution is very rapid at first,
then gradually settling down to the same relative rate as that of redistribu-
tion of vis viva.
If then we take as before a — 3X (a)/d1 (t), &c. to represent the small
angular inequalities instituted by the distortion in the mean system during
the time d2 (t) ; the rates of redistribution to which these are subjected will
approximate to that to which the vis viva is subjected as p approximates
to zero. Thus the law of redistribution has an asymptote
m=~^m (0)-
Then if we take f{G) as expressing a coefficient by which the upper
limit of p must be multiplied to bring it to unity
d2(a) 2 l-f(G)p \
dM = " vvx m TT^- {1 ~{ {G) p]> &c" &c-
.(202)
are expressions which give the rates of redistribution correctly except,
perhaps, in the immediate region of the higher limit.
156. The rates of probable redistribution of angular inequalities in the
rates of conduction.
156]
REDISTRIBUTION OF ANGULAR INEQUALITIES.
129
Any angular inequalities in the rates of conduction result, solely, from
angular inequalities in the distribution of mass, but the coefficients of the
rates of redistribution are not the same for rates of redistribution of mass as
for the redistribution of conduction.
The mean time of continued existence of the path of a grain
i=n£
V
.(203),
is not the mean time for the continued existence of the product of the mean
path multiplied by the vis viva. If however the mean time for the mean path
be multiplied by the factor
we have
I
V2 V2 I
.(204),
which is the same coefficient as for the time of continued existence of
vis viva.
To obtain the expressions for the probable relative rates of angular
redistribution of angular inequalities in the rates of conduction correspond-
ing to the rates of angular redistribution of angular inequalities in the
distribution of mass, we have to multiply the relative rates of redistribution
of mass by the factor
37T
IT-
Then substituting the actual inequalities in the angular rates of con-
duction
{Pxx'-p"), pyx', Pzx", &C.,
for a, f, &c, the expressions for the rates of redistribution of these
inequalities of conduction are
~ (Pxx" - p") = - 1 vtt 1 1 l~/}ZlL(p™ - p")> &c-> H
2 / " \
d~t {Fyx )
dTt(p™ )
V II ~ -. av.^ [Jyx >
?v
= -|a/7T
X 1 + e-»(w(ff)p)
a l-f(G)P
Xi+e-»(l-/(0)p)
Pzx
&C, &C.
&C, &C
V.. .(205).
In these equations (204) for the rates of angular dispersion of the dis-
tortional inequality, and the two rotational inequalities in conduction, as well
as in the corresponding equations for the rates of angular dispersion of the
corresponding inequalities in the vis viva of relative motion (195), the analysis
r. 9
130
ON THE SUB-MECHANICS OF THE UNIVERSE.
[156
for each inequality has been effected separately in terms of the quantities
which define the state of the medium.
These six rates of dispersion for each of the components in directions
x, y, and z added together constitute the rate of increase of the energy of the
component of relative motion received from the other components of the same
system. And thus it appears that the expressions for these six rates of
redistribution are the analytical equivalent, in terms of the quantities which
define the condition of the medium, of the fourth term in the equation (117 a);
which may be expressed as
1 fdu' dv' dw'\ |
S\dx dy dzJ\
&c, &c.
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),
lid }"
2 )T~, [(Pu'u' +Pxx)'u'] + &cl , &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'jdx + &c.)" 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
* Pogg. Ann. 1860.
9—2
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, V1 ; 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, <x,
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 X are of the same order, or
in which X/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 a ; so as to obtain expressions for the mean-ranges of mass, momentum,
and vis viva, as determined by conduction as well as by convection.
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 is obviously expedient to restrict the arrange-
ment of the grains, to be considered, to those which have three axes, at right
angles, of similar arrangement, as in the octahedral formation; in which cases,
* "On the Dynamical Theory of Gases," Phil. Trans. Royal Soc., p. 49, I860,
t Phil. Trans. Royal Soc, 1879, Part u.
161] LINEAR DISPERSION OF MASS, MOMENTUM AND ENERGY, ETC. 133
whatever may be the formation, equilibrium is secured when the internal
arrangement of the medium is uniform along each of the three axes ; and the
external actions on the medium over planes which are perpendicular to the
axes are also uniform.
159. Mean-ranges.
Having obtained expressions for the rates of flux of mass, momentum, and
vis viva, respectively, by conduction as well as by convection, for any group of
grains in any direction, in a uniform medium, it remains to analyse these
expressions so as to obtain the component mean-ranges of mass, momentum,
and vis viva.
It is to be noticed that mass and vis viva are scalar, while momentum or
velocity is vector ; and that this fact gives the mean-ranges of momentum and
velocity a different significance from those of mass, and vis viva or energy.
The mean-range of convection by grains in the direction of their actual
motion, whatever they may convey, is A. And the mean-range of conduction,
at encounters between pairs of grains in the direction of the normal, whatever
is conducted, is a.
160. The component mean-ranges.
The respective component mean-ranges of conduction and convection are
obtained by multiplying the components of the rate of flux by convection, in
the direction of the elementary group, by the component of A in that direction,
and the component rate of flux by conduction, in the direction of the elemen-
tary group, by the component of a in that direction, respectively, integrating
for the general group and dividing by the integral flux for the same group.
The component mean-range of mass.
As mass is not conductible the mean-range of conduction is zero. The
component mean-range — that of convection — is then obtained from equation
(175) as
/;/>- n>
= |X (206).
161. The component mean-range of momentum or component velocity.
In equations (158) and (163) if the factors for convection and conduction
under the signs of integration are multiplied respectively by A, cos 9 and
a cos 6, and integrated with respect to 6 from 6 = 0 to 6 = tt/2, (f> — 0 to
</> = 7r/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
| A and §o\
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, bv convection and conduction, respectively, are
| A, and §o\
162. The mean-range of vis viva.
Multiplying the convections and conductions, under the signs of integra-
tion, in the three equations (172), (171), (174) respectively by \cosd and
acoscf) and dividing by the respective integral rates of flux, the respective
mean-ranges are found to be, for convection and conduction,
For actual energy §A and |cr, coefficient §.
Direct displacement f A. „ |cr, „ §§.
Lateral „ fg\ „ fza,
35'
The mean-ranges of momentum and vis viva, inasmuch as they are
expressed in terms of A. and cr, are general when A 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 A/cr 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) o-3/v/2, the mass of a grain,
(2) the constants in the expression /(c-)> Art. 102,
(3) u" , v", w", the mean velocities of the medium,
(4) N, the number of grains in unit volume,
(5) a, where 3x^2 = (V1'V')'\
Of these five mean characteristics (1) and (2) stand in different position
from the rest, (1) being constant in time and (2) depending on the ultimate
arrangement of the grains, and the consideration of these may be deferred.
165] LINEAR DISPERSION OF MASS, MOMENTUM AND ENERGY, ETC. 135
The mean characteristics (3), (4) and (5) all enter into the definition of
the state of a medium in uniform condition.
164. Characteristic velocities, densities and mean-velocities of the grains.
From equation (136) it appears that, referred to axes moving with the
mean motion of the medium (u", &c), the number of grains having velocities
between F/ and V}' + 8V/ in directions which referred to the centre meet the
surface of a sphere of unit radius in the small element d (cos 6) d (<£), is
n N /FA2 -(— V fV'\
-dcos0# = ^f( ±)e \*'d(^)de.d<l> (207).
Dividing by iV
n 1 'V'\2 -(—)' /V'\
^dcoB0d4> = ^(-±) e ^'dfflde.d* (208).
If then in one state of the medium a has the value al3 and in another state
has the value O2 = oti(l +da1/a1), the characteristic velocities, for which
F, = F8 <209>'
will be VI and F2' = Vx' ( 1 + da,/ a,).
The inequality between the characteristics is :
In the same way for the characteristic densities if the numbers of grains
in the two states are jVj and ^0 = ^(1+ ~) the characteristic numbers of
the two states are
ih and njl + -^M ,
with the inequality iij -^-.
And if u" and u" (1 + du"/u") are mean component velocities in the two
states the characteristics are
«/' and <' = < (l + 9^Q (210)
165. Characteristic rates of flux when the axes are fixed.
Putting I = cos 6, m = sin 6 cos (f>, n = sin 6 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 A, while that of conduction is a, that for
momentum the rates of the fluxes are ^2 crf( - ) / 3 A and for vis viva ofl- )/ 3A,
and putting d(cQ)xx, &c, and d(pQ)xx for the respective rates of convection
136 ON THE SUB-MECHANICS OF THE UNIVERSE. [166
and conduction of an elementary group in direction denned by — d (cos 6) d<j),
with respect to fixed axes; for the flux of mass we have by equation (175)
d(eQ1U = p(u''+^\o8 0^d(^^jd<f>, &c, &c. ...(211).
And by the last Art.
a (cQS™ = d LQX* +(S(a)~ + 8 (>") ^ + 8 (N) ^ d (,&)**> '
whence the inequality of flux is
a (cQd™ -dicQO™ = (S (a) ~ + 8{u") ^ + S (iV) A) d(cQl)xx . . .(21 2).
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 6)f V-{ while the conduction is not altered by the mean-
motion — omitting the square of the mean-motion and dividing out the A,,
we have : —
For direct action referred to fixed axes
a (CQ, ).,, + a (M.vx = p {(«' + v; cos ey +^ £/ (?) v^ co**} | d ( - cos 0) d<f> '
&c. &c.
&c. &c.
(213).
For lateral action
a (cQi)^ + a (M!IX = P l(u" + v{ cos 6) (v" + v; sin e cos </>)
+ ^^/(^F1'cos0sin0cos<J^(-cos0)<ty (214).
167. For the rate of displacement of vis viva by an elementary group
referred to fixed axes.
Taking, as before, X(u" + V-[ cos 6) ju for X and omitting, for the sake of
simplicity, all quantities of the second order, such as u"-/X and Xa2, we have
for the direct rate of displacement
a (&)** = p {(»" + Vi cos 0) {u" + v; cos ey
sT f GD Fl'2 (*" cos ^ + v" sin dc0S(f>+ w" sin ^ sin ^) cos30 } ■ • •('
+
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,
•Su,'V1,2cos"d+V1'3cos3d.
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/ cos- 0) by V-[.
The second of the two terms, which changes sign with cos 0, expresses the
displacement ( F/2 cos- 0) by Vx' cos 0.
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
3 (Qi)m = P lO" + Vx cus 0) (v" + V; sin 0 cos <£)2
+
,3 / '( - j F/2(M"cos#+v''sin0cos</>-H</'sin#sin(£)cos#sin-acos2</>
+ £/(?) J72 cos 0 sin20 cos20 \%d(- cos 0) d<f>
.(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 (\/N)(dN/dx),
and (A./a) dajadt 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 A. + a,
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,
7« = /- + '-('<S + '"| + "<5)/' (217)-
If, then, r is the range of /, whether it is \, \f2<rf(-)/S or a f 'f -- J /3,
138 ON THE SUB-MECHANICS OF THE UNIVERSE. [109
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
/«-/'-r('=+'"S+"s)/j (21<S)
for d (I) and integrating 1 1 d (I) sin 6 dd d<j> for the partial groups.
There is however nothing in the definition of the mean characteristics,
at a point, in a varying medium, as stated above, to warrant the assumption
that the grains arriving from the direction Q will arrive at P with the mean
characteristics of the medium at Q.
The mean characteristics are the means of all the groups at Q, whereas
the grains arriving at P from Q must, unless PQ is at right angles to the
direction in which the medium varies, differ from the mean at Q taken in all
directions ; and therefore cannot have the mean characteristics at Q. It is
necessary therefore to obtain further evidence before we can determine what
are the characteristics of the elementary groups in different directions, which
evidence is found in the conditions of equilibrium of the varying medium.
169. The conditions between the variations in the mean characteristics,
a, u", &c, N or p, in order that a medium, in which a and the constants
in fir) &re constant, may be in steady condition with respect to all the
characteristics.
The condition of equilibrium of a medium in mean uniform condition
requires that u", a. and N should each be constant for all positions and all
directions ; so that in a medium in which any one of these mean character-
istics varies, the rest being constant, the equilibrium would be disturbed.
But it does not follow that equilibrium would be impossible if two or more
of the mean characteristics vary.
For the case where <r/X is small these general conditions have been
already determined, in the study of the conduction of heat by Clausius*,
and more generally, in the study of the dimensional properties of matter in
the gaseous state f. In the latter instance, this was accomplished by the
recognition that if the mean characteristics, u", a, N, of flux by a mean
group of molecules arriving at P were the mean characteristics of the
medium at Q, PQ being the range of the characteristics, the three conditions
* Pogg. Ann., Jan. 1862 ; Phil. Mag., June 1862.
t Phil. Trans. Royal Soc, 1897, Part n. pp. 786—803.
170] LINEAR DISPERSION OF MASS, MOMENTUM AND ENERGY, ETC. 139
of steady density, steady momentum and steady vis viva, could not be
satisfied ; whereas if the characteristics, a and JST, of the flux arriving at P
from Q were the characteristics at Q, while instead of the characteristics
n", v", w" at Q arbitrary functions of x, y, z (U, V, W) are taken for the
mean velocities of the arriving group, all the conditions could be satisfied;
and the values of U, V, W be determined in terms of it", v", iv", a and N.
This method may be applied for the determination of the conditions
between the mean characteristics, U, a, N and u" , when - is large as when
A
small, now that the expressions for the mean rates of flux and mean ranges,
resulting from conduction, have been determined, as well as those resulting
from convection, in a uniform medium.
170. The equation for the mean flux.
Substituting U for u", &c. in the expressions for the characteristic rates
of flux by au elementary group ( ), remembering that \ is the range
of convection and <x the range of conduction, that
\
, dN dN dJS
dN--=\ (I , + m •- , - + n j-
dx dy dz
, da da da
da = A [I -r- + lib -y- + 11 -y-
dx dy dz
, d d , d '
da = cr ( I -,- + in -y- + n -y-
dx dy dzj
For convection
For conduction
\
•(219),
in the expression for the inequality of the flux, and integrating from 6=0
to 6 = ir and from $ = 0 to <£ = 2tt, 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
.(220).
Equation (220) has reference to fixed axes, for moving axes the equations
become
0 = P"(^-O-l^{«f + />"§•« ^221>-
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 \ to be the same in all directions,
V2 o- JarW \
p (uu) + p
■ + 7i/U
„ a2 4 _\_
9 2+3vV
aj-[p(U-u")} + p(U+u")da
dx
, &c, &c.
jPo?' lateral flux.
From the second of (213) the equations become
-f(-)\-~
V ...(222).
»/ i i\,i .a i , v- °"
p (« v ) + p ^ = j 1 + 3
«(£jA*(F-">]
+ 1 [/>" ( p - «")]} +p"(y- "") a + p" ( u - "")
da
dyvtJ /jj ■ r ' doe ' r ' ' dy_
For the 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
[P" (*V + pxx) u>)» = | {l + ^/ (£)} ( U - u") P»*
"5U +
3*/ W
For lateral flux.
From equation (216)
V («v +iw) «r = ^ (i + i|(V GDI ( ^- m,/) '"'
y ...(223).
u\x+kf{l
a
dp da3
dx " dx
I
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 <r 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" CaV)" = p" ? , &c, &c-
p (u v')" = 0, &c, &c,
and by conduction, equation (159),
p ** " T \J {x)p 2 ' &c" &c-'
p"xy = 0, &C, &C
.(224).
171] LINEAR DISPERSION OF MASS, MOMENTUM AND ENERGY, ETC. 141
The total rates of flux of momentum being
P"«*+p"(uur=\i + ^^f(')\p*°
3 V7 U/l ...(225).
p"xy + p" (u'v')" = 0, &c, &c.
Substituting in equations (223) the values of U-u", &c, as obtained
from equations (221), the rates of flux of the vis viva of the component
motions become by transformation :
\
, // / / ,Y, ^ a (^ dp 21 da?\
{PxxU)" = ~
15 vV
[£-*)'
,6a _1_\ „ da?)
1 A. V2/P dx]
{pu'v'v')" = £ (pu'u'u')
by equation (223)
(P*yv')" = HP**"')"
And for the rate of flux of the total vis viva
, &c.,&c. ...(226).
puuu +pxxu'
pu'v'v +PxyV
{- pu'iu'w' + pxziv
1 a
-puv'v' + pxyV =9^.
i2 dp _ 21 da2
cfo; 2 " dx
'4c
&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 A. ; 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
dp
dx
= P\
6JA
v/2o-/("-)-v/2A.a262e
\
IdX I dor
\ dx a2 dx
3\ + V2<r/(
and from equation (146), differentiating and transforming,
, &c, &c. ...(228),
b-\
dp SX + \s/2a2b2e « 1 dX
dx
= P
SX + ^,/(£)
A. dx
, &c, &c (229).
142
ON THE SUB-MECHANICS OF THE UNIVERSE.
[171
Adding the equations (228) and (229) the condition of equilibrium is
o-/>(*£-x£),fa.*° <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 :
0 3 + V2 orbH
a
p" (u'u'u)" = — z-^-r- 6X2
r 15 \/TT '
3X + V2o-/
-1 + 21x1 P\ ^,&c,&c.
<t\ y 1 ax
\) J
62A
+
•(231),
3\+V2<r/(?
the respective rates of lateral convection and conduction being one-third of
the corresponding direct rates.
Adding the respective members of the equations (231) the expression for
the total rate of direct flux of vis viva by convection and conduction is :
kav +1u «r - - 13^ j(ex> - 2 (*- V2 ,)./(0) ;;V;"'^;;
Then, since the rates of flux of lateral vis viva are each one-third of the
normal rate, the total rate becomes
(p uu + pxx)u
ip'liV +pxy)V
+ (p"u'w'+pxz)w>
6*A
a
Hp"u'V +Pxy) v \ =q^\ [W + 2 (4o- - V2 X) cr/^-j
c-\\3 + V2as&3e "
+ m+,'^-V2)./g)},^,*o.,&c.
3X + V2o-/g)
....(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 ft- and b~ are constant.
173] LINEAR DISPERSION OF MASS, MOMENTUM AND ENERGY, ETC. 143
172. The coefficients of the component rates of flux of (a3 . otr/2 \J2) the
mean component vis viva of the grain.
By equation (129 b) Section VIII.
o-3
Substituting this in equations (233), dividing by N and putting (C.22 + D22)
for the product of the first two factors of the member on the right, these
members take the form :
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 (022 + A2) express the rate of
flux under the condition of steady motion.
But as long as the scales of the variation of a2 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,, a,
i.e. if the scales of mean motion are sufficiently large and the inequalities
sufficiently small, the coefficients C','J and D22, 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
F Ft*- =¥ Tx [{p** + pu'll'} "' " (Px" + puV) v' " (Pxz + puW) w'1 ■ '(2m
Nd,t *x~ K 2 + 2V'2^2V2
we have Xr */«* = -( G? + A2) 775 ;v-2 «
ds j _ (r2,D*\?t & f*!\\ (935)
Ndst x>l v 2 ' ~"V2<ty
NottIxz = ~(G** + I)^ V2 dz*\2t
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
1 d3 ? ,„m . _ <r2_/a2
Ngi-W+Mji*[i) ■■(*»>
And multiplying by N
dj
g-W+iv>,>(f)
174. The expressions for the coefficients C22 andD2-' involve the arbitrary
constant b-, so that the general expression cannot be completely interpreted
until b2 is defined. But the terms which depend upon b are very small
except for states of the medium in which X is greater than o-/10 or less than
lOo-; so that outside these limits the coefficients are independent of b'~
within the limits of approximation.
Then, outside these limits, the expressions for C,2 and D22, as appears from
equation (233), when <r/\ is small, are, within the limits of approximation,
3\a
CI2 =
V7T
D92=0
.(237).
And when <r/\ is large
C2 = 0
"I
n,_4_o-a/<M (238).
-~3v/7rx.UyJ
And these values become infinite in the limit.
175. Summary and conclusions as to the rates of redistribution by
relative motion.
The equations (202) express, in terms of the quantities which define
the relative motion of the medium, the rates of angular rearrangement
of the relative-mass, by institution of relative motion, corresponding to the
last term in equations (119) Section VI.
Equations (235) Section XII. express the linear redistribution of in-
equalities in vis viva of relative motion by the actions of convection and
conduction corresponding to the second and third terms of equations (117 a)
Section VI.
Equations (195) and (205) express the respective rates of angular redis-
tribution of angular inequalities in the vis viva of relative motion, resulting
from convections and conductions respectively, corresponding to the fourth
term in equations (117 a).
The second term in the equations (119) Section VI. is the only term
in the equations of mass which does not become zero when p" is constant in
175] LINEAR DISPERSION OF MASS, MOMENTUM AND ENERGY, ETC. 145
time and space. Therefore equations (202) express the only redistributive
actions on mass, equation (204), resulting from relative motion. These
redistributions of relative-mass are essentially positive dispersions of un-
sym metrical arrangement, at rates which are proportional to the inequalities
in the arrangement of the mass. But subject to the same limit as the
permanent diffusion, as \ja becomes small.
Thus the action of relative-motion on the mass is that of positive
dispersion of all inequalities.
The second, third and fourth terms in equations (117 a) are the only
terms in the equation which depend on relative motion only ; that is, are
the only terms in these equations that do not necessarily vanish when the
vis viva of mean motion is constant.
Therefore the equations (195) and (204), Section XL, express the only
redistributive actions on the vis viva resulting from relative motion.
From these equations it appears that all these actions are essentially
dispersive of inequalities, at rates proportional to the inequalities multiplied
by coefficients depending on the characteristics of the medium ; the only
limit being that imposed by the nearness of the grains, which is the same
limit as that of permanent diffusion as expressed in equation (205).
It thus appears that to a first approximation the action of the relative
motion on relative mass and relative vis viva is essentially that of positive
dispersion of inequalities ; in which the rates of linear dispersion, and of
angular dispersion of vis viva, by convection, are subject to no limit, while
those of angular rearrangement of mass and of angular dispersion of vis viva
by conduction are subject to a finite limit as the grains become closer.
The generalization of the dispersive actions.
The numerical coefficients of the several rates of redistribution expressed
in the equations (202), (195), (205) relate to a medium consisting of uniform
spherical grains. But if, for these numerical coefficients, arbitrary constants
are substituted, these equations become general, that is to say, they include
all discontinuous media in which the separate members do not alter their
shape or size.
Whence the conclusion follows, that discontinuous, purely mechanical
media satisfy the condition for the maintenance of the state of relative
motion.
K.
10
SECTION XIII.
THE EXCHANGES BETWEEN THE MEAN- AND
RELATIVE-SYSTEMS.
176. It has been shown (Sections XI. and XII.) that the effect of the
relative motion is to disperse all inequalities in the mean vis viva of
relative motion and in the arrangement of the mean-mass ; the rates and the
limits of these actions having been expressed in terms of the quantities
which define the relative motion.
It remains therefore (1) to effect such analysis of the terms in the
equations which express the effect of inequalities, in the mean-system, in
instituting inequalities in the relative-system, as is necessary to define the
actions they express, in terms similar to those in which the rates of redistri-
bution are expressed ; and (2), by combining the effects of the respective
actions of institution and redistribution, to arrive at expressions for the
resultant inequalities which may be maintained.
The only terms, which remain to be considered in the members on the right,
of the equations of component vis viva of mean- and relative-motion (123)
after transferring the first term on the right, which is the convection term :
to the left member, are those terms which are concisely expressed as the fifth
and sixth terms in equations (117 a).
Therefore these terms are the only terms which express exchanges of
vis viva between the two systems taken as a whole. And since these terms
do not become surface integrals they express the exchange, at points, of vis
viva from the mean-system to the relative-system. And further, these terms
are transformation terms solely ; so that they each express, under the
opposite sign, the exact rates of exchange as the corresponding terms in the
equations (116 a). Thus the fifth term in equations (117 a) expresses the
rate at which vis viva is received by the relative-system from the mean-
system on account of the diminution of the abstract resilience in that
system, while the sixth term in (117 a) expresses the rate of exchange of
178] THE EXCHANGES BETWEEN THE MEAN- AND RELATIVE-SYSTEMS. 147
kinetic energy necessary in order to satisfy the condition of no energy in the
residual system, the expressions under opposite signs being identical in the
two systems.
177. The initiation of inequalities in the state of the medium.
Since the terms in (117 a) express the only actual rates of exchange of
energy between the two systems, and the effects of the relative-system are
purely dispersive, it at once appears that in a medium, in a state of general
equilibrium, inequalities can be initiated only by acceleration of mean-
motion, and whatever the state of the medium may be, all initiation of
inequalities springs from acceleration of mean-motion as the prime cause.
This being so, any rate of change which may result by transformation from
inequalities in the mean-motion will be expressed as :
d,( ) or 3i( )
d,t dxt '
according to whether or not the rate of convection dc» ( )/dt is or is not
included in the action.
In this way the joint actions of institution and redistribution are ex-
pressed as
d,( ) }d2( )
dxt d2t
178. As presenting by far the greatest difficulty, and thus entailing the
most discussion, the rates of institution of angular inequalities in the rates
of conduction through the grains demand first consideration. These rates,
it would seem, have not hitherto been the subject of analytical treatment ;
and although the expressions for these rates of institution are clearly dis-
tinguishable, now that the conductions are separated from the convections,
the interpretation of these terms presents difficulties owing, partly, to the
novelty of the conceptions involved.
It appears that the analysis of these conductions constitutes the kinetic
theory of the abstract elastic properties in the mean-system of a granular
medium, that is to say, properties of distortional elasticity.
The terms which express the rates of increase of abstract resilience in
the mean-system are included in the last term but one in the right members
of equations (116 a).
In a purely mechanical medium there is no resilience in the resultant
system, so that these terms in the mean-system have their identical counter-
part under the opposite sign in the corresponding equations of the relative-
system. But that which has rendered this subject obscure, is that the
counterpart is under different expressions.
This is owing to the generality of the equations, which are not confined
to a purely mechanical medium. However, on changing the signs of the
10—2
148 ON THE SUB-MECHANICS OF THE UNIVERSE. [178
terms in (116 a) we have the interpretation of the corresponding terms in
(117 a). These terms,
>" /du" dv" dw"\ _v»)^
1 f „ /du" dv"\ „ /du" , dw"
, &c, &c,
represent the rate at which kinetic energy in directions x, &c. is being
abstracted from the relative-motion to supply the abstract mean resilience,
depending on conduction, to the mean-system of motion. This is obvious, as
regards the first of the terms within the brackets, for the components in
directions x, y and z. But as these represent uniform expansion multiplied
by uniform pressure, both the expansion and pressure being equal in all
directions, it introduces no angular inequalities in the relative vis viva. It is
however these terms, or more strictly, the three corresponding term's for the
directions x, y and z taken together, that, owing to their simplicity, reveal
the modus operandi by which the conduction through grains, of changeless
shape or volume, can affect the work done in contracting the space in which
they exist.
It is not the conductions that are the active agents. But these conduc-
tions are a passive necessity of the space occupied by the grains ; and thus
measure the contraction of the freedom of the grains, owing to their volume.
Whence, it is at once realized that the amount of increase of kinetic energy,
which would result from a contraction of the entire space occupied, would
not be the same as it would be if the grains, while conserving their mass,
ceased to occupy volume. For in the latter case, taking V the velocity of
the grains and p for the density, and supposing the action were what is
called "isothermal," the velocity V remaining constant, the rate of displace-
ment of momentum would not be pV2/S, as it would be if the volumes of the
grains were zero.
Neither would this stress vary with p but with p{l + <f>(p)} where <f>(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,
1 . ,, tt. ecu
»<*--*)■&■
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
|v2(n'v/(|),
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 W.
Therefore the ratio of the corresponding normal conductions and normal
convections is
2V2(F')"(r,/(7\ sJ2cr,(a\
■f®-Hif® (239)-
3 2(V)"X
And the rates of institution of relative angular inequalities in the arrange-
ment of the mass are represented by
9X V2 a f*\ LduT_2(duT <W <M\) &c*_(240).
dj 3 XJ \XJ \ dx S\dx dy dz ))' v '
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 grains 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, b, 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,
V2 <t /./oA (du dv dw^
~S\*\XJ \dx + dy+dz
and tIKI
du 2 (du dv dw\
" dx 3 \dx dy dz )
express, respectively, the rate of relative increase of X, the mean path,
in all directions, and the rate of increase of the inequality in the mean value
of the mean paths of the pairs of grains having motion in the direction of
x only. This at first may appear paradoxical ; but the explanation becomes
clear when it is remembered that a rate of strain does not represent a strain,
however small.
For a finite rate of strain to cause a strain it must exist for a finite time.
And to convert the expression for a rate of strain into the expression for a
strain it must be multiplied by the expression for a time; recognising this,
the difference between the effects of volumetric strains and distortional
strains is at once seen. In the uniform volumetric strain the effects on the
path of every pair of grains, whatever the direction of the paths, are the
same ; whereas in the distortional strain, if the strain in direction of one
of the principal axes is positive, the sum of the strains in the other two axes
is equal and negative, and thus they neutralise each other except for such
effects as result from rearrangement of the grains.
Noticing this, it is seen that the rates of strain in the directions of the
principal axes on the pairs of grains with relative motion only, in one or
other of these axes, are perfectly independent. And, assuming that there
are no initial inequalities, these independent rates express the initial rates of
increase of the initial inequalities in the mean relative paths, with relative-
motion in the directions of the principal axes of rates of distortion. And,
as long as the relative inequalities are very small, this independence will
be approximately maintained.
Taking St as an indefinitely small increment of time and multiplying both
members of equations (146) by this time we have, putting a = da'Bt/dt, as a
first approximation to the effects of the rates of institution,
, V2 a- .fa\ (adu" 2 (du" dv" dw"\) . 0 0 ,ft.,,v
or since A, is not affected by the distortional strains we may put for the actual
rates
\ (1 + a') = X
■ </2<r ,/oA f,du" 2 ,du" dv" dw"\\ ,
, &c, &c.
(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,
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,
-*-$£'© {"£-5(S+J*S)}* (->•
Having thus obtained to a first approximation expressions for the effect
of rates of institution of inequalities in the pairs of grains having relative
motion in the directions of the principal axes, we may proceed as in Art. 149
to find, to a like approximation, the effect of these inequalities in the numbers
of encounters on the normal conductions in the directions of the principal
axes of distortion.
180. The initiation of angular inequalities in the distribution of the
probable rates of conduction resulting from angular redistribution of the mass.
Taking x, y ', z as measured in the directions of the principal axes of
the distortional strains, and — a, — b', - c respectively for the relative in-
equalities in numbers of encounters between pairs of grains having relative
velocities in the directions of x, y', z respectively, where a -\-b' + c = 0, we
have for the probable relative inequality in the number of encounters of pairs
of grains having relative motion in the directions defined by I', m', n referred
to the principal axes,
- (Pa' + m"-b' + ri*c'), since /' = g = h' = 0.
Then, taking l1} m^, nx as the direction cosines of the principal axis
measured in direction x, with respect to any arbitrary system of axes
measured in directions of x, y, z; L, m2) n2 and l3, m3, n3 being the direction
cosines of the principal axes of y and z respectively referred to the arbitrary
system, the inequalities in encounters between pairs in directions x, y, z
respectively are expressed by
-(l^a' + l.fb' + l/c), &c, &c (245)
respectively. Then using — aly — bly — cx to express these inequalities, we may
also take, in the usual way, /, g, h, the probable tangential inequalities,
. [dv diu^
dy,
— (wj?i1a'+ m2n.J)' + m3n3c), &c, &c.
Then to find the inequality in the number of encounters having normals
in the directions of the axes of x, y, z, respectively, resulting from encounters
between pairs of grains in all directions, we must express the probable
number of pairs having relative velocities in a direction defined by I, in, n
referred to the directions of x, y, z ; such an expression is
ax = Pa + m2b + n2c + 2mnf + 2nlg + 2lmh (247).
>-iffi+a-fc-to ^
154 ON THE SUB-MECHANICS OF THE UNIVERSE. [181
Then the angular distances of the direction of a^ from this line to the axes
of x, y, z respectively are defined by I, m, n respectively ; and the probability
of the normal at encounter being in the direction of x is lalt in the direction
of y is malt and in the direction of z is na?. These are the inequalities in
the numbers of encounters of which the directions of the normals are in
the directions x, y, z, respectively, resulting from encounters between pairs
having relative motion defined by I, m, n. Then integrating — aj,, — axm,
— a{ii over hemispheres having axes in the directions of x, y, z, respectively,
we obtain, respectively, on dividing by ir the mean inequalities in the proba-
bility of encounters having normals in the directions of the axes x, y, z.
Thus putting I = cos 6, m = sin 6 sin c/>, n = sin 6 cos <f>,
a f2(cZ cos4 6> 1 id cos2 d ldcos40\/7 J
= | + j(6 + c) (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, b, 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, lax
must be multiplied by -^ V^/f-) Vxl, and then integrated. Thus
2 ,lTrff,/(r\ f* (d cos5 0 1 fdcos3d d cos5 6\ ., .) _ p
(249),
reduce to
-V*W()9 (g« + £(* + .))- -s Avsf^j
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 V1 and a, in the members on the right of these
equations, in the directions of x , y\ z respectively, integrating over a
sphere of unit radius and dividing by 47r. Thus since a' + b' + c = 0,
=-^l^^/©^' + r5(6'^')}l'&c-'&c (251)-
= -0-32/(^2V2<TFia'
,\J 9
It will be observed that these expressions are for inequalities of the
probable component of conduction in the directions of the principal axes,
taking into account the relative inequalities in probable normal conduction
in all directions ; and that they do not express rates of conduction corre-
sponding to the expressions in equations (158) and (159), but if multiplied by
o-3/\/2 the mass of a grain, they express inequalities of conduction corre-
sponding to the conductions expressed in equation (148).
To obtain the expressions fur the inequalities in the rates of the relative
component conductions in the directions of the principal axes of distortion,
the expressions for the corresponding component conductions must be multi-
plied severally by the number of encounters each grain undergoes in unit
time, and by the number of grains in unit space, as expressed by the integral
of equation (157).
Comparing the expressions thus obtained with the rates of conduction,
equation (158), it is at once seen that the inequalities in the probable rate of
component conduction in the directions of the principal axes of distortion
are, remembering that a expresses d1(a')d1t/d1t, &c,
0:32 7 1 "f © f I <a'>3-(= h W" -*">3A &c" &c <232>-
Then although the significance of the a and a, &c, used to express
relative inequalities in mean paths have no relation to the a and a, &c, used
to express inequalities in the vis viva, in equations (192 — 194) they are of
similar significance and admit of similar transformation, whence it follows
that by a process strictly corresponding to that followed in Art. 152, these
rates of conduction transformed to any system of rectangular fixed axes x, y, z,
a4P 8lt = Un^ + rn^^+wM M, &c.
Oxt { Oxt OjC Oxt )
.(253),
156
ON THE SUB-MECHANICS OF THE UNIVERSE.
[181
7\ / f\
then dividing by St and substituting the values of -^ — - &c. from equations
dit
(146)
d1(a)
V2o- (a\(adu" 2 fdu" dv" dw"\) \
3 \f[\)\ dx S\dx ' dy ' dz)]'
V2o- ,(<t\ (dv" dw"\ 1 0
f.
d1(f)_
(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 d1(a')/d1t in equations (252) which by equations (159) may be expressed
as: 0~'S2p"; the coefficients of the right members of equations (254) may
also be expressed by 2p"/pa2. Therefore
9 2
Q-32p"2 (dvT dvf
a- " \dz dy
p2
2dxt
y
.(255)
(p"yz), &C, &C.
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 \'\\J a2'
Thus for the relative rate of increase of p"
1 d1(p")= p" fdu" dvT du/'}\
p" dj a2 \ dx dy dz
the actual rate of increase being
3i (/>")
f
p"2 fdu" dv" dw'
Bit a2 \dx dy dz
p 2
.(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
v { (-')» % + c vr % + K«.y d4 } , &c, &e.
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
a2 Bxa
' ■S-ftS.+S+T)}..*.*-™
Putting km^, l2m,7i.2, l3m3n3 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
a = lsa' + l.22b' + l32c
b = nil a' + m£b' + ni32c'
c = n{-a' + n<?b' + n3c'
f= nixii^a' + m2n2b' + m3n3c'
&c. &c.
where a + b + c = a +b' + c'
.(258).
Then
Bxa _]0da' ]ndV 7 „ dc'
B,t~k' dt + 2 dt + 3' dt
.(259),
and substituting for the values of da/dt, &c, from (257)
a2 B,a „ „ fdu" 1 fdu" dv" dw'
-r • -7T-. = P aM -i ^ —i — I- T-
9 2 * dxt H "~ \dx 3 \'dx ' dy ' dz J
+
)|, &c.,&c (260),
a2 d1(f) a a2 dv" dw"\ 0 . /0,.1X
p - . Vy ' = 2p- h- + -r- , &c., &c (261).
r 9 a* ^ 4 \ ds dy J' '
Then putting a2/2 for
2' d^ r 4<\dz dy
{uxi + v'v + w'w')"-
aJ
3
= -p
3a2 fdu
dx
dv" dw"
dy dz
•(262),
P (!tl() —
9:
aJ
„ „a?\du" \ fdu" dv" dw"\) 0 „
wb*W]~vf.J(£+fW*
(263).
2 ' 2 V dz
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 con vection 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 (a2)"/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 (201) and the last of
equations (256)
d1 I a2\ 2 a" fdu" dv" dw"\\
^Vr 2/ 3r 2 \dx dy dz
3i ,/v 2 p"2 fdu" dv" div" \f (264).
di t L 3 a2 \ dx dy dz
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.,
d,t
= 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 d^'/dxt = 0, the institution of du" jdx, 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
// / / l\H . //
P (UV) +P Xy
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" (uV)" + p"xy depends on the mean inequalities
du" dv" #
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
9i / /// > f\tt // n / // «2 0-64 „\(du" dv"\ „„„v
^(p'(uv) +y'w) = -(p'- + ^-,^(- + s) ...(266).
Then changing the sign and differentiating with respect to y we have for
the rate of increase of reaction from this inequality,
" ^2 / »\ ( >>a\ °'64 A(d"u" ■ d'v"\ /o^x
P W(U) = \P 2+27V^'j(^ + ^J ^267>-
Differentiating this last equation with respect to y the acceleration of the
rate of increase of the inequality in the mean motion is
„dx* (du"\ /„a2 0-64 „\d2(du"clv"\ ,__fi,
p WV*y-) = \p 2+2pV? )dtf[dj + te) (268)-
This equation expresses the partial effect of the inequality p" (u'v')" + p"xy
on du"/dy. And proceeding in a similar manner we have for the other
partial effect on dv"/dx
„ d,2 (dv"\ ( „a2 0-64 „\ d2 (du" dv"\ /a/lrtN
p m^)=\p 2+v^ )^{iy-+d^) <269>
Then adding, the total effect becomes
„d1a/dufdv"\ ( „a2 0-64 „\ / d* d2\ fdu" dv"\ /aHns
P WK~dy- + ^)==[P 2+2p^P JW + dy-2)[dy-+d^h^70)-
It is at once seen that this equation represents a positive acceleration, to
dispersion of the inequality in the mean motion, du'jdy 4- dv"/dx, as the
result of the rate of institution of the inequality p" (u'v')" -\- p"xy.
In a similar manner it may be shown that the effects of the five
distortional inequalities, in the rates of convection and conduction, are
accelerations to the dispersion of the five remaining inequalities in the
rates of increase of mean motion. These, together with rates of dispersion
of the volumetric inequalities, admit of expression in a general form.
186. The inequalities in the component of mean motion.
du" dv"
dvT_l fdu^ dvT dw"\) ~dy~+dx~
dx S\dx dy + dz )] ' 2
du" duf\
dz + dx) 1 fdu" dv" dw
1 ■ -s{-dx^+dJ+-dT)^c-'&c"
admit of expression after the manner of expression of component stresses by
simply substituting I"xx for p"xx, &c, &c, and we may further simplify the
expressions by putting /"„ for (I"xx + I"yy + I"zz)/S.
187] THE EXCHANGES BETWEEN THE MEAN- AND RELATIVE-SYSTEMS. 161
V
In the same way we may take Ixx 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
PW(I XX~I v)={p a~+p"tfp 'Jd^1 --7 v)
9i2 ,v, ,T„ v / //«2, 0-64 „A/d2 d2\/r" 7-,/ v
(271).
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 :
Tlie 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'"di{u'u')ld1{t), &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
(\
P ^(uu) =-
„ U U +VV + w w
P -Q- +P
3
+
it r r
p uu
lilt' + v'v + w'w'
3
du" do" dw"
dx dy dz
dx
\
+ pxx~P
fdu" dv"
+9 [p'w+iVr^+
dy dx
„ , , y,(dii" dw"\)
.(272),
and two corresponding expressions for the other components.
By equation (265) dxu" fi^, &c, &c. as well as all inequalities of relative
motion are initially zero; so that, initially, both members are zero. Then
performing the operation djdit on both members and observing that by
equation (265) this operation has no effect on the mean inequalities,
Si „ 9i
d,tP dj
/ / r\l>
P crAuu) =
3 9^
,,ct- ,
P 2+P
du" dv" dw''
dx dy dz
+
or
u u - 2" ) f P™ "
7 "
du
lx~
3i
+ ri[p'VM'+J)at]'
+
dw"\
•(273),
„ / du"
djt Lr ' /'Z!CJ V ds ' d* /
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
dl /3 .A t „ „ . 0-64 „.,\ {fdu'y- /dv"\" . (div"
3^1.2
dw"\2
+ W) +
p
1 [fdu" dv"
2 |\ dy dx
+
dx J
dv"
v dz
dz
dw" du"
dx dz
.(274).
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
=f{+Jp'* + ^P')(t-y) (275),
P
which expresses two reciprocal inequalities of mean motion proceeding in
opposite directions uniformly at velocities
, / " „ _ 0-64 „2
±V p a+y^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
reclistribidion 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
11—2
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 XL, 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 necessaiy, 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
(I860), 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 a/X is small ; or, of far greater interest, local disarrangement
of the grains when so close together that diffusion is impossible, except at
inclosed spaces or surfaces of disarrangement, depending, as already ex-
plained, on the value of G being greater than 6/V2 . ir. Under which con-
dition it is possible that, about the local centres, there may be singular
surfaces of freedom, which admit of their motion in any direction through
the medium by propagation, combined with convection, together with strains
throughout the medium which result from the local disarrangement, without
any change in the mean arrangement of the grains about the local centres ;
the grains moving so as to preserve the mean arrangement.
191. Steady continuous uniform strains or undulations extending through-
out the medium otherwise in normal condition.
We have :
(1) Equations for the angular inequalities maintained in the vis viva of
relative motion.
166 ON THE SUB-MECHANICS OF THE UNIVERSE. [192
(2) Equations for the angular inequalities maintained in the rates of
conduction.
(3) Equations for the component linear inequalities maintained in the
'mean vis viva.
(4) Equations for the linear inequalities maintained in the rates of
conduction.
(5) Equations for the rates of increase of mean vis viva — a2/2 — resulting
from angular dispersion by convection.
(6) Equations for the rates of increase of mean vis viva resulting from
angular dispersion by conduction.
(7) Equations for the rates of increase of mean vis viva by linear dis-
placement resulting from inequalities in the mean vis viva.
(8) Equations for the rates of increase of mean vis viva by linear dis-
placements resulting from inequalities in the mean pressures.
192. Theorem. To a first approximation the first four of these eight
equations all have the same general form as long as the space and time
variations of the mean motion are constant, simple harmonic, or logarithmic
functions of time and space, in which case ilie constants of frequency and the
hyperbolic variations are such as may be neglected as compared with cr/X
and 1/X. And the same for the last four equations.
It is to be noticed that the condition in the theorem as to smallness
of the constants is necessary when treating the variations of the mean
motion as arbitrary, since the condition is, as shown in Section V., a necessity
for the maintenance of the mean and relative systems.
To prove the first part of the theorem :
The equations for any one of the six partial angular inequalities in vis viva
of relative motion.
Putting
V
/ for the inequality in vis viva of relative motion.
I" „ „ „ „ „ in mean motion.
A? for the coefficient by which /" is multiplied to represent the rate of
institution.
A22 for the coefficient by which / must be multiplied to express re-
distribution.
192] THE EXCHANGES BETWEEN THE MEAN- AND RELATIVE- SYSTEMS. 1G7
- , j , - , to represent distances in directions x, y, z, which are the
parameters of the component harmonic inequalities in the mean motion ; the
equation for the maintenance of I becomes:
% + A2*i=2Ax*r
V
In this case where /" and / are component inequalities in the mean-
motion, and in the vis viva of relative-motion, the coefficients A-?, A.?,
are respectively, as in equation (203) Section XIII. and (195) Section
XL:
^-vf. ^~§£« (2"»-
V
Then if /" is as before, and I is taken for the inequality in conduction
corresponding to the inequality in convection in the same direction, the
equation will become the equation for the inequality in conduction. If
Bx2, B.,- are put for the coefficients of conduction corresponding to A-?
and A22,
Q-32j>"3 R2_3Vtt l-f(G)P
B^' = — ' B* 4 ~\ a 1 + r~*-fim (278)'
p2
as in equation (205) Section XI.
Also, if /" is taken to express the linear inequality in mean-motion in
any direction, say that of x, in the rate of volumetric strain in the rnean-
V ... . .
motion, and I is taken to express the linear inequality in the mean vis viva
V V V V
of relative-motion, since d"I / 'doc- , &c. take the forms — a2Ixx> — b'2Iyu> -c2Izz,
where 1/a, 1/6, 1/c are components of some constant parameter, the equation
will become the equation for the linear inequality maintained in direction x
in the mean vis viva when X/a is large.
Putting CV and a2C22 to correspond to Ats and A22 in (277),
Gl - 3 P 2 ' ° ' " Vtt '
-CWI=-C?^{I) (279).
And /" being the linear inequality in the same direction in the rate of
volumetric strain of mean-motion; if / is taken to express the linear
inequality in the rate of mean-conductivity (p"), equal m all directions,
168 ON THE SUB-MECHANICS OF THE UNIVERSE. [193
the equation becomes the equation for the inequality in the mean-conduction
if Di2, a2D22 correspond to A? and A.r in equation (277),
Dt-\%, *iV-«^££! (280),
P2
V f£2 V
since, as in equation (270), a"I = ^„ (I).
Thus as long as the inequalities in the mean-motion can be expressed
as simple finite harmonic or logarithmic functions of time and displacement,
the equations for the dispersive inequalities have the common form as in
equation (276).
The second part of the theorem follows as a consequence of the first
for, since the equations for the dispersive inequalities have the same form,
the general solution of this form of equation will apply to all the in-
equalities.
Then if such solution can be found for the dispersive inequalities,
since the rate of increase of the mean vis viva at a point, at any instant,
is the result of the action of the inequality on the space rate of variation
of the mean strain which institutes the inequality, the rates of increase
V
of the mean vis viva (a2/2) are the products of the inequalities (I) by the
corresponding inequalities (/") in the mean-motion. And these are ex-
pressed in a general form.
193. The approximate solution of the general differential equation J or
the inequalities in mean vis viva, of relative-motion and rate of conduction
resulting from steady or periodic inequalities in the mean-motion.
In all probability the equation (276) does admit of complete solution.
But the analysis is greatly simplified by recognising that any secondary
effects, resulting from the existence of inequalities, to vary the mean vis
viva of relative-motion (oc2/2) by transformation from mean-motion, and thus
to vary the coefficients A-? and A22, are proportional to d2I". And con-
sequently, since by definition a2 is finite, by taking /" sufficiently small the
secondary effects of I" and a" may be rendered as small as we please, and
the integral effects indefinitely small as compared with the finite value of a2.
In this way the coefficients A? and A% may be taken as constant, and
there is no loss of generality in the solution ; while the expression for the
rate of increase of a2, as determined by the approximate solution of the equa-
tion of transformation, may be subsequently introduced as a small quantity.
Solution to a first approximation, I" small.
Since according to the theorem the space and time variations of /" are
constant or periodic, we may transform the equation (276) by putting
193] THE EXCHANGES BETWEEN THE MEAN- AND RELATIVE-SYSTEMS. 169
qxx, &c. for the maximum values of I"xx, &c, which are constant. And
I"xxla is the maximum value of u". Hence
I"xx = qzx sin{mt-ax),
where qxx is constant in time and space.
We then have for the angular inequalities and linear inequalities re-
spectively :
d v v \
=- (/) + A22I = Afqzx sin {mt — ax), &c. |
a « v . \ wn
^- (I) + d2C.?I = G^qxx sin {mt — ax), &c.
The introduction of the two forms is only a matter of convenience in
keeping the partial constants distinct.
V . .....
Then if we put / = CeM and eliminate by differentiation with respect to
time, A{2, A.,2 being constant, it can be shown that for steady or periodic motion
1
or
V
V
1=
{A2f + rri
Al_
CA* + m2
.A;2
A.rl'
»<'">
**- Wt (I»)
(282),
and that this is the only solution if Ai2, A22, &c. are constant. The
analysis is somewhat long. But if we recognise that all the terms in the
equation (281) must have the same frequency m, the same result is obtained
by differentiating both members of (281) and substituting the result from
A.?I - ^~ (/) = A ;2qxx [A22 sin {mt - ax) - m cos {mt - ax)) . . .(283),
V V
whence, since d'2I/dt2 = — m2t is of the same form as equation (282),
v ^
/ = A?qxx {A./ sin {mt — ax) — m cos {mt — ax)} . . .(284),
A.., "T" 111
winch will be the general form on substituting B^2, 2?22 for Cj2, a2G.?, and
A2, «2A2 for A,2, Ai. Q. E. D.
The equation for the rate of increase of the mean vis viva {a2/ 2).
Multiplying the expression for /, equation (284), by the corresponding
expression for /", it at once appears that / consists of two parts, the one
being continuously positive and the other periodic.
Thus
//'
m2 + A2*
1
wzHTZ74
A-fq A2° sin {mt — ax)
Afqm cos {mt — ax) (285),
170 ON THE SUB-MECHANICS OF THE UNIVERSE. [194
from which it appears that the dispersive inequality in equation (284) is
expressed by
— ■ — A^qAf sin (mt — ax) :
m? + AS 1 - v '
the remaining part of /,
1
m2 + AS
Afqm cos(m£ - 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
II" = , 1 A 4 AfqA.? sin (mt - ax) (286),
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 < . ,. * ,«w„(
6 * (conduction, „ „ (278),
e ,. • ,.,. • (convection, ;, „ (279),
lor linear inequalities in - _ . ,n '
(conduction, „ „ (280).
They are, for angular inequalities in convection :
I Xx = L / N , Q it — a sm (m^ — ax) ~ m cos (m* — ax)\ • ■ -(287) ;
for angular inequalities in conduction :
J- = — IFvV l-f(G)P Y qU X * 1 + r^iSm(^"M)
iri- -I- <-. -— a
[4 k "l + e— fl-/<G)ri
— m cos (mt — ax)
...(288);
195] THE EXCHANGES BETWEEN THE MEAN- AND RELATIVE-SYSTEMS. 171
for linear inequalities in a2/2 in convection :
oJ
v 3P2 fa23Aa . ± . . . J /aom
/ya; = , 7 { — j — sin (/«£ — a#) — m cos (/mc — ax)\ . . .(z»y);
m + ( v^")
for linear inequalities in or/2 in conduction :
4"* = / 4 ** g V fy VTVtTa 4 tt" sin (w* " ^ ~ m cos (m* " clx)\
(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 \/<r is less than the limit at
which diffusion ceases, or is greater than that at which diffusion is general.
[See Art. 145 and Art. 155, Section XL]
In considering periodic inequalities in a medium of unlimited extent,
which is, except for the inequalities, uniform and isotropic, it will simplify
the analysis to recognise, that such inequalities as can be propagated through
the medium, must have directions of propagation which are normal to con-
tinuous surfaces which are either spherical closed surfaces, or of such extent
that their boundaries are at distances large compared with the periodic
parameters.
This in the first instance confines our attention to directions of propaga-
tion everywhere normal to an infinite plane. We notice that the classes of
inequalities in the mean motion are reduced to two: those in which the
mean motion is in the direction of propagation, and those in which the mean
motion is normal to this direction.
We also notice that these two resultant inequalities are to a first
approximation independent, although they may have the same direction
of propagation, and therefore may be dealt with separately.
195. Exp7%essions for the resultant institutions of inequalities of mean
motion luhen the motion is in the direction of propagation.
Putting xx and n" as the direction of propagation and motion for institu-
tion of angular inequalities we have, since
/du,
\da
duT dv" dw^y
x dy dz /
172
ON THE SUB-MECHANICS OF THE UNIVERSE.
[195
is an invariant, for the Inequalities of moan motion for the inequalities
(u'u, v'r'. w'w*), &c.
3 V dxj ' dyr dzj
1 d>^" 1 du,"
+ - — — + —
3 di/l 3 d::
Then, taking ./•, . y,, &, as principal axes, /,, ///,, », as the direction cosin< -
ylt z, referred to any rectangular system ./. y, e, the components are,
since
1 (/»," da," efo," dir '
! = _ - -I - -f — ,
3 (irj d:i\ dzr
s rfo," 1 </«," dw
1 ( 3 V (t'./j di/i d:
&
&c.
2 10 a).
Fur the linear inequality of moan motion, taking the principal ;ixr> the
same as lor the angular inequality, we have
dui" dw"y
where
/diii di\ du\ \
\ d.*\ rfy, (/.-, / '
dy, dzl '
And transforming to the axes x, y, z, we have for the components in directi
x, y, *,
(du" dv" dw"
- U + </;/ + 7-- •*" 'v
Exp ess - ':-/- fc&e resultant institutions of ineq ' mmti motion
when the direction of propagation - pendicular to the direction of motion.
If ./-, . >/ . are measured in the directions of propagation and moan motion
bively, the resultant rate of shear strain is expressed by
dvf
dxt,
Then taku _ s for the principal axes, I:. w,, it, for the direction-
ines of the principal axes referred to ; . we have, resolving for the
principal strai -
diii'
«i' , d»„ rfi-r , rfi-, dw, n
rf./j
(£% t&c,
196] THE EXCHANGES BETWEEN THE MEAN- AND RELATIVE- SYSTEMS. 173
And since
TO, = v., = Mj, = 0, If - Lr = )il{2 = in,r = \l^mx = — Liu., — ± £,
dnl/dx1 = — dvljdx1 = ± ^dv0/dxn,
and referring to any rectangular axes x, y, z, the partial inequalities are
duT _ fdu" dv" dw"\) 1 , dv" du"\
' { dx " \ doc dy dz )) 2\da> dy )
I 1/-
) 2 \ dx +
- !(£■•■£)•*-*? <290B>-
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 Af, Bf, &c, for the respective constant-.
and introducing the actual expressions for the general expressions /", or the
harmonic expressions p(u'u'), &c, for the inequalities, we have, for angular
and linear inequalities in vis viva,
P(n") = -«S+Ar
6?
[ dl^duT 1 fdu" dv" dw"\
[_ *~dt\ [dx" S\dx + dy + dz )
m? + (aC2)4
A?
a?Cf-
dt
(In" dv" dv/'}
dx dy dz )
p (v ii ) = —
H v ' m2 + A24
A?
*-s]iffi+?P'*-fc (292>'
p(w'u')=-
m' + AJ
A* dt
±i^+df\,&c.,&c (293).
2 | dx dz )
174 ON THE SUB-MECHANICS OF THE UNIVERSE. [196
And for angular and linear inequalities in conductions
dz
P . - ^ jf mP dt = -^Bs\B»-ft\ I'd, ~ sUt + ^ +
A2 ( ari, 3 ] {da" dv" dw") 0 0
VA2 -^ , + -r- + j— h , &c, &c.
ra2 + («D2)4 J 9iJ [ d!# fZy cfc
(294),
and two corresponding equations in directions ?/ and £ for convections and
conductions.
N.B. The linear inequalities which form the second member of equations
(291) and (294), and the corresponding terms of the equations for directions
y and z, do not include such linear inequalities in the vis viva and con-
ductions as are instituted by dispersion of angular inequalities, since these,
being secondary effects of the mean inequalities which are themselves small,
are altogether negligible. And thus equations (291 ) to (296) are the
equations for the inequalities in vis viva of relative motion to a first
approximation. Q. E. F.
As to these inequalities it may be well at this stage to point out :
(1) That if m2 and a2, b2, c2, which express the frequencies in time and
space are zero, the angular inequalities in the mean motion are severally
constant, while the linear inequalities are zero.
(2) If the direction of propagation is in the direction of motion, or is
normal to a shearing motion, all the inequalities in mean motion are zero
except that one, whether it be
du du du
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?Qf/m2 is small. But treating
197 j 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 adC.2/m2 as
a first approximation, and putting (a2 + b2 + c2)2GV/4m2, the mean value of
a4C22, in the second approximation, the terms expressing component linear
inequalities take the form
V (Mt - l\ I" (l - ^ + ^ +.^'-CA , &c, &c....(297),
m
dt
4>m2
and these obviously satisfy the conditions of resolution for inequalities in
both vis viva and conduction :
Cr / ,„ , 3 \ (du" dv" dw"\ f (a2 + ¥ + c2)2 C24)
Av^-|
TO'
dw" cfo" rfw"
cfoe cfa/ dz
;i-(^±^^j.&c.,&o....(299),
which satisfy the conditions of resolution, and the second approximation may
be neglected.
(6) The proof that these — a2C22/m2 — are small, is not possible as long as
m2 and a' are considered as arbitrary, and subject only to the conditions of
being small as compared with a/X 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"__\ A2
9 dt " \m2 + A24
A
_
+ &c. &c.
+ \ G'
W + iaCo)
4
dt
+
B2
to2 + B,*
B2-
dt
(wo+scS
dvT dw_
dx dy dz
a2C2 -
dt
+
A2
a2D2-
dt
aW dv" duT
dx dy dz
m:-+(aI)o)4
(300),
with two similar partial equations for the rates of increase of dv" /dt and
diu'jdt, and the conditions
dw dv . .
j + -j- + &c. = 0.
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 A^, A22, &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 :
\P
dt
2 / m? + (aA2f
a2Ai-~)
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
2Pdt
A?
m- + Ao*
A.?-
i a
2 dt
+
A2
m? + B,4
B2
II"
2dt
du 1 fd\
■u dv dw
dx 3 \dx dy dz
-if.
1 fdv_ duV 1
+ 4,[dx + dy) +4
G-
+
m2 + (aC2y
A2^
m2 + (ai),)4
-w-ii;
"^-Ui_
du
dx
dv" dw'
+ -T- +
dy dz
dw du\'2
dx dz
(301),
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),
1 3
2pdt
3a2
2
A,
13
2 dt
m2 + B,*
^ 2dt
du"
dx
fdv"\2
dw"s
1 (duT_ djT du/'y
3 \ dx dy dz
+
du"
dy
dv"V (dv'_ dvf'y
dx J \dz dy J
dw" du"
dx dz
.\9i
m2
G22(a2 + b2+(f)-^~
D.? (a2 + b2 + c2) -
3 9/
2dt
) (du" dv" d
+
w
\ \ dx dy dz
.(302).
And putting for the right-hand member its equivalent
1 1 [p" (u'2 + v'2 + w'2)] - \ I [p" {u* + v'2 + w'%
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
lUh')+Hi^+v'"+w"i)=0-
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
(1*7-0, W = 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, A-?, B{2, C\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 or ; 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 x for
that of propagation and the axis of y for that of shear, so that the
inequalities (/") in mean motion are expressed by
die" , dv"
-j— and - , - .
(jjQu \ajOG
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 ;
I1 the coefficient by which mean energy of mean motion must be
multiplied to express the rate of dissipation.
Then, the bar indicating the mean,
ai — 111 ■ — Hi , 1'0\ i -lit i — //o i 7TT.//-'\ \
u - + v - + w 2\ ™^ (u 2 + V 2 + w 2\ \
pKTt\ 2 )=T*
The logarithmic rate of decrement is
-T
\/u"* + v"2 + w"2 = e %R
.(303).
The values of T are all to be obtained from equation (302) omitting
the 3/dt.
The values of R are a little more complex. But as in the first
* No connection with r (tau) — the rate of propagation of light.
12—2
180 ON THE SUB-MECHANICS OF THE UNIVERSE. [201
approximation the motions are a simple harmonic function of t and as 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 cr/X is small in this case it enables us to compare this approxi-
mation, and, as will appear, to show that the results are identical. In
this case total mean energy is the same as the energy of mean motion.
The only values of R which are not included in the list above are the
values of R for transverse waves for the region between the state of no
diffusion and that at which diffusion becomes easy, and in this case the
value of R varies, very rapidly at first, but at a diminishing rate, from
2 to 1.
201. The rates of decrement in a normal wave.
Taking x for the direction of propagation and motion, the motion
harmonic and v^"2 for the maximum value of u"2 ; the mean value is u"2\%
and the mean energy u"2/4>.
The two rates of irreversible dissipation of energy by angular inequalities
and linear inequalities are obtained by 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
3 fu"'2\ ,„ * t, fdi
4(¥)=-«^>(^)s <w
And we have for the mean square of the inequality, mean energy of
motion, and total energy,
q2/2, q2/2a2, and q2\a? respectively.
Thus R = 2 and g = =| (*A + L) a2,
r = -(lA+*;)a2 (305).
R V3 2.
And the equation for the normal wave is
u" = ^e-QA+^a°~f sin (mt-ax) (306)
a
201] THE EXCHANGES BETWEEN THE MEAN- AND RELATIVE-SYSTEMS. 181
In a similar manner for the transverse wave
.(307).
The mean values of - (dv"/dxf, (v")2, and total energy are, when cr/X is
large, and since there is no linear inequality,
2V/2, q%x/2a\ and tf\a\
T=-Aa\ E = 2,
and the equation for the transverse wave becomes
a -J±t
v»=ivxe 2Tam(mt-ax) (308).
If a/X is small, R is 1 and
v" = qyx e-AaH sin (mt - ax) (309).
When a/X is large the equation for undulations in the direction of the
propagation is
_ji ,15 ^.4of_o_G «n
u" = {ile l» '+8 3 p2| 3WH m*J CQS (mf _ a^ (310);
and the equation for transverse undulations
v" = £^e-(^«2)* cos(mt-ax) (311).
In the same way if a/X is small the equation for the normal un-
dulations is
n 1/4 p\ 5 A.a%2 \ ,.
u" = li e~ pM> vVa+ipv^W cos (m<- ax) (312),
and for transverse undulations
v" = qyx e~(t^ai)f Cos (mt- ax) (313).
From equation (310) the coefficients A. B, L, are
. 4 Xa D
and for — small
X
and for - large
X
L =
1 5Xa3a2,
m2 2 V^ '
j. 5 £>2 4 tr2a G .,
3^2 ~ 3 \7w- 4 ""
^2
.(314).
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, A,, <x which define the condition
of the medium.
These equations are thus available for obtaining the rates of propagation
and the rates of decrement for normal as well as transverse undulations for
any specified values of a, X, a.
Also if the rates of propagation together with the rates of decrement for
both the normal and transverse waves are known, the values of a, \, <t may
be found from the equations.
At this stage of the analysis, however, we have not before us all the data
necessary to make a complete determination of the values of a, X, a, so that
the equations would be the equations of light, as this would require a know-
ledge of actual rates of decrement as to which we have no certain knowledge,
and further, these equations have been obtained by neglecting all secondary
actions (see note, Art. 196). And thus these equations afford no evidence as
to the limits of the possible magnitudes of the undulations.
The conditions which limit the possible magnitudes of the undulatory
strains have been generally discussed in Art. 91, Section VII. From which
discussion it appears that, when the medium in normal piling has relative
motion, however small \/a may be, the medium yields in proportion to the
stress when subject to indefinitely small variations of 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
as small compared with the first term ; and in this case the medium has the
properties of an elastic solid within the limits of such strain. It has no
finite stability and only such dilatation as would correspond to the elastic
solid as long as the terms involving the square and higher powers of the
strain are small.
On account of both these the further consideration of the undulations is
continued in the section next but one to this — after the consideration of
the possible strains, other than the undulatory strains, which afford further
evidence.
SECTION XIV.
THE CONSERVATION OF MEAN INEQUALITIES, AND THEIR
MOTIONS ABOUT LOCAL CENTRES, IN THE MEAN MASS.
202. In the last section we obtained the equations for continuous steady
undulations, including the rates of decrement, for normal and transverse
waves in terms of a", X." and a, the only quantity undetermined being the
superior limit to the amplitude ; while from the same section it is evident
that undulatory strains have characteristics which differentiate them from
strains other than undulatory, and that they are essentially elastic strains
maintained only by the inequalities of the mean motion, and independent of
motion by propagation. It remains to effect such analysis of the strains
other than undulatory, the possibility of which has been pointed out in
Art. 190, Section XIII. These are:
(i) Some local disarrangement of the medium together with some dis-
placement of portions of the medium from their previous neighbourhood,
such as vortex rings, which may have a temporary existence if X" fcr is large.
(ii) Local abnormal arrangements of the grains when so close that
diffusion is impossible except in spaces or at closed surfaces of disarrange-
ment, depending, as already explained, on the value of G being greater than
6/*J2tt, under which conditions it is possible that, about the local centres,
there may be singular surfaces of freedom, which admit of their motion in
any direction through the medium by propagation, combined with strains
throughout the medium, which strains result from the local disarrange-
ment 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.
203. The character of these two general classes of strain must depend
primarily on the state of the medium, where uniform, as indicated by the
value of cr/X".
When crjX" is small there is no dilatation, and there is diffusion, hence
there are no singular surfaces except such temporary surfaces as result from
vortex motion. Therefore this class of strain may be considered as belonging
to the undulatory class which does not concern us in this section.
184 ON THE SUB-MECHANICS OF THE UNIVEKSE. [203
The second of these classes of local disturbance, in which <r/\ is large, so
that there is no diffusion except about centres of disturbance, includes all
local disarrangement of the normal piling that can under any circumstances
be permanent.
(i) 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 disturbance may be
called centres of negative disturbance, or centres of negative inequalities in
the mean density.
(ii) We can also conceive disarrangement resulting from excess of grains
in the otherwise uniform medium — a definite number of grains over and
above the number which constitute the uniform piling, and such, whether or
not capable of independent existence, will be called a positive disturbance.
These positive and negative centres are the principal centres of distur-
bance, as well as the simple centres of disturbance.
There are other classes of disturbance which, although more or less com-
plex, are to some extent permanent.
(iii) If by any action on the medium in normal piling a number (n)
grains were displaced from their previous neighbourhood when in normal
piling, to some other neighbourhood previously in normal piling, the distur-
bance would be reciprocal, and, if there were no further displacement, would
be permanent if there were no further action.
It should be noticed that such displacement might correspond exactly
with that of a negative disturbance resulting from the absence of (n) grains,
and a positive disturbance from introduction of (n) grains in positions corre-
sponding to those from and to which the (n) grains were displaced.
It should be noticed however that, assuming the possibility of the
displacement and that of the simultaneous existence of equal negative
disturbances, this in no way proves the possibility of the existence of a
solitary positive disturbance.
(iv) Another class of possible local disarrangement of the normal piling
in an otherwise uniform medium is that class which does not depend on the
absence, presence, or linear displacement of grains, but does depend on the
rotational displacement of the grains about some axis.
If we conceive a finite spherical surface in the medium, and further
conceive that for .30° on either side of a diametral plane the medium im-
mediately external to this surface is, owing to rotational disarrangement,
resisting positive rotation of the surface, while the medium immediately
internal to the surface, that which extends from each of the poles to within
204] CONSERVATION OF MEAN INEQUALITIES AND THEIR MOTIONS. 185
30° of the diametral plane, is resisting negative rotation, then it will appear,
since owing to the relative motion the medium is to some degree elastic,
there will be positive rotational strains extending outwards in the external
medium within 30° of the equator, and negative rotational strains extending
outwards over both the surfaces from the poles to within 30° of the diametral
plane.
These represent a state of polarisation in the strains of the medium,
inside and outside, and if we had two such polarising surfaces with similar
poles in contact the strains would superimpose, while if the opposite poles
were in contact the strains would cancel.
204. With regard to the conservation of similarity in the arrangement
of the grains within and without singular surfaces, we may prove the follow-
ing theorem.
Theorem 1. When the condition of the medium is such that there is no
diffusion except at a singular surface, where G is greater than 6/\/2 -w as
a result of the absence of n grains, the replacement of which would restore the
uniformity of the medium to that of unstrained normal piling, there will result
inward strains extending from an infinite distance to some spherical surface
within the singidar surface; then ivhatsoever may be the inward strains in
the normal piling and the disarrangement of the grains, with the surface at
which the strained normal piling ceased and abnormal piling commenced, the
number of grains absent would be the same (n) and the strains in normal
piling tvould be the same.
To prove this we have only to consider that, owing to the pressure from
the outside and the mobility of the grains due to the relative motion, a",
however small, would secure that in the first instance the arrangement
of the grains was such as to cause the minimum dilatation, and hence
would secure the maximum normal inward strain and then would be in
equilibrium. Then since there would be no outside disturbance, if there are
to be any exchanges of neighbourhood owing to relative motion, these ex-
changes must be such as do not entail any increase in the mean dilatation.
Whence it follows either that all the grains within the singular surface must
maintain their neighbourhood, in which case the centre of disturbance
would remain unchanged, following whatever uniform motion the medium
might have, or the arrangement of the grains immediately inside and
outside the singular surface must be such that the dilatation caused by any
influx of grains into the singular surface from one side would be simul-
taneously compensated by the contraction caused by the efflux of the same
number of grains from the opposite side, in which case the centre of dis-
turbance, together with its attendant strains extending from infinity to the
abnormal piling, would be free to move in any direction and maintain the
same minimum dilatation, q. E. l>.
186 ON THE SUB-MECHANICS OF THE UNIVERSE. [205
It is to be noticed that the second alternative requires conditions as to
the possibility of which nothing has been affirmed in the proof of the theorem,
while the first is general.
Then again we have as a corollary to the last theorem : If two negative
centres of disturbance exist within any finite distance of each other, the
numbers of the grains absent in each of the centres would remain the same.
But it does not follow, as a necessity, that the strains in the normal piling in
the respective centres should be the same as if the other centre of disturbance
was absent.
Then again we have a theorem with respect to a more complex dis-
turbance :
Theorem 2. When the disturbance is such as would result from the
removal of n grains from one place in a uniform medium and their introduc-
tion to another place at any finite distance, which is the same thing as two
equal centres of disturbance at a finite distance, one negative as the result of
n grains being absent, and one positive as the result of n grains in excess.
Then whatever may be the resulting strain or motion in and about the
two centres, the number of grains absent in the negative disturbance must
always be the same as the number of grains in excess in the positive dis-
turbance however this number may be changed by exchanges between the
centres.
This theorem being self-evident needs no demonstration.
205. The dilatations which result from strains in the normal piling in
the otherwise uniform continuous granular medium have been subjected to
somewhat full discussion in Arts. 86 to 92, Section VII. This discussion
includes the ideal case (a" = 0), in which there is no relative-motion, as well
as that (a" finite) in which there is relative relative-motion.
It is with the second of these cases that we are directly concerned, but
it appears that the only process of effecting the analysis necessary for
determining the coefficients for the dilatations in the medium with relative
motion is, in the first instance, to determine the coefficients of dilatation,
when a" = 0, for small strains in the directions of the axes of distortion.
Then by examining the effects of relative motion on these to arrive at the
general coefficients of dilatation for small strains in all directions in the
medium with relative motion.
206. In Art. 90, Section VII. it appears that in the uniform kinematical
medium (A. = 0) there are six axes symmetrically placed, which are axes of
no contraction, and bisect the middle points of the edges of the cube of
reference, and all pass through the centre. Between these axes and at angles
207] CONSERVATION OF MEAN INEQUALITIES AND THEIR MOTIONS. 187
of 45° to them, that is in directions parallel to the axes of reference, or the
edges of the cube, there are three axes of possible symmetrical distortion ;
hence this medium under any mean stress p", equal in all directions, has
stability and crystalline properties. If however the stability resulting from
uniform stress is overcome, say by uniform superimposed stress in the
direction of one of the axes of reference, the dilatation resulting from the
initial small strain is positive, and can be shown to be equal to the normal
contraction, i.e. the result of the normal contraction and lateral extensions
is to increase the volume by a quantity equal to the small normal strain
multiplied by the initial volume. Hence the coefficient is unity.
As the strain increases the coefficient diminishes according to a definite
law (which will be expressed) slowly at first, then more rapidly until maxi-
mum dilatation is reached, when the coefficient is zero, and G = Qjir. The
medium is then unstable, and under the mean pressure equal in all directions
would revert to some second state of normal piling.
207. To prove the statements in the a
last article as to the coefficients of the
dilatations resulting from small strain in
the direction of one of the axes of dila-
tation in a kinematical medium : /-""I
Let OA, OB, 00=0,!, bly c1} respectively
be the principal axes of strain. b^- -:fQ—
Let AB, AG, &c. the generating lines of
the conical surface be the lines of. no con-
traction.
Put
6 = OAB, <£ = OA G, LB = AB, Lc = AG.
Then «* 2-
a = Lj; cos 6 = Lc cos <£ )
b=L]ism6=G = Lvmncj>\ (315)'
da t ■ n da r
_ = -LBsm6, ^7 = -Z,,sm</> (316),
V=-.a.b.c = -.a. LltL,< sin 6 sin <£ (317),
a dV
— r- . -y=-l + cot- 6 + cot2 $ (318).
Then, since dV/V is the dilatation and - da/a the strain, the coefficient
of dilatation is by equation (318)
-^- /y =-l +cot20-f cut-'c/> (319).
188
ON THE SUB-MECHANICS OF THE UNIVERSE.
[207
Whence it appears, since 6 = cj> and cot 0 diminishes as 0 increases, we have
for the maximum coefficient
cot2 0 + cot2 0-1 = 1,
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°, cot2 6 diminishes until
dilatation is zero, when the condition of the medium is unstable.
This may be demonstrated graphically. In Figs. 3 and 4ii, BB and GO
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 4 A A' in plan.
208] CONSERVATION OF MEAN INEQUALITIES AND THEIR MOTIONS. 189
These losses and gains are taken on the three axes at right angles of
which the dimensions are A A, BB, CC.
The normal strain is 2AA'/AA.
The volume is AA . BB . CC or (AAf.
The increase of volume (AA)2 . 4>AA' - (AA)"- . 2AA' = (A A)- . 2AA'.
Whence we have the dilatation
dV_(AAf.2AA'
V ~ ~ (AA)3
And dividing by the strain — 2AA'/AA and changing the sign, we have for
the coefficient of dilatation
AA (AAf . 2AA' _
2AA' ' " (AAf 7
207 A. Then as regards the inequalities of pressure pr = 2pt = %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, pr for the normal in the direction of the strain,
and pt for either one of the tangential since these are principal stresses
p,.+ 2pt = 3p" (320),
and since there is no work done on the grains,
Pr = 2pt (321),
whence by (320)
JPr = fp", Pt = %p" (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 unsymmetrical 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 ivhatever 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
othemvise 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, pr 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 pr + 2pt = 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 pT + 2pt = dp".
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. Tli ere fore
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 tJian 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 a is indefinitely
small as compared with the reciprocal of the curvature.
192 ON THE SUB-MECHANICS OF THE UNIVERSE. [210
For since <r is indefinitely small when the scale of disturbance is finite,
if we conceive all dimensions including a to be exaggerated so that <r
becomes finite, and the distances between the grains exaggerated on the
same scale, then, since the mean strains before exaggeration vary continuously
without crossing, so that in the strains the finite paths of two grains which
were neighbours before the strain would still be neighbours after the finite
strain although separated by any distance which is less than the finite
distance a, their two paths would still be parallel lines of infinite length
and at any finite distance apart.
It is thus shown that if the grains are indefinitely small as compared
with the dimensions of the disturbance, the only dilatations would be those
resulting from uniform parallel distortional strains. Q. E. D.
Again in the case of the medium in which the grains are finite it has
been shown, Art. 207, that when the grains are finite, however small as
compared with the dimensions of the finite volume from which grains are
absent, that the effects must differ from those resulting from uniform parallel
distortion.
And by The last theorem, putting 4nrr03/3 for the volume the absent
grains would occupy in normal piling, it appears, since cr/r0 is indefinitely
small, that the dilatations result solely from uniform parallel distortional
strains. And hence whatever finite curvature may result from finite strains,
this curvature does not, as curvature, produce any effect on the dilatation ; so
that there are no curvature effects.
Then since it is shown that when a is finite, however small compared
with the reciprocal of the curvature in the strained normal piling, the
dilatation resulting from curvature depends solely on the existence of a
finite value of the product of a multiplied by the curvature, the dilatation
will equal a multiplied by the curvature.
Further, it follows that for any given strain, this dilatation resulting
from curvature will be in excess of the dilatations resulting from uniform
parallel strains.
210. The analytical separation of the dilatation resulting from uniform
strain and that resulting from the curvature would be perfectly general if a
might have any value as compared with the curvature. But, in that case,
any analytical separation of the dilatation resulting from distortions from
that resulting from the size of the grains would be different on account of
the reaction of the dilatation resulting from the size of the grains on that
resulting from distortion. But we are only concerned with cases in which
a is such that a multiplied by the curvature is so small that to a first
approximation any reaction from the dilatation resulting from the curvature
may be neglected.
211] CONSERVATION OF MEAN INEQUALITIES AND THEIR MOTIONS. 193
Whence it appears that, to a first approximation the only curvature is
that institnted by a uniform distortional strain — as if a multiplied by the
curvature were indefinitely small — the dilatation resulting from small inward
radial displacements about a centre being of necessity equal to the curvature
at each point. It follows as a necessity that, taking A as the dilatation
resulting from the uniform distortional strains, the dilatation resulting
from curvature owing to the finite size of the grains at the same point is
expressed by A<t\%\, where i\ is the radius of the singular surface, whence
we have for the total dilatation
211. Granular media with relative motion have this fundamental
difference from media without relative motion, that when in normal piling
the medium with relative motion is within certain limits perfectly elastic
without crystalline properties, that without relative motion is perfectly rigid
and crystalline.
When the media are both under strain this difference is not so apparent,
as the medium without relative motion is then also without rigidity. But
the difference is still fundamental, and the fundamentally of the difference
in no way depends upon the degree of relative motion. For in the one the
medium satisfies the condition of virtual velocities, while in the other state,
owing to its elasticity, this condition cannot be absolutely satisfied however
near the approximation may be.
The crucial difference between the two states is virtually reduced to the
existence of a state of absolute rigidity in the one, however limited, when
the piling is normal, and the absence of such rigidity in the other however
small may be the relative motion.
For as has been shown in Art. 207 the medium without relative motion
while satisfying the condition of virtual velocities when strained from the
normal piling, will also satisfy the condition of equilibrium— that the sum
of the normal and tangential pressures equals three times the mean pressure,
or that
pr + 2pt = Sp" (323).
Another medium will also satisfy the conditions that the pressure between
the grains cannot be negative, and that every grain is in contact with at
least four grains, whence it follows (since the last three of the four preceding
conditions are satisfied in the strained medium without relative motion they
are of necessity satisfied by the strained or unstrained medium with relative
motion) that if, as has been shown, the condition of virtual velocities can be
satisfied to any degree of approximation in the medium with relative motion,
such medium has to any degree of approximation all the properties of the
r. 13
194 ON THE SUB-MECHANICS OF THE UNIVERSE. [212
medium without relative motion, except those depending on the limited
stability on which the crystalline properties depend.
It is thus shown that the necessary distinction between the two states is
that of finite rigidity when there is no relative motion.
In regard to this statement it is perhaps necessary to call attention to
the fact already demonstrated, that in the case of a medium with relative
motion, the relative motion as expressed by a in a steady state of strain
must be constant, since any inequalities in a are subject to redistribution,
so that the mean energy of every grain remains constant. Therefore the
energy of the medium after the grain has been removed and the inward
strain established would be constant, and there would be no change in the
mean relative kinetic energy of the grains 0 . — , and it is the state after
the grains have been removed with which we are alone concerned.
This although, for the purpose of analysis, an ideal action — that of
removing grains from a medium in otherwise uniform normal piling — such
action has no existence. This appears from Theorem 1 in this section, from
which it follows that whatever may be the volume occupied by the absent
grains when in normal piling such accident is permanent.
It has thus been shown that the inward strains resulting from the
absence of grains which would occupy the volume 4nrrQ3/S in normal piling
about any centre in the infinite, elastic medium, must cause dilatations
extending from an infinite distance to the singular surface about the centre
of disturbance, which dilatations occupy a volume equal to 47rr0"/3, the
volume from which the grains are absent ; and they are such as satisfy the
conditions of equilibrium under the same mean pressures normal and tan-
gential expressed by
pr + 2pt = Sp" (324),
p" being the mean pressure equal in all directions.
212. It also follows from Art. 210 that these dilatations, notwithstanding
the relative motion of the medium, admit of analytical separation into the
two classes :
(i) Dilatation resulting from uniform distortional strains such as would
result if a were indefinitely small.
(ii) Dilatation which results from the finite value of a and the curvature
induced by the uniform distortional strains.
The relations of these dilatations are those expressed in Art. 210 by
. / a \ (the total dilatation per unit]
V 2rJ ( of volume at the point j ^ h
21-3] CONSERVATION OF MEAN INEQUALITIES AND THEIR MOTIONS. 195
for the only difference resulting from the relative motion is the absence of
any limited stability.
213. From the conclusions arrived at in Art. 211 it follows, if p" is
constant, that the total dilatation resulting from the inward strains does not
depend in any degree upon the coefficients of dilatation, nor upon the relative
motion a, as long as crjX is within the limits of no diffusion, whatever may be
the value of a.
It does not however follow from this that the distribution of the strains
is independent of the variations in the coefficients of dilatation, since it has
been shown (Art. 207) that if there is no relative motion the coefficients of
dilatation must increase with the distance from the centre of disturbance.
But in the absence of any limited stability as in the case of a beiug finite,
since we need consider those cases only in which the coefficients of dilatation
from small strains are unity, the circumstances may be so chosen that the
strains follow some regular law.
However, before discussing these circumstances, we may with advantage
consider what further conclusions as to the relation between the strains and
dilatations, as well as the relation between the normal and tangential
pressures, are afforded by the adoption of unity as the general coefficient
of dilatation in the medium with relative motion.
Since the coefficients are constant and equal to unity, the mean strains
resulting from the absence of a volume of grains expressed both in magnitude
and shape by the sphere 47rr03/3, will be radial and symmetrical. Then by
the theorem of Art. 212, if a is small compared with r0) since the strains
must be everywhere very small, the relations between the inward strain and
the dilatation will be such (if at any point we take a* for the principal
strain in the direction of any radius and /3 and 7 for the principal strains
tangential to the surface of the sphere, since the strains are inwards ft and 7
are negative and equal) as are expressed by
/3 + 7 = -4a, or -^ ' = -1 (o2b).
jd +7
Then adding (ft + 7) the negative or contraction strains to a the positive or
expansion strain, we have the dilatation
-08+7) = !
.(327).
a=-2(/3 + 7)J
Then we have from these equations the general relation that the dilatation
resulting from tangential contraction - (ft + 7) is equal to half, and can only
be half, the normal elongation resulting from the tangential contraction,
together with the dilatation caused by the contraction strain.
* a, /3, 7 are here used to express principal strains.
13—2
196
ON THE SUB-MECHANICS OF THE UNIVERSE.
[213
The dilatation expressed by either member of equation (327) is the total
dilatation resulting from the uniform distortional strains, as well as that
resulting from the curvature on account of the finite size of the grains. And
to complete the analysis of the relations between the dilatations and the
strains it is necessary to effect the analytical separation of these two
dilatations.
The separation of the dilatations follows at once from equation (324).
By equation (327) we have for the total dilatation per unit of volume at
a point
-(/3 + 7)-
And from equation (325) the total dilatation is
a(i +
Therefore
A
A a -
2r,
-(£+ 7)
2^
- 03 + 7)
2r}
.(328).
The first and second of equations (328) are respectively for the dilatations
resulting from uniform strains and from the size of the grains.
These involve the squares of c/2r1; neglecting this term we have as
approximations :
For the dilatations resulting from uniform strains
-0+*>(i-£)-
And for the dilatations resulting from the size of the grains
Adding these two last expressions we have
-(£ + 7) (329>>
which expresses the total dilatation per unit of volume at a point in the
medium.
Then integrating the partial dilatations from go to rx over the medium,
since the total integral dilatation is 4nrrQ3/3 we have for the integral dilatation
resulting from uniform distortion
£)£* (330).
-G8 + 7)[i_^_)
z/-,
3
214] CONSERVATION OF MEAN INEQUALITIES AND THEIR MOTIONS. 197
And for the dilatation resulting from the size of the grains
i^-08 + 7).^^*-(l-7l)«^Xr•, (881)-
The relations between the strains and the resulting dilatations, as expressed
in equations (326 to 331), are the complete relations to a first approximation
as long as there is no other disturbance in the normal piling than the
spherical disturbance which gives rise to the radial inward strains. And they
have been obtained by taking the coefficients of dilatation as unity.
The relations between the principal stresses are such as satisfy the
equation of equilibrium
pr"+2pt"=3p" (332),
and are also such as satisfy the condition of virtual velocities approximately,
which on the assumption that the coefficients of dilatation are unity, since
the contraction strains are tangential, requires that
Pt" = 2pr" (333).
Therefore from (332) and (333) we have
pt" = ip" and p;.' = \p" (334).
Equations (332) and (333) express completely, to a first approximation, the
relations between the constant mean pressure, equal in all directions, and the
constant mean tangential and normal principal stresses resulting from a
negative spherical disturbance about an only centre on the supposition that
the coefficients of dilatation are unity.
214. Having in the last article effected the analysis of the relations
between the dilatations and strains, as well as between the mean tangential
and normal principal stresses and the mean pressures, equal in all directions,
about an only negative centre, on the supposition that the coefficients of
dilatation are unity, it remains to consider that choice pointed out (Art. 213)
of the circumstances under which this condition can be realised.
The definition of a negative local disturbance (Theorem (i), Art. 203) in-
volves the absence of a certain number of grains, which if present in normal
piling would render the piling in the medium normal, reverse the strains,
and so obliterate all trace of disturbance about the centre.
There is nothing in the definition of such local centres that defines the
mean distance from the local centre at which the grains may be absent, nor
is there any obligation that the space from which the grains are absent shall
be continuous, as long as there is some symmetry about the centre.
It is therefore open for us to consider such arrangement of the position
198 ON THE SUB-MECHANICS OF THE UNIVERSE. [214
about the centre from which the grains are absent as will result in the least
analytical complexity.
It would seem at first sight that the greatest simplicity would be secured
by assuming that the grains were removed from a spherical space. But in
that case it at once appears that the inward radial displacement would
extend to the centre of the sphere. And it also appears (Art. 207) that
the contraction strains as the centre was approached would be such that
instability would come in, and the arrangement near the centre would revert
to some more nearly normal piling, forming a nucleus of grains in normal
piling without dilatation. In this case the dilatation would commence in the
grains outside the spherical nucleus, there being a spherical shell of grains in
abnormal piling constituting a broken joint between the nucleus and the
medium outside, which, although strained inwards, would still be such that
the grains had not changed their neighbourhood. Thus it appears that the
abstraction of grains from a spherical space would not entail that this
strained normal piling would reach the centre.
The arrangement instituted as a result of this abstraction from a
spherical space seems most natural and, with a little modification, such
arrangement presents the least analytical difficulty.
If we adopt the nucleus in an exaggerated form and the spherical shell
of grains in abnormal piling, no matter how thin, also take 1\ for the radius of
the singular surface which is somewhere within the spherical shell of grains
in abnormal piling, since the volume of grains absent is 47rr03/3 which volume
as a spherical shell of radius 1\ would have a thickness approximating to
?,03/3r12, we have as an expression for the inward radial displacement of the
grains in strained normal piling which are adjacent to the singular surface
'o _ 'i'o /'3'}t\
3n2 3n3 K >'
Then since this is the greatest possible radial displacement, and being
adjacent to the singular surface is independent of dilatation, the contraction
strain, owing to the displacement, would be the largest contraction strain
possible. Whence, if this is small, all the contraction strains will be very
small, and as the dilatations are equal to the contraction strains, though
of opposite sign, the dilatation would be very small, and by Art. 207 the
coefficients of dilatation would approximate to unity.
In order to show that the contraction strains at the singular surface
resulting from radial displacement
3n2
would be very small ; let the outer circle (Fig. 4 a) represent a section
215] CONSERVATION OF MEAN INEQUALITIES AND THEIR MOTIONS. 199
through the centre of disturbance before the volume 47rr03/3 is removed,
and the inner circle represent the section through the centre after the
Fig. 4 a.
volume is removed. Then if the inner circle is taken to represent the
section of the singular surface through the centre of disturbance, since the
radial displacement [a = — 2 ((3 + 7)] of the grains at that surface has been
shown to be (equation 335) ^/Sr^, the contraction at the singular surface is
rA —r
fir2 ' #1f
Wl (336).
(3>y
Then since ruh\ is small, according to powers of rnh\, we get a rapidly
converging series, the first term of which is
-^=# + 7 <3S7>
Then by equation (327) we have as a first approximation to the dilatation
resulting from the contraction at the singular surface r^jSr^. And as this is,
approximately, the greatest possible dilatation, it follows that under the
conditions as stated above the radial displacement and inward strains are
such that the coefficients of dilatation would to a first approximation be
unity.
It is thus shown that the conditions assiuned in the present article are
not only possible but are also the most probable.
215. In order to complete the analysis for an only negative centre it
remains to obtain the expressions for the contraction strains and dilatations
at any distance from the singular surface corresponding to those found in
the last article for the contraction strains and dilatations at the singular
surface.
Thia problem differs essentially from that of determining the strains at
the singular surface; this difference appears at once when we realise, as
already pointed out, that the radial displacement which the grains at the
singular surface have undergone is definitely expressed by rQ*/Qrf, since
it is subject to no displacement from dilatation, whereas the radial displace-
ment which the grains at an arbitrary distance r from the centre have
undergone depends on the dilatation between r and
r.
200 ON THE SUB-MECHANICS OF THE UNIVERSE. [215
There are however two definite conditions that the radial displacements
must satisfy to a first approximation.
(1) The condition (Art. 207) that whatever the radial displacement may
be it must be such that the integral of the dilatations taken from r, to x
shall be equal to the volume from which the grains are absent.
(2) That the radial displacement must be such that at any distance
greater than i\ the resulting tangential contractions will cause dilatation
which, integrated over the volume of the spherical shell 4>tt (r,3 — r03)/3, will
express when divided by ?y radial displacements corresponding to those
assumed.
If instead of taking — r03/3r2 or —r^/S^r2 we take
for the radial displacement, we have for the contraction strains, since they are
negative and only half the total elongation,
?Vo3
6r3
r\ — r
r 3
'0
From which to a first approximation we have for the contraction strain
1 r^3
Then changing the sign, multiplying by it and integrating from 1\ to r
[r 1 , 4<7rr03 47r?v03
10 ]r 3? rdr==—s 3" r~ ( )"
The result arrived at in equation (338) admits of more general proof,
from which it appears that this result is the only result possible.
Putting X for the radial displacement ; since the dilatation is expressed
by X/r we have to obtain the expression for X satisfying the condition
whence it appears that
^7r\ri-r'dr = ^r03 (339),
r 6
X=~WT (840).
Also dividing the last term in equation (338) by r- we have for the radial
displacement at a distance r
7VV5
3r3 '
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,
<7i 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 au and the singular surfaces both of radii i\,
4<7r
but in one of which the volume of the grains absent is — ra3, and in the
3
other — »y.
o
Then by equation (331) we have for the dilatation at a distance r for
the centre a
3 a r* \ 2rx
and for the centre b
4-7T 9 O-! ( ^ cr1
T
r*3 ? i1 - ft)
202 ON THE SUB-MECHANICS OF THE UNIVERSE. [216
and neglecting <r,/2r1 for the present, as small, multiplying by r-dr and
integrating from ?\ to r = oo we have for the dilatation, taking w to express it,
.(341).
4<7rra3 aA
6 r
4
wh =
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 a into the curvature is directly propor-
tional to the volume occupied in normal piling by the grains absent. And
in the second place from the form of the expressions obtained, that the total
dilatation is inversely as the radius of the singular surface.
It is this fact, that whatever may be the volume occupied by the absent
grains in normal piling, the dilatation will be inversely as the radius of the
singular surface, which proves the effect of dissimilarity between the constant
value of a and the different values of rl3 namely that for any particular
volume of grains absent the dilatation resulting from the small centre will be
greater than that resulting from the large centre in the inverse ratio of the
radii of the centres.
So far we have only considered the effect of dissimilarity in ajr^ on the
supposition that each centre is the only centre within finite distance.
We may now proceed to prove that negative centres at finite distances
attract each other.
Taking &> to express the total dilatation from i\ to r = oo resulting from a
single negative centre, then as has just been shown
°>. = 4f^ (342).
Then the number of such singular surfaces which would occupy an
empty spherical shell of radius rB when arranged in closest order would be
approximately
07 or,
3
N' = "'iiB (343).
And by equation (341) the total dilatation of each of the N' surfaces outside
the surface 47rr02 is
47rrn3 a
••-V* (nH)-
216] CONSERVATION OF MEAN INEQUALITIES AND THEIR MOTIONS. 203
Multiplying co1 and wB by N' we have for the respective total dilatations
and
N'co^N'^- (345)
3 i\
N'coB=N'~p3 - (346).
6 7'B
Subtracting these equations as they stand we have
*'<--« >-*-^£-3 <*«>•
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
„, 47rr03 a
3 n
while if these surfaces are arranged in closest order the dilatation will be
Ar/47rr03 a
6 rB
Whence since rB is greater than i\ it is shown that, no matter how
accomplished, the dilatation resulting from negative centres diminishes in
the ratio
n
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
„ 4>7rr03
^Ta-s)
as the expression for the difference of the energies in the media when the
N' singular surfaces of radius rx are at an infinite distance from each other,
and when the N' singular surfaces of radius i\ are arranged in closest order
within the surface rB.
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 a2 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 rB to r = oo , as in equation
(346), we take the dilatation from rB to r, where r is greater than rB, the
dilatation from the N' singular surfaces in closest order is
3 \rB r
Then if there is another singular surface of radius r3 in which the volume
of grains absent is 4>7rr03/S at the distance r the variations of the strains of
the outside singular surfaces interfere with those from the centre rB\ and
multiplying the dilatation outside rB less the dilatation outside r b}? minus
the volume of the grains absent in the outside centre, we have the expression
and differentiating this expression with respect to r we have
-N
, f4nrr
V 3
J r2'
218] CONSERVATION OF MEAN INEQUALITIES AND THEIR MOTIONS. 205
whence multiplying by p", since aj\ is large so that the density within the
singular surfaces is unity, we have for the acceleration
^•Wife-S-^W? ^
This expresses the space rate of variation in the work, or energy in the
system, with the distance, that is the effort to bring the centres together
whatever may be the passive resistance.
It is thus shown that the law of attraction, that is the effort to bring the
surfaces together, whatever may be the passive resistance, is the product ol
the masses of the grains absent multiplied by a and again by minus the
reciprocal of the square of the distance.
This law of attraction, which satisfies all the conditions of gravitation, is
now shown by definite analysis to result from negative local inequalities in
an otherwise uniform granular medium under a mean pressure equal in all
directions, as a consequence of the property of dilatancy in such media
when the grains are so close that there is no diffusion and infinite relative
motion ; and further it is shown to be the only attraction which satisfies the
conditions of gravitation in a purely mechanical system.
The mechanical actions on which this attraction depends are completely
exposed in the foregoing analysis, and offer a complete explanation of the
cause of gravitation.
In this explanation of the cause of gravitation there are some things
which are at variance with previous conceptions, besides the fundamental
facts, (i) that the attraction of the singular surface which corresponds to
that of gravitation is not the effect of masses present but of masses absent,
which has already been revealed in the previous analysis, and (ii) that the
volume enclosed within the singular surfaces, which is the volume from
which the singular surfaces shut each other out, has no proportional relation
to the number of grains absent, but, as will at a later stage appear, depends
on the possibility of some one definite arrangement of the grains absent, out
of a finite number of possible different arrangements.
218. In the analyses of Newton, Laplace, Poisson, and Green, for defining
the consequence which would result if distant masses attracted each other
according to the product of the masses divided by the squares of the distances,
the attraction is taken as inherent in the masses. This assumption assumed
that there was something that was not force, but which varied with the
distance from a solitary mass, and this something after various names is now
generally called the potential. That any of the philosophers named believed
in force at a distance is more than doubtful, as Hooke and Newton and
Faraday repudiated any such idea. Maxwell went a stage further and
206 ON THE SUB-MECHANICS OF THE UNIVERSE. [218
showed that such attractions might be a result of a certain law of varying
stresses in a medium — as to this he writes, " It must be carefully borne in
mind we have made only one step in the theory of the medium. We have
supposed it to be in a state of stress, but we have not in any way accounted
for the stress or explained how it could be maintained." " I have not been
able to make the next step, namely to account by mechanical considerations
for the stresses in the dielectric*."
Maxwell is here writing of electricity, which is not the same thing as
gravitation, as will presently appear.
This second step, namely that of accounting by mechanical considerations
for the stresses in the medium, has now been overcome; as we have the
mechanical interpretation of the potential as the product of the uniform
pressure p" multiplied by the integral of the dilatation over the medium
rB to rlf or
V = p"N'^f<r{^-^\ (349),
or, omitting the constants,
]\J'4—r 3
V=-p"a 3 ° (350).
This is entirely rational and when multiplied by — 4>7rr03/S and differ-
entiated gives us the attraction hitherto expressed by ttf.
And it thus appears that the thing to which the name potential has been
applied is the product of p" multiplied by the total dilatation between the
surface of radius rB and the surface of radius r (greater than rB).
It is to be noticed that in so far as we are concerned with the effort of
attraction and not with acceleration, it is only the volume of the space from
which the grains are absent, and not the mass within the space, that we
have to take into account.
And it is for this reason that in the foregoing analysis, in this section,
p has not been introduced. But since, in states of the medium under
consideration, in our present notation p is, to a first approximation, equal to
unity, it would have made no difference if we had taken it into account
(when we have to. consider the displacement of mass owing to the effort, the
fact that p" is unity is of primary importance), since whatever the effort to
acceleration, the acceleration is inversely proportional to the density — and
^his will appear at a later stage.
In order to render the expression for attraction intelligible it should here
be noticed that strains, and consequent dilatations in the medium, which have
* Electricity and Magnetism, Vol. i. Arts. 110 and 111.
t This R has no connection with the R used in Arts. 200 and '201.
220] CONSERVATION OF MEAN INEQUALITIES AND THEIR MOTIONS. 207
no dimension, and which are the only actions, are outside the singular
surfaces ; so that we are not dealing with two or more independent masses,
but with the variations in the displacements in the entire medium, all the
mechanism, so to speak, being in elastic connection controlled by the pressures,
as conditioned by the positions of negative inequalities in the mean mass
represented by 4>7rr03/S.
There is no complete freedom of inequalities as long as there are other
inequalities within a finite distance.
Thus it appears that the singular surfaces are virtually the handles of
the mechanical train.
219. Having effected the analysis for the attractions and the potential,
we may now return to the inequalities in mass as mentioned in the schedule,
Art. 203.
The second inequality in the mean mass in that schedule is that which
may be conceived to result from an excess of grains, instituting a positive
centre.
The analysis for the effects of such positive centres is precisely similar
to that already effected for the negative centre, except that in the case of
the positive centre the curvature would be reversed, the curvature being
away from instead of towards the centre.
The effect of this appears to be to cause positive centres to repel instead
of attract each other. Such repulsions would as in the case of negative
centres depend on the product a multiplied by the curvature, which is of
opposite sign to that for positive centres, and thus the effort of repulsion
between two positive centres would be expressed by
„/47rr03\ a
The coefficient of dilatation is the same — unity. There is thus no
necessity to repeat the analysis. This concludes the approximate analysis of
the actions between centres having similar signs.
It may however be remarked that there are reasons why it is probable
that positive centres should exist, as will appear at a later stage.
220. The first of the class of complex local inequalities ((iii), Art. 203) is
that which would be instituted if by action on the medium in normal piling
a number of grains (n) were displaced from their previous neighbourhood
when in normal piling to some other neighbourhood previously in normal
piling.
Such complex inequalities are only second in importance to groups of
208 ON THE SUB-MECHANICS OF THE UNIVERSE. [220
negative inequalities at finite distances, such as have already been discussed.
In the case of complex inequalities there is no difficulty in conceiving that
owing to the mean pressure there would be an effort to reverse the displace-
ment, as nothing would seem more natural if we have an absence of grains in
one place and an excess in another, under pressure, than that there should
be strains from the place of excess to the place in which the grains are absent,
and vice versa.
It also appears at once as pointed out in Art. 203 that the case is identi-
cal with that which would result from the existence at finite distance of equal
positive and negative centres, having the same number of grains absent and
present respectively.
This identity indicates the direction of the analysis necessary in order to
obtain the expressions for the effort to reverse the displacement.
We have already obtained the expressions for the dilatations per unit of
volume at any point distant r from a negative centre resulting both from
the distortional strain and from the curvature owing to the finite size of the
grain
47rr03 r, , 47rr03 a
and — -.
3 r4 3 r4
And it has also been shown that there is no diminution in the dilatations in
the former as the centres approach.
It has also been shown, Art. 217, that multiplying the dilatations at a
point resulting from a negative centre by p"r2dr and integrating from r\
to r, we have the equation
4-mv [r<r . , „47rr03 fa
V
J p.**-* -rk—r) (30lX
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
/47rr03Y
And again, although not previously noticed, it appears at once from equation
(351) that, if instead of the limits of integration being from r^ to r, they are
taken from r to r = go , we have
P 3 J ^.fdr = p 3 •- (353).
This integral must have some significance as a potential. And it appears
on multiplying equation (353) by 47rr03/3, 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 :
1
dr
r4
V*-) Ir
= -P [-3-) r* (3°4)-
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\ 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 :
„ /4tt7'03\ o"
which differentiated with respect to r is:
/47rr03\ a
P
\TT;
& ■
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>irr03/3 in place of the negative inequality
— 47rr03/3 in the last expression for the attraction, which becomes
3 J f
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. 14
210 ON THE SUB-MECHANICS OF THE UNIVERSE. [223
(Art. 221) is as a to i\ 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 <r in the last expression (Art. 221).
Thus for the total effort, in the complex inequality resulting from the
displacement of a volume of grains 47rr03/3 through a distance r, to restore
the normal piling we have
R = -p"{^)2? (355>
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 ML3T2, 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 :
„ 47rr03 a
-P
,2:
3 ' r
and the effort to reverse the displacement in the complex inequality, as in
equation (355), to be :
„ /47rr03\2 ?'i
-p
•■> I ~& '
and then integrating each of these expressions from rt 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 1\,
'47T/VY 1
pa
and
„ /4irr0y 1
/ r.
-pr-{-s
Then to obtain the expressions for the potential of attraction for either
of these respective energies, the factor \jr 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 :
/ „ MthvY 1
and for the attraction :
\A'v (
47rr0:
"3"
3\2
And in the case of the complex centre, since the product of the centres is
negative, we have for the potential :
p n
47rr03\2 1
3 I r
and for the attraction
— i
47T?VY
-§-;■
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 denned 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 i\ = rQ and r= 1, we get :
Vp"(4/)V=i.
14—2
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
P"(yJV = l (356),
from which if p" is known r0 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 H h, 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 rjcr, 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 rQ3/rx3 —
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 and 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/X
is large, on the arrangement of the grains and the distribution of the strains
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 ajX 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 7r?\rQ3
3 r4
in which r is greater than rlf while 7'0/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 47r/'1r03/3r4, which can under no circumstances be
greater than 47rr03/3r13 are almost indefinitely small.
236. Taking U8 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 Us.
Of course this does not affect the integral displacement of mass integrated
over the medium from 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<7rr03/3.
For if we integrate u"'2 over the medium it is small compared with
4>7rrn3
U,
2
0
3
This apparent paradox, however, is explained on recognising that the grains
being uniform, since crjX 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"2p but depends also on the conductions
- au p,
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 (47rr03/3) 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 (r\) 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) :
3/r
And thus if we consider N — the number of such negative centres within a
distance r3 — to be indefinitely large as compared with rlt 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 '
and one complex inequality in which the numbers of grains absent in the
positive and negative centres are :
A-B B-A
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 reiustitution between the strained aggregate inequalities, correspond-
ing to electric attractions. But as the attraction would be normal to the
surface of weakness, while for reiustitution 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, B/2, 0/2
in the arrangement :
, B C
4+2+2
a B G
A+2+2
2
alities :
A (B G\
2
B G .
2+2"^
and two complex inequalities :
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, 5/2, 0/2 ; for since the work done by the strains might be
sufficient to overcome the resistance to combination of B/2 and 0/2, the recoil
from the breakdown would cause a total or partial combination of B/2, 0/2,
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 densit}^ 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 :
4"rrr0s „
--3-. P.
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 7T . 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 :
3 r,3
4'
r
p" = U (357).
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
p" _ 4 r*
n 3r3
.(358).
0
In equations (357) and (358) IT 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 n 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"/H must have a minimum value.
Then taking II 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 H, 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 :
1_
12'
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:
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 Q 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", X", <r, 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", a, \".
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, X", 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
22H,
where the constant number 12 has still to be determined.
245] THE VALUES OF a", A," a AND G BY EXPERIENCE. 225
The change of units of density, from that in which the density of the
medium was taken as unity, to the density as measured in units of matter,
has thus been effected.
244. From the last article it follows that, measured in c.G.s. units of
matter, the mean pressure in the medium, equal in all directions, becomes
p = 22nP" (.359).
Also the mean density of the medium p" or unity becomes
p = 22nP" (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
'-B5 <361>-
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 ajX" is large.
From equation (311) the rate of degradation of transverse waves is
expressed by
1 dv" 2 X"a" /q«o\
-77 . — r- = — - — : . Of,2 (362).
v" dt 3 Vtt v !
Then if tt is the time taken to reduce v0" to v1
where v± = - . v0 ,
e
vv, 3 yV 1 (363)j
2 a? tt
which gives one equation between the three quantities a", \" and tt.
A second equation is obtained from the dynamical condition of undulation
-=r = A/- (364),
a V P
and n = 8^'' * bein& i 2ir\") ' P ' ~2 (365).
Therefore, reducing,
' 4 ' 2tt " X"
T = ^3.^ " (366),
X" a/3 <r /o«7\
15
226 ON THE SUB-MECHANICS OF THE UNIVERSE. [245
Then, L being the wave-length, if we put
n2 . <r = L
t 2tt
since -^ = — ,
2_
substituting — for a in equation (367),
^' = ^?. J- (368).
Then eliminating a" from equations (367) and (368) to find \"
\" = Sl . (n2tt)-* (369),
the value of the constant coefficient being
Sl ~ 8aV '
Then substituting from equation (369) in equation (367)
„ 1 3 *J7T , , 1]
or a" = s2{n2tt)^ . -
n"2 <?2 1
V=|(^>^ (371>-
The equations (369) and (371) define the values of the constants \" and a"
which enter into the expression p" in equation (159) in terms of a, r, n2 and
tt which define the wave-length and rate of propagation for any particular
rate of degradation.
Thus substituting in the equation (159) which is
.(370),
V2 a_ aT2 fa_
V 3 ' A" ' P * 2 J U'
and which, under the condition aj\" large, is, taking the density of the
medium as unity,
//_V2 <r a"2 6 ( 72)
the equation becomes
„ \/2 L \/nJt s22 1 6 7.
* =t-»/ — • T^^tf-4^? (373)-
Then transforming we have
4<7T S-,
V
rv7! (374)-
246]
THE VALUES OF a", \", a AND G BY EXPERIENCE.
227
If the constants sx and s2 are taken to correspond with the rate of propa-
gation of light and with the wave-length of the ultra-violet light in the C.G.S.
units
«, = 9-7005 x 10-14
52= 10738 x 103,
from which substituting in equations (309) and (371)
1 X
*Jn,t
2H
n2
.(375).
9-7005 x 10~14
a" =1-0738 x 103
a"2 _H~,~ -i^ffhU
T = 5755xl0(#
And since the wave-length L is 3'933 x 10~5 we have, dividing by s1 and
substituting in the second expression for p",
p" = 1-8574 x 10" (^Y (376),
which becomes in C.G.S. units of matter (by equation 359)
220p"=22Q x 1-8574 x 10u(^) (377).
For convenience the expression for a"\" may be here included :
1
tt
a
"\"= 1-0418 xlO-10
.(378).
246. Having effected the translation of units and obtained an expression
for the mean pressure in the uniform medium in terms of n2/tt, we now
proceed to the evidence as to the absolute density, or, what is the same
thing, the value of the number expressed by X2.
The density of the luminiferous ether, thus far, has been an unknown
quantity. Such views as have been expressed range from a density in-
definitely greater than that of the heaviest material — Hooke — to a density
indefinitely smaller than that of the lightest solid material — Sir Gabriel
Stokes and Lord Kelvin.
But as pointed out in Art. 242 we have now the two sources of evidence —
that arising from the known law of gravitation, which includes the existence
of permanent negative inequalities, or molecules with surfaces of freedom,
and that resulting from the limits to the intensity of waves of light ; besides
such evidence as may accrue from the determination made by Lord Kelvin as
to the dimensions of the molecules, and such evidence as has been obtained as
to the rates of degradation of the transverse and normal waves.
15—2
228 ON THE SUB-MECHANICS OF THE UNIVERSE. [247
The equations (376) and (377) define the pressure in terms of
J or 22H?|2,
according to whether the density of the uniform medium is taken as unity, or
is expressed in C.G.s. units of matter.
247. As measured in c.G.s. units, the matter in the earth, assuming
Baily's value, 5"67, for the mean density, is
6-14 x 1027,
the mean radius is 6'3702 x 108 and the attraction of the earth on a unit of
matter at the surface is
# = 981 (379).
To compare with this evidence we have the expressions for the correspond-
ing quantities as obtained from equations (348) for corresponding conditions
when translated into the same units.
In the general expression for the attraction of negative centres in closest
order, equation (348), where p" = 1 :
r/ /47T \2 a
r8'
where N' = *75 ( — J and r = rB;
substituting, the expression for the attraction of unit mass becomes, if the
,. n3 4 5-67
ratio -i = 3 x 22^ when p = 1,
-'TP <r[-J -rB-
r0\s
Then, supposing that rfjr? is a maximum, we have from equation (358)
T 3 1
(380).
1\3 75H
And as the density of the mean negative inequality is 5*67/22 of the
maximum inequality, we have for the attraction
5-67
which becomes, on substituting from equation (380) and reducing,
4 „ 4 „ 5-67
Then transforming so that the density of the medium is 22H, since rB is
6"37 x 108, we have for g
4 ^-67
981 = 22%/Vg 7T ~ . 637 x 108 (381).
248] THE VALUES OF a", X", a AND G BY EXPERIENCE. 229
Then substituting the value of 22%/' in equation (377) we have
-7TO-67 x 6-37 x 108 x 1-8574 x 10" x (j*\ <r = 981 (382).
Then, cancelling and reducing the numerical factors, since
a-(7i2ftt)i=L/^/n2tt,
we have
981-M0BJLlff1
(383).
981 = 1105xwn
vn2tt
whence */n2tt = 1126 x JO14,
And thus we have obtained the value of
n2tt,
which satisfies the condition g = 981.
248. The evidence afforded by the limits of the intensity of light and
heat does not appear to have hitherto demanded much attention. But it
now appears that, if we can find a fair estimate of the maximum intensity of
transverse undulations, it would afford important evidence.
For the rate of displacement of energy by the transverse waves in the
uniform medium we have, taking U for the rate at which energy must be
supplied to maintain the waves, and t for the rate of propagation : since the
velocity of light is independent of the wave-length, the maximum energy of
mean motion over a unit surface
P-T*
is, by equation (308), the mean energy of the undulation ; and
U=r.p"-2 and «"=(4tH ■ (384).
It must be noticed that in these expressions for U and v" no account is
taken of the secondary effects imposed by the dilatation in the granular
medium. This was noticed in the last paragraph, Section XIII., as sho wing-
that there is a limit to the intensity of harmonic institutions.
Put definitely, the condition to be satisfied for harmonic undulations is
that, taking x and y for the directions of propagation and mean motion
respectively,
-K-^is small as compared with p".
Thus if the amplitude of the transverse motions is considerable, the
action will not be confined to the institution of simple harmonic waves,
but will include compound harmonic waves, and probably normal waves,
which would proceed faster than the simple transverse harmonic waves,
until, by divergence or degradation, their intensity was reduced.
230 ON THE SUB-MECHANICS OF THE UNIVERSE. [249
Evidence from which we may form an estimate of the limit to the
amplitude at which the waves cease to be sensibly harmonic may, it appears,
be found. The greatest intensity of transverse waves is obtained from the
carbons of the electric arc. If then we assume that U, the work expended
in producing the light, is all spent in radiation of heat and light from the
carbons, we have only to measure the radiation area of the carbons to obtain
an outside estimate of the mean value of v" .
Thus if U per sq. cm. is 2*29 x 109 ergs
2-29x109 = i/o</2.t (t = 3x1010) (385),
whence we have
152
Vi'a = — xlO-1 in c.g.s. units (386),
P
where p is 220/)" and where p" is unity.
249. From this value of v" we may obtain the expressions for y the
amplitude of the undulations, and for x.
Taking r as an arbitrary amplitude
y = r cos 0 and dyjdt = — r sin 0 . dd/dt.
Then since the periodic time is 27r/m, differentiating 0 with respect to
time dOjdt = in, and
v" = — mr sin 0 and v" is a maximum when
0 =-w/2.
.". r = — , y = — cos 0,
and x — - ,
a
'• dx-a-T0-~a mSmd~V (887)-
Then multiplying this by n or pr2 we have for the shearing stress
22n§--l="»"T <388>'
and these are in gravitation units.
Then from equation (386) we have, for the maximum value of the
transverse velocity v",
-"A <*»
and multiplying by 220 we have for the maximum shearing stress
p .v". t = 1172 x 1010 x V22fi (390).
250] THE VALUES OF a", \" , a AND G BY EXPERIENCE. 231
Taking s (= 10-2) as the coefficient of the limit within which 22ft . 3/c/8
may approach 22Qp", we have, substituting the expression on the right of
equation (377) for 22ftp",
22ft x 1-8574 x 1011 (~) = V22ft x 1172 x 1012,
logs
whenc
e follows
:
(!)*=
6-31
V'«2^=
= 1126 x
1014
7108 x
1014
1U =
equation (390), '8000 - log V22ft (391),
(384), -0517 + 14 (392),
^=p , -8517 + 14- log V22H... (393),
tt = 1-785 x 1013 x \/22ft, -2517 + 13 + log V22H . . .(394),
o- = 5-534 x 10--° XV22H (372), -7430 - 20 + log V22ft ...(395),
G'777 x 103 '
a"=-^mr (370)' -83io+3-iogV22n (396)
X"= 8-612 xlO-28 (375), '9351-28
250. So far we have obtained the expressions for the limiting values of
a", A/', a and the logarithmic decrements for transverse and normal waves
in terms of the constant coefficient ft which enters as a factor into the
expressions for the density of the medium and the potential of attraction.
Substituting from the equations (391 — 393) in equation (375) we have
gmxiop
V22fl
\"= 8612 xlO-23 (398),
a = 5-534 x 10-20 x V22ft (399).
Then for logarithmic decrement of the transverse undulations, aj\" large,
substituting in equation (311) the values as given above for a" and \" we have
as in equation (362), tt being the time required to reduce v" from v0 to v0/e,
tt=l /^= 1-784 xl013V22ft (400).
2 A. a cr
N.B. This result checks the calculation, since this value corresponds
with equation (394) in the first three significant figures, which is the limit
of the arithmetical approximation attempted.
The value of tt thus found in terms of the coefficient V22ft expresses the
time the transverse waves would travel before their amplitude was reduced
in the ratio from 1 to 1/e, or their energy in the ratio 1/e2.
232 ON THE SUB-MECHANICS OF THE UNIVERSE. [251
The values of a", X", <r cannot be defined except by further evidence. Such
might be obtained if we could completely solve the dilatation problem and
so obtain the value of ft. Failing this, however, there remains one source of
evidence from which we may obtain a close approximation to the value of the
ratio V22ft.
251. The conclusions to be drawn from, the absence of evidence of any
normal waves in the medium of space until very recent times.
From equations (310) and (311) it appears that in a granular medium
normal as well as tangential waves may exist, the only difference being in
their rates of propagation and in their rates of degradation.
From this it would seem that, if the medium of space is purely mechanical,
either such waves did not exist for lack of incitement or the normal waves had
no effect upon our senses or on the physical properties of matter. The recent
remarkable discovery of Rontgen that under certain intense electrical actions
a system of waves which have the properties of normal waves in a uniform
medium subject neither to refraction nor reflection, can be produced, has
opened the door to different conclusions. The first suggestion by Rontgen
was that these were normal waves. And although various special explana-
tions have been attempted to avoid the admission of their being normal
waves, every one of these explanations involves normal action.
It appears, from the definite analysis of the granular medium, that when
the uniform medium is in the state to propagate transverse waves the degra-
dation of which is such that the diminution from loss of energy by degradation
in some millions of years is in the ratio 1/e2, the rate of degradation of the
normal wave is such as would occupy something less than the millionth (10~6)
part of a second to reduce it in the same ratio ; so that the normal wave
would lose nine-tenths of its energy before it had traversed some thousands
of metres, say x metres, and this affords crucial evidence of the purely
mechanical granular structure of the medium of space. The coincidence
is such, that in the absence of any definite proof to the contrary, it should
carry conviction notwithstanding those things which cannot be defined for
want of evidence.
252. Without attempting any general discussion of X-rays there are
several very significant characteristics which afford evidence besides that
of not being subject to refraction or reflection. In the first place the rays
in their production are attended with very intense light, that is they are
attended with transverse waves. In the second place, after the light waves
have been filtered out, they can again be transformed into visible transverse
waves by their passage through certain earthy substances. And in the third
place, in passing through any matter they are subjected to rapid degradation
253]
THE VALUES OF a", A", <x AND G BY EXPERIENCE.
233
which is proportional to the density and thickness of the matter through
which they pass.
Thus it has been so far impossible to study these rays except by their
passage through matter, while it is shown that in two wTays their passage
through matter is attended by degradation other than the degradation of
the normal waves in vacuo.
Any estimate as to what might be the rate of degradation of these waves
in vacuo is at best very difficult. But the fact that these waves, which are
subject to divergence as well as the three sources of degradation, have
sufficient range to permit of experiment through a distance of some metres,
shows that if they are normal waves their rate of degradation in vacuo would
be much less than it appears to be in the experiments. It thus appears that
x, the distance the waves must travel in vacuo to reduce the energy in the
ratio lje2, cannot be less than some thousand odd metres.
253. To find the rate of decrement of the normal wave under the limits
defined by equations (221) to (224) in terms of the ratio l/v/22H.
From equation (310) we have, neglecting as small the first term in the
index, and substituting 6/a/27t for G,
1^
U
du"
~dt
a*
1 /5 p2 4 <r2 a
2\3~7^ + 3\'JirJ2 4^m2
= e
.(401).
The index in the right member of this equation represents the logarithmic
rate of decrement of the normal wave.
Transforming this index and substituting the values of a, A and er as defined
in equations (221) to (225) for the transverse wave, and of m and a for the
normal wave, taking the time frequency m to have the same value as for the
transverse wave and the linear frequency a to be a'/2-387 where a is the same
as for the transverse wave [2-387 being V 3/c + 4m/3n]. Then taking A as
expressing the numerical constant in the expression for the decrement, w^e
find as the values of the several factors and their logarithms,
A = 1-567 x 10-2
t-2 =1-111 x 10~21
a- = 2-553 x 1010
I = 3-102 x 10~2
o-4 = 9-376 x 10~82 x (22ft)2
a"3 = 3-113 x 1011 x (22fl)-f
A"-3= 6-387 x 108,i
log -1952 - 2
•0457 - 21
•4068 + 10
•4916-2
•9720 -82 + log (22H)2
•4930 + 11 + log(22fl)-
•8053 + 86
....(402).
234 ON THE SUB-MECHANICS OF THE UNIVERSE. [254
The logarithm of this product being
•4076 + 3 + 1 log (22.Q) (403),
log decrement log (log decrement)
- 2-556 xl03xV22O, - [-4076 + 3 + £ log (22ft)] (404).
Then if tn is the time to reduce u"'2 in the ratio 1/e2 we have
tn = 3-923 xl0-4/\/ 2211", log tn = '5924> -4 (405).
The product of the time tn multiplied by the rate of propagation of the
normal wave is the linear distance which the normal wave must travel so
that the energy is reduced in the ratio 1/e2.
The rate of normal propagation is 2387 x 3 x 1010 as above.
Therefore taking x as the distance the normal wave must travel to
diminish the energy in the ratio 1/e2 we have
# = 2-801 x 107x— L= (406).
V22H
Q. E. F.
254. Then to find the inferior limit to the value of the raiio ex-
pressed by
n.
From the evidence furnished by Rontgen rays we have in Art. 253
denned this ratio to be such that the value of x (in c.G.S. units) shall not
be less than some thousand odd metres. And from the absence of any
evidence of normal waves other than Rontgen it follows that there must
be a superior limit ; but this depends on the value of II and cannot be
defined without further evidence.
To find the superior limit of fl, putting for simplicity
# = 2-801 x 107~'i (407),
we have by equation (406) from the evidence of Rontgen rays
V22H = 109 where q is not less than 2,
whence we have for the value of Q,
102?
fl= ^- = 4-546 x 102?-1 (408),
and for the density of the uniform medium
22fl = 102« (409).
255. It is pointed out (Art. 254) that the superior limit to the value
of fl cannot be obtained except on further evidence ; evidence which has
as yet not been taken into account, and is exactly to the point, is
available.
This is the evidence as determined by Lord Kelvin (and confirmed by
255] THE VALUES OF a", \", a AND G BY EXPERIENCE. 235
the observation as to the area over which a definite volume of oil would
destroy the ripple caused by a moderate wind on the surface of water), that
the diameters of the molecules or singular surfaces are of the order of the
ratio of the wave-lengths of the ultra-violet light multiplied by some ten
thousandths, say 4 x 10~10, and this evidence comes in as directly bearing
on the value of q.
Although there is a degree of uncertainty about the relative value of the
"atomic- volumes" of the elementary molecules, it appears certain that there
is no great difference, that is to say, no difference greater than from 1 to 10
in the relative volume of the molecules, and for our purpose it is sufficient
to consider that, assuming the relative volumes equal, the greatest difference
of the grains absent is from 1 to 1/200.
It has been shown (Art. 230) that the probable arrangement of the grains
in a negative local inequality, which has a surface of freedom, is that of
a nucleus in normal piling, that is to say, a permanent nucleus on which
the inward strained normal piling reaches, forming a broken joint in
abnormal piling, whence it appears, in order that the singular surface may
be a surface of freedom, the maximum inward strain, that is, the inward
strain at the singular surface, must be greater than a the diameter of a grain,
and probably some five times a.
In this way we have a limit to the diameter of the singular surface,
4 x 10-10,
and by the last paragraph, taking 10 to be the inferior limit to the maximum
inward strain, we can find a value for q which is quite independent of any
evidence already adduced.
Taking 22D = ] 02q for the density of the medium,
7\ = 2 x 10-10 for the radius of the singular surface,
4nrr03/3 = volume of grains absent.
By equation (380)
r»,=I6*-ri' (410)-
Then since by equation (396)
0-3 = (5-534)3 x 10-54+3?,
and r0/<r = n0/2, also r1/<r = n1/2 (411),
^Y=n03, and 6 (^Y = 6 «>2 (412),
3
n0' \ a
2r^3
0
6V 6 /2r,y
.(413).
236 ON THE SUB-MECHANICS OF THE UNIVERSE. [256
Equation (413) expresses the number of diameters of a grain which would
measure the inward strain at the singular surface of the maximum inequality
as of platinum or 22.
Then reducing
n,s
6^-2 = 1-602 x 10 <9-3?> (414).
For the minimum inequality, n1 remains the same, and n0s is divided by
200, and we have from equation (414),
T x 10"2
— — - — = 8-013 x 106-3? (415).
brii
Then if we take the number of the diameters of a grain which measure
the inward strain at the singular surface of the minimum inequality to be
8-013,
q = 2 (416).
We have thus found the superior limit of the square root of the density
of the uniform medium to be
V22H = 100.
256. Comparing the inferior limit of V22I2 in Art. 254, obtained from
the evidence of Rontgen rays, with the superior limit in Art. 255 obtained
from the evidence as to the size of the molecules, we see they are identical.
Too much weight must not be attached to this identity since the
estimates on which they are based are somewhat wide approximations, so
that they must be considered as relating rather to the order of the quantities
than the actual numbers. Yet considering that the evidence of the size
of the molecule, and that of the Rontgen rays, are perfectly independent,
the result, which, taken as a wide approximation, would be almost infinitely
improbable as a mere coincidence, when substituted in the equations (390)
and (396), and (402) and (409) enables us to obtain, in c.G.S. units, the values
of all the arbitrary constants which define the condition of the purely
mechanical medium, and they are such as correspond with the experience —
as to the rates of propagation and as to their rates of decrement — of both
transverse and normal waves ; they also correspond with experience as to
the existence of molecules and gravitation, the limit of the intensity of the
energy of light and radiant heat, besides the absence of normal waves, and
the evidence of Rontgen rays.
The numerical values of these constants are for convenience given in the
following table.
256]
THE VALUES OF a", \", a AND G BY EXPERIENCE.
237
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238 ON THE SUB-MECHANICS OF THE UNIVERSE. [257
It is thus shown by definite analysis that an infinite, purely mechanical,
medium consisting of uniform spherical grains, in relative motion, the grains
being in normal piling, except for local inequalities in the mean density, and
so close that there is no diffusion, affords a complete account by purely
mechanical considerations of potential energy, the propagation of transverse
waves of light and the apparent absence of any rate of degradation, the
lack of evidence of normal waves, the gravitation of matter and electricity,
as the result of the dilatation which follows from the strains caused by local
inequalities in the density of the medium.
It is also shown, by definite analysis, that this is the only explanation
possible by purely mechanical considerations.
257. Having arrived at the conclusion stated in Art. 256 we might
make this the end of this research, having every confidence that the evidence
which has not already been adduced would confirm that which has been
adduced. It is not, however, the sole purpose in undertaking this research
merely to show that there is a mechanical explanation of such parts of the
universe as shall render the mechanical structure of the remainder in-
definitely probable, but also to obtain as much light as may accrue from the
purely mechanical analysis. The analysis is therefore continued so far as it
relates to effects in the medium, that is to say, it does not include electro-
dynamics or electro-magnetics, since the institution of complex centres, that
is, the magnetic conditions, is not a primary effect, for it results in separating
the molecules, after combination, the reunion of which results in electric
currents.
258. The blackness of the sky on a clear dark night would be explained
if the light waves were subject to viscosity however small, or nearly so.
It has been so far a moot question whether there is such viscosity. But
it now appears from the rate of decrement of the transverse waves, Art. 256
(5'603 x 10-lfi), that the time taken to reduce the energy of the wave in
the ratio 1/e2, or 1/8, would be more than fifty-six million years. This rate
of decrement, although affording an ample account by mechanical considera-
tions of the absence of uniform brilliance in the sky, such as would result
in an infinite space from an infinite number of stars, however sparsely
scattered, if there were no rate of decrement as the result of viscosity, is
such as has baffled all attempts to obtain any evidence of decrement by
observation.
259. The dissipation of the inequalities in the mean energy of the
medium resulting from the rates of decrement of transverse and normal
waves which, as shown in Art. 256, affords a complete mechanical explanation
of the blackness of the sky, differs fundamentally from that dissipation which
results in the increase of energy of the molecules, or singular surfaces. This
261] THE VALUES OF a", \", a AND G BY EXPERIENCE. 239
is at once apparent since the degradation of the energy of the normal and
transverse waves can only be a dissipation from the energy of the molecules,
or mean motion, to increase the irreversible energy of the mean relative
motion of the medium.
It thus appears that the dissipation of the mean motions of matter, such
as the motions of the sun and planets, or vortices in fluids, until all motion
ceases, does not complete the dissipation of energy, for this would go on
until the only energy was irreversible relative motion of the grains, which is
expressed by a"2.
260. The electrostatic unit, or more correctly the unit corresponding to
the electrostatic unit, is defined (Art. 224) by the condition
W^)V=1 (417).
3
This definition is on the supposition that the density of the medium
is taken as unity.
Thus if the density is taken as 22fl, we have as the condition
„ /47rV
2211/' (^j rj = l (418).
Then reducing the member on the left by the table (Art. 256) it is found
that the complex inequality in which the number of grains is displaced is
1-615 x 1045,
and in which the displacement is unity ; the effort to institute the normal
piling is unity and thus corresponds to the electrostatic unit.
Comparing the effort to revert to the effort of attraction between two
negative centres, each having the number of grains as above, since the radius
of the shell which would contain the grains is
r0=6-493 x lO"3 (419),
the ratio of the effort to reinstitute the normal piling, to the effort of
attraction between gravitating mass, is approximately
1-2 x 1015.
Thus the effort of attraction between the two gravitating masses, the
grains absent in each of which are the same as the grains which constitute
the electrostatic unit, is eighty-one thousand billion times less than unity.
261. The conclusion arrived at in Art. 256, as to the density of the
medium, does not exhaust the conclusions to be drawn from the size of
the molecules. Coupled with the evidence afforded by the effects in dis-
sociating certain compound molecules, possessed by the transverse waves
240 ON THE SUB-MECHANICS OF THE UNIVERSE. [262
of shorter length and greater frequency, it appears that there must exist
certain coincidences of periods between the possible internal vibration periods
of compound molecules and the periods of the shorter waves.
262. From the evidence, Art. 261, it follows that the compound mole-
cules which are dissociated by the waves of light must have been in a state
of limited stability : so that
(1) by the breakdown the total potential energy is reduced,
(2) a sudden disturbance in the medium is produced causing waves,
which are of undefined length, in the medium.
263. Comparing the evidences as to the effects of waves of greater
frequency in dissociating certain compound molecules, adduced in Arts.
253, 254, with the conclusions arrived at in Arts. 238 — 241 as to the
effects of collisions between compound singular surfaces, rotational strains,
and the institution of complex inequalities corresponding to electrostatic
induction, it appears that the latter account for the former by mechanical
considerations as will appear in the following articles.
264. Accepting the statement in Art. 263, we find ourselves face to face
with the question, What is the source of light ?
From the mechanical analysis it follows, Art. 238, that undulations in
the medium can arise from nothiug else than the relative motion of the
singular surfaces. The collisions of these surfaces would set up disturbances
which would be propagated through the medium with the velocity of light,
and which would correspond to the waves of heat. But from Arts. 238 — 241
it appears that there is another effect than that of simple collision, by which
undulations may be instituted.
In Art. 241 it appears that when two aggregate inequalities, separated by
a surface of weakness, in which the numbers of grains absent in the primary
inequalities differ, are subjected to rotational strain, parallel to the surface
of weakness, the strain will cause the total aggregate inequalities to reform,
instituting two fresh aggregate inequalities with limited stability, which, as
the strain is gradually reduced, do not gradually revert but, owing to the
limited stability, are maintained until the strain has been relaxed sufficiently
to overcome the limited stability and then break down under the nearly
full effort of the complex inequality; which, by Art. 260, is more than two
hundred billion times greater than what would be the effort of attraction
of the two equal negative inequalities at the same distance.
Such a transverse reversion as that considered would not result merely
in reinstituting the normal piling. But, as it involves the absolute displace-
ment of mass, the recoil by reversing the strain would institute a complex
265] THE VALUES OF a", \", a AND 0 BY EXPERIENCE. 241
inequality of the opposite sign ; and this would be repeated, in a gradually
diminishing degree, until all the energy was spent in setting up undulations
which would be transverse.
We have thus two, more or less distinct, sources of undulations; and
from the evidence it appears that, whatever undulations result from the
collisions of singular surfaces, the undulations corresponding to those of
polarised light are those caused by the reversion of the complex inequalities.
265. Since, from Art. 264, it appears that the institution of light
depends on the existence, in the medium, of compound molecules with
limited stability, and it also appears that these compound molecules dis-
sociate in the production of light, it follows that either the source of
light must be continually diminishing or that there must exist some action
which results in thus reassociating the primary inequalities, and as the
first alternative is contrary to experience we must accept the second as
a fact.
The reassociation of the primary molecules which, when associated, form
compound molecules with limited stability, receives its explanation from
the mechanical analysis on the same lines as that of their dissociation.
Thus if we have two aggregate inequalities in one of which the primary
inequalities are not combined the differing primary inequalities are combined.
These may be analysed by putting
a + a for the combined total aggregate inequality, and
b + b' for the total aggregate inequality uncombined, then
a + a' + b + b' a + a'+b + b'
~2~ "' ~2~ ~'
a + a'-(b + b') b + b'- (a + af)
2 " ' 2
These if added together constitute the total aggregate inequalities ; they
express two equal total negative aggregates together with one complex
aggregate inequality.
Thus putting a + a = A the total aggregate inequality in which the
primary inequalities are combined, we have
A + b + V A + b + b'
2 2
A-(b + b') b + b'-A
2 2
Then if the strains were sufficient the normal attraction might overcome
the normal stability, i.e. the stability in the direction of the normal, of the
r. 16
242 ON THE SUB-MECHANICS OF THE UNIVERSE. [266
complex inequality, causing a reversion. In this case, however, it does not
follow that the reversion would be complete and so reinstitute A, b and b',
for since the work done by the strains might be sufficient to overcome the
resistance to combination of b and b', the recoil from the breakdown would
cause a total or partial combination of b and b', thus instituting B, the total
aggregate inequality, and so diminish the energy available for the institution
of undulations.
We have thus an explanation by mechanical considerations of the part
played by electricity in instituting the combinations of molecules which
differ into compound molecules with limited stability.
266. The absorption of the waves of light, let us say by lamp-black,
presents a problem, the explanation of which, by the assumption that the
molecules are capable of internal vibrations in various periods, is altogether
sufficient. Thus, supposing the molecules in the lamp-black are so various
that there are molecules the internal vibrations of which coincide with
all periods of the incident wave, they would be set in periodic motion
and absorb the energy of the waves ; but this is not all. For supposing
the absorption of the light continuous, the energy in the molecules would
continually increase, and this is not in accordance with experience. There
must therefore be some means by which the energy absorbed by the
molecules may escape. This cannot be by radiation, since in that case
it would only escape as light, which it does not. It is mechanically
impossible that it should escape by radiation in the form of the long
dark waves. And the only other mode of escape for the energy is by
transmission — by convection and conduction through the molecules to the
surface of the lamp-black. Nor does this altogether solve the problem — for in
such an experiment as we are considering, it may be possible that the lamp-
black is in vacuo ; in which, having reached the surface, it would be arrested.
And the absorption continuing the energy of the molecules would con-
tinually increase indefinitely. Since any such indefinite increase of the
absorbed energy is outside experience it follows that within the limits of
experience such perfect vacuum as contains no free molecules is impossible.
The evidence which follows from the theoretical explanation of Sir William
Crookes' radiometer* at once illustrates the fact mentioned above, for when
the light is turned on the receiver which contains the vanes, the latter
almost instantly acquire a steady speed which shows that the lamp-blacked
surfaces as well as the opposite surfaces, which are white, have acquired
a steady difference of temperature, so that there is no further increase of
temperature from the absorption of the light ; the energy received from the
light wave by the black surfaces of the vanes, taking the form of energy
* " Certain dimensional properties of matter in the gaseous state." Phil. Trans. R. S., 1879,
p. 823.
267] THE VALUES OF a", X" , a AND G BY EXPERIENCE. 243
of vibration of the molecules, is transmitted to the surface beyond which
the vibrating molecules do not pass, but, as the molecules at the surface
are vibrating, the energy of this vibration is communicated by contact to
any free molecules whose paths bring them in contact with the molecules
at the surfaces of the vanes, causing reaction and conveying the energy to
the inner surface of the receiver.
Thus if there were no free molecules there would be no motion imparted
to the vanes, and as the stage of exhaustion at which the vanes do not
revolve in unlimited light has not yet been attained, it follows that on the
assumption that the waves of light are capable of communicating energy
to the molecules in the mode of internal vibration, the production of an
unlimited intensity of energy by the absorption of light is outside
experience.
267. The assumption on which the absorption of light is based, Art. 266,
has not as yet been subjected to the further analysis necessary for a
mechanical explanation of the actions involved.
It therefore remains to show that, in spaces where negative inequalities
exist, the state of the granular medium is so far affected by these in-
equalities that it no longer transmits waves which pass through the medium
at the same velocity as when there are no inequalities, undisturbed, other-
wise than by divergence.
To show this :
We have (Art. 230) the fundamental misfit between the nucleus in the
singular surface with the grains in strained normal piling, instituting in the
medium a shell of grains in abnormal piling which constitutes a shell about
each singular surface which offers little or no resistance to strains tangential
to the singular surface.
We have also (Art. 255) the diameter of the singular surface some ten
thousand times less than the wave-length. Thus we have a free singular
surface through which the medium is free to move by propagation, the
diameter of which is 10000 times less than the transverse wave, but which
is still subject to the undulatory motion of the medium corresponding to the
light waves.
Consider next what must happen from the existence of a single negative
inequality in a space through which transverse waves are passing : —
In the first place, since the surface of the inequality is a surface of
freedom there would be a certain small area of the surface about an axis
through the centre of the inequality which presents a nearly plane surface
perpendicular to the direction of propagation, and this small surface, owing
to the freedom of the inequality, offers no resistance to the transverse wave.
16—2
244 ON THE SUB-MECHANICS OF THE UNIVERSE. [267
This area of freedom would relieve the stress in the medium iu the plane
normal to the direction of propagation, and so cause an increase of the
undulatory motion at the small surface, the recoil from which would reverse
the direction of propagation over the small area, thus instituting a partial
reflection. (N.B. The amount of this reflection would admit of quantitative
determination, but the analysis is long and it does not appear to be
necessary.)
The reflection considered does not constitute the entire reflection which
would result, for there would be similar reflections at the opposite surface of
the inequality, and besides the reflections on the small surfaces nearly plane,
there would be reflections resulting from the relaxation of the components
of the transverse stress all over the surface of the inequality, causing re-
flections in all directions except in planes normal to the direction of
propagation. So that there would be a general but varying scattering of the
transverse wave in all directions greater than 7r/2 from the direction of
propagation, varying from a maximum at ir to nothing at tt/2.
The proportion of undulations within a distance 7\ of the axis iu the
direction of propagation scattered by the passage of a wave by a single
inequality is extremely small, for, although the small surfaces of freedom do
relax, to some extent, the stresses consequent on the undulations in the
medium, a singular surface is so small as compared with the wave-length,
that they follow the motions of undulation, and are subject to nearly the
same stresses as if there were no inequalities.
Then if we consider a space, through which the waves are passing, to
be occupied with negative inequalities in somewhat close order it does
not appear that the rate of propagation would be greatly altered owing to
relaxation of the elasticity of the medium.
But the rates of propagation do not, as it seems, depend solely on the
elasticity ; for the singular surfaces, owing to their cohesion, introduce
another system of possible vibrations — the internal vibrations of the negative
inequalities.
That the vibrations possible in the inequalities may be instituted as
the result of undulatory stresses requires only a coincidence in the periods
of the waves and the vibrations of the inequalities. Then since the evidence
of the existence of a considerable number of periods of vibration in all
inequalities is according to evidence, and it has been shown that however
small the effects of the undulations solitary grains do cause a certain dis-
turbance in the negative inequalities, it follows that the passage of a wave
through a space in which the inequalities are somewhat close will result,
if continued for a sufficient time, in imparting periodic motions to the
inequalities having periods coinciding with the wave periods.
268] THE VALUES OF a", X", a AND G BY EXPERIENCE. 245
Then supposing the regular undulation to cease, the vibrations of the
inequalities would institute waves of the same period until their energy was
exhausted. Whence it follows that in the case in which waves are passing
steadily into and through a space occupied by inequalities in somewhat close
order, they will maintain the vibration of the molecules and at the same
time pass through the medium, and then the energy of the waves and the
vibration of the inequalities together would be greater than that of the
inequalities alone in the ratio
energy of wave motion + energy of inequalities
energy of wave
Then supposing a steady state to have been reached, if either of these
actions were diminished it would receive assistance from the other; and
from this it follows directly that, while the energy in a wave-length before
entering the space containing the inequalities is the only energy of the
undulation, the energy in a wave-length in the space would be the energy
of the undulation before passing plus the energy of the inequalities.
Then again if the mean rate of the motion of the energy of both
undulation and inequality were that of the undulation, there would be more
energy passing out of the space than that entering, and the state could
not be maintained steady. But if, on the other hand, after entering the
space with inequalities, the rate of passage of the total energy was that
given by
energy of wave
energy of wave + energy of inequalities '
the state would be steady, and the rate of propagation diminished in the
same ratio.
It has thus been shown that in the granular medium waves corresponding
to light waves are capable of communicating energy to the negative
inequalities corresponding to molecules, which was the object in this some-
what long article.
268. Refraction of waves in the granular medium, when passing from
one space to another which differs as to the closeness of the arrangement,
follows directly from the paragraph last but one, Art. 267, in which it is
shown that the waves pass from a space in which there are no inequalities
into a space in which the inequalities are in some close order; the ratio
of the rate in the space without inequalities to the rate in the space with
inequalities is as
energy of propagation 4- energy of inequalities
energy of propagation
and this is the expression which corresponds with the index of re-
fraction.
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THE TAL V, 9 ASD <x ~T ZXFZBtfES
The rertkal harmonic motions, in the mediam EF in fhmv parallel v>
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at the surface earned hj the inequalities. T ix?^i-,r i* prop
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at which the reflection is a nmximnm 5* of necesatr such that
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248 ON THE SUB-MECHANICS OF THE UNIVERSE. [270
inequalities, the wave will be reflected according to the laws of reflection,
such reflection being strictly parallel to the paper.
On the other hand, it follows that the transverse waves in which the
motion is normal to the paper can, in a granular medium, be instituted only
by rotational stress in which the rotation is normal to the paper ; such waves
propagate parallel to the paper in the direction to which their planes are
normal, and are not subject to reflection at an inclined surface perpendicular
to the paper, as shown in Fig. 5, since the motion in these transverse waves
is entirely normal to the paper, as is shown by the line GH, turned
through 90° in Fig. 6. Thus it is seen that the only reflection resulting
from both components of the motion in the medium when the waves pass
from a space without inequalities into a space with inequalities is the
reflection resulting from the inclination of the surface parallel to the plane
of incidence, as shown in Fig. 5.
It may appear from what precedes that there is a difference besides that
of the motion of one of the rays being parallel and the other normal to
the paper, since so far no mention has been made of any reflection of the
ray in which the motion is perpendicular to the paper. This apparent
difference disappears, however, since if the reflecting surface in the plane
of incidence were removed and replaced by a surface normal to the paper
inclined at a corresponding angle to the direction of propagation, then
the reflection would be from the waves perpendicular to the plane of
incidence, and there would be no reflection from the plane of incidence.
It is thus shown that in the granular medium when the transverse stresses
in the medium are equal in all directions normal to the direction of propa-
gation, when waves proceed from a space in which there are no inequalities
into a space in which there are inequalities, if the separating surface is
inclined to the direction of propagation there will be reflection in the plane
of incidence of that component of the wave which is in the plane of incidence,
in a degree depending on the closeness of the inequalities and the angle of
incidence, while the other component of the wave-motion will not be subject
to any reflection resulting from the inclination. And as this applies whatever
the direction of propagation may be, it affords a definite proof that the
motion in the medium which is reflected is in the plane of incidence.
This result in the granular medium corresponds in every particular with
the experiences of polarisation except that heretofore it seems to have been
a moot question whether or not the motion in the ether which is polarised by
reflection was parallel or perpendicular to the plane of the medium*.
* "In the theories of Fresnel and Canchy the vibrations are assumed to be perpendicular to the
plane of polarisation — in those of MacCullagh and Neumann to be parallel to it. Stokes arrived
at the conclusion that they are parallel, while by a similar experiment Holtzman arrived at the
opposite conclusion." Lloyd, Wave Theory of Light, 1857.
271] THE VALUES OF a", \" , <T AND G BY EXPERIENCE. 249
Thus not only does the analysis of the granular medium account by purely
mechanical considerations for the phenomena of polarisation, but also removes
all doubt, if the explanation is mechanical, as to the fundamental necessity
that the motion in the medium that can be reflected must be in a plane
parallel to the plane of incidence.
The foregoing proof that that component of the motion of the medium
which is reflected is that parallel to the plane of incidence has been based on
the relaxation of the mean coefficient of rotational elasticity owing to the
presence of negative inequalities, as discussed in Art. 267. This was all
that was required, as the relaxation in translucent matter is comparatively
very small. When, however, we come to metallic reflection, which in the
case of mercury at perpendicular incidence is 0*666 as against 0"()018 for
water, it appears that the relaxation is altogether of another order than in
translucent substances.
In the mechanical medium such difference is accounted for by the
extremely small size of the singular surfaces, the radii of which are about
2 x 10~10 or 2 x 10-5 of the length of the shorter waves. These singular
surfaces as long as their arrangement is in open order will cause relaxation
which is small but which increases somewhat proportionally to the number of
such surfaces in unit space, each surface being, as it were, independent, so that
the abnormal pilings which embrace every grain will only meet at a few
points. But as the inequalities approach the closest order the rate of
decrease of the relaxation increases very rapidly until the normal piling
of the singular surface becomes nearly continuous. The surface of the space
enclosing the inequalities then becomes a singular surface of the aggregation
of inequalities outside of which the piling is abnormal.
To realise the evenness of such a boundary surface embracing the whole
or any part of the aggregate inequalities it is only necessary to remember
that the radii of the singular surfaces are less than one ten-thousandth of the
wave-length, whence the roughness which would be less than 1 x 10~9 cm. and
thus would be smoother than any artificial polish which can be imparted to
metal, and hence could only compare with the surface of mercury.
It is thus shown that the granular medium not only affords an explana-
tion of the polarisation of light but also affords an explanation of metallic
reflection. And these explanations being accomplished it appears that the
mechanical explanation of the rest of the phenomena of light must of
necessity follow.
271. The aberration of light admits of an explanation so simple and the
coincidence of the value of the velocity of light thence deduced with that
derived from the observations of the eclipses of Jupiter's satellites is so re-
markable as to leave no doubt in the mind as to the truth of the explanation.
250 ON THE SUB-MECHANICS OF THE UNIVERSE. [271
But when the aberration is subjected to closer examination the explana-
tion is found to rest on the heretofore unexplained absence of any resistance
to the motion of the ether through matter ; for notwithstanding the efforts
made to rest the explanation on another basis this has not been completely
accomplished.
The difficulties in conceiving the free motion of the ether through matter
do not present themselves in the analysis of the properties of the granular
medium as now accomplished. This follows from the analysis which has been
effected in this and the previous section.
It is shown : —
(1) That the motions of the singular surfaces are independent of the
mean-motion of the grains in the medium (Art. 233).
(2) That the institution of undulations depends on the varying strains
resulting from relative motion of the singular surfaces (Art. 264).
(3) That the energy of the wave is absorbed by the singular surfaces,
and that the energy thus absorbed is conducted and conveyed through the
aggregate singular surfaces (Art. 266).
Whence it follows that the singular surfaces which correspond to matter
are free to move in any direction through the medium without resistance, and
vice versa the medium is free to move in any direction through the singular
surfaces without resistance. And that the waves corresponding to those of
light are instituted and absorbed by the singular surfaces only. So that after
institution at the place where the singular surfaces are, the motion of the
waves depends solely on the mean motion of the medium, and the rate of
propagation is equal in all directions until they again come to singular
surfaces. Thus all paradox is removed and the explanation of aberration
is established on the basis of the absence of any appreciable resistance to
the medium in passing through matter.
Thus besides the explanations by definite analysis of:
the potential energy,
the propagation of transverse waves of light,
the apparent absence of any rate of degradation of light,
the lack of evidence of normal waves,
the gravitation of matter,
electricity,
which explanations render the purely mechanical substructure of the universe
indefinitely probable, we have by further analysis obtained : —
271] THE VALUES OF a", \", a AND G BY EXPERIENCE. 251
The explanation of the blackness of the sky on a clear night. (Art. 258.)
The definite proof of the fundamental dissipation of the energy of the
waves of light and the relative energy of the molecules to increase the mean
irreversible relative motion of the grains ; which dissipation is independent
of that which tends to the equalisation of the mean energy of the molecules.
(Art. 259.)
The number of grains, the displacement of which through a unit distance
represents the electrostatic unit. (Art. 260.)
The proof of the coincidences between the periods of vibration of the
molecules and the periods of the waves. (Art. 261.)
Proof that dissociation of compound molecules proves the previous state
to have been one of limited stability. (Art. 262.)
Proof that light is produced by the reversion of complex inequalities.
(Arts. 263—264.)
Proof that the reassociation of compound molecules results from the
reversion of complex inequalities. (Art. 265.)
Proof of the absorption of the energy of light by inequalities. (Art. 266.)
Proof that negative inequalities affect the waves passing through.
(Art. 267.)
Proof that refraction is caused by the vibrations of the inequalities having
the same periods as the waves. (Art. 268.)
Proof that dispersion results from the greater number of coincidences as
the waves get shorter. (Art. 269.)
Proof that the polarisation of light 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
inequalities through a surface into a space in which there are inequalities.
(Art. 270.)
Proof that metallic reflection results from the relative smallness of the
dimensions of the molecules compared with the wave-length, and the close-
ness of their piling, when the waves pass from a space without inequalities
across the surface beyond which the inequalities are in closest order.
(Art. 270.)
Proof that the aberration of light results from the absence of any
appreciable resistance to the motion of the medium when passing through
matter. (Art. 271.)
INDEX.
a", X", 0-, and G, determination of, 224 —
30
aberration of light, 249
angular inequalities in relative system,
redistribution of, 110 — 28
attraction, law of, 204 — 5
blackness of sky, 238
Clausius' explanation of redistribution
after encounters, 131
cohesion, 212
component systems of mean and relative
motion :
approximate equations of, 55 — 66
conditions for continuance of, 69 —
85
equations of continuity for, 32—41
conducting properties of the absolutely
rigid grain, 71 — 3, 87 — 8
conduction of momentum, 19—22
rates of, through the grains, 95 —
109
conservation of inequalities, 183 — 221
continuity of mass, 16
convection, 11
Crookes, Sir W., radiometer, 242
decrement of normal and transverse
waves, 179—82, 233
of inequalities, 78—9
density of medium, 234
dilatation, coefficients of, 186—92, 195—
200
discontinuity of the medium, 74—82
dispersion of mass, momentum, and energy
of relative motion, 131 — 44
dispersion of light, 246
distribution of velocities of grains result-
ing from encounters, 89 — 93
electricity, 209—12, 220, 239
equations for any entity, 9 — 13
of momentum and energy, 19 — 30
of continuity for component sys-
tems, 32—41
of motion for component systems,
55—66
for the mean system in terms of
a", X", o", G, 173
for the mean path of a grain,
96—7, 101—2
for rates of conduction through
the grains, 95 — 109
for redistribution of angular in-
equalities, 110 — 28
for linear dispersion of mass, mo-
mentum and energy, 131 — 44
for the exchanges between mean
and relative systems, 146 — 80
for the conservation of mean in-
equalities, 183—221
general integration of, 224—36
exchanges between mean and relative
systems, 146 — 80
fluid, reasons for rejecting the perfect,
71, 74
gases, effects neglected in kinetic theory
of, 113
grains, rigid, conducting properties of,
87—8
case of uniform spherical, 77
254
INDEX.
grains, points of contact of, 83
distribution of velocities- of, 89 — 93
mean path of, 95—8, 101—2
gravitation, law of, 204 — 5
Hooke's views of density of medium, 227
inequalities in the relative system, re-
distribution of, 110—28
in the mean mass, conservation of,
183—221
institution of, 146 — 70
negative, 204
positive, 207
complex, 207 — 11
Kelvin, Lord, views of density of medium,
227
determination of diameters of mole-
cules, 234
kinetic theory of gases, effects neglected
in, 113
light, production of, 234
aberration of, 249
refraction of, 245 — 6
dispersion of, 246
polarisation of, 246
magnetism, 213
mass, continuity of position of, 16
matter — the absence of mass, 221 — 2
Maxwell's theory of hard particles, 89
extension of, 93
Maxwell's law of redistribution of veloci-
ties, 89
Maxwell's theory of the stresses in the
medium, 206
mean and relative motions, 42 — 53
systems, exchanges between, 146 —
80
media, the only possible, 71 — 6
medium, meaning of purely mechanical,
14, 15
medium, density of purely mechanical,
88, 224, 234—6
discontinuity of, 74 — 82
characteristics of the state of,
134—5
diffusion in the, 79
pressure in, 189, 194 — 7
momentum, conduction of, 19 — 24
displacement of, in uniform medium,
104
normal waves, 231, 233
perfect fluid, 71, 74
piling of grains, 83
potential, 205
pressure in the medium, 189, 194 — 7
purely mechanical medium, definition and
necessary properties of, 15
Rankine's expressions for intensities of
component stresses, 22
" Outlines of Energetics "—the pas-
sive and active complex accidents,
111
redistribution of angular inequalities,
110—28
redistribution of mass &c, by convection
and conduction, 131 — 44
Rontgen rays, 232, 236
singular surfaces, 214 — 15, 216 — 18
Stokes, Sir G., views of density of
medium, 227
angular distribution of the grains,
150
surface tension, 212
surfaces, boundary, 24
velocity of grains resulting from encoun-
ters, 89—94
waves, normal and transverse, 231 — 3
CAMBRIDGE: PRINTED BY J. AND C. F. CLAY, AT THE UNIVERSITY PRESS.
QB Reynolds, Osborne
351 The sub-mechanics of
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