76 REPORT— 1894. 



that if there are a very large number of systems exactly like the one de- 

 [icribecl, and if the proportion of these systems in any given phase is 

 measured by an expression of the form 



f{x + iu'^ + b^'-)di('du' dxdx' . . . (17) 



the same law of distribution will hold good at any subsequent time. Also 

 the mean kinetic energies with which the two particles pass simultaneously 

 through two given configurations are 



\[/{x + W + lu'^) \'»^du du' [/(x + -2«' + ^^'") \^<''^du du' 

 JJ and JJ 



I f/(x + \ '^' + * w'^) du du' f/(x + W + \u"^) du du' 



and are therefore equal. 



28. The test case derives an additional interest because it forms a sort 

 of transition between those systems in which the form of f is indeter- 

 minate and the systems of colliding bodies with which we have to deal in 

 the Kinetic Theory of Gases where /=e~''^. 



For in general (17) will represent a law according to which the dis- 

 tribution of the A particles depends on the energies of the coi-responding 

 C particles in the systems to which they belong. If, however, the dis- 

 tribution of the A particles is independent of that of the C particles, their 

 separate laws of distribution being 



fi^du dx a,nd /edit,' dx', 

 then we must have 



Axfo=fix + W+hu'^-h 



the most general solution of which assumes the well-known form 



f^=^nu.m /c=e-"-5«'^ . . _. . (18) 

 h being any constant whatever. In that case the mean kinetic energy of 

 all the A's in the neighbourhood of a given point irrespective of the corre- 

 -sponding position of the C's is 



e-''-i-'{^tt^)du 



=i^ =^^^ . . . (19) 



f»oo 



e-'''-"'du 



as in the Kinetic Theory of Gases. This accoi-ds exactly with the re- 

 marks of Mr. Culverwell already mentioned. The A's do not all reach K 

 every time, those that are moving slowly only penetrating a small distance 

 into the region KH, and the depth of penetration increasing with the 

 velocity at H or B. Hence the density of distribution of the A's diminishes 

 ■as we approach K, but out of the whole number at any point the proj)or- 

 tiofi having kinetic energies within certain limits will be the same every- 

 where. 



Summary. 



29. The conclusions so far arrived at may be summed up as follows : — • 

 (i) If there exist a very (infinitely) great number of independent conser- 

 vative dynamical systems, the equations of motion of each system having 



