in connexion with the Kinetic Theory of Gases. ' 63 



place in % except as the equivalent of work done by or against 

 the forces on change of position of the particles. Then SQ 

 consists (1) of alteration of kinetic energy — that is, 



S (S) 0r -|p 8/l; 

 (2) of alteration 8% of % ; (3) of external work done, TSv — that 



is, 



We may here observe that 



SQ-P8,= -gs/ 4 + Sx, 



which is an exact differential of a function of two variables, h 

 and v. This proposition is proved in a different way by Ran- 

 kine (see ' The Steam-Engine/ eighth edition, pp. 304-313). 

 Again, 



^=- r aA-i-yS % +yPo>; ... (A) 



and it is required to prove that this expression is an exact dif- 

 ferential. 



III. Consider the expression 



in which the integration extends over all configurations ; and 

 let 



log f J f . . . e h * dx 1 dy x . . . dz K = u. 



Then 8u denotes the whole change which takes place in u con- 

 sequent on h becoming h + SA, and v becoming v + $v ; that is, 



5> du <v 7 , du ^ 

 ou= -^011+ -j—bv. 

 an av 



Now for any given configuration the ergal % is not altered by 

 alteration of h, though the comparative frequency of the occur- 

 rence of such configuration is altered ; therefore 



d, u J g.-.X^cto,^.. ^ - 



& J Jj . . . e h * dx x dy x ... dz K X 



We have now to find -*- Bv. Both in the original and in 



the altered volume the integration is to be extended over all 

 possible configurations. Now for every configuration M in 

 the original volume there is in the altered volume a correspond- 

 ing configuration, which we may call M'; and M 7 is to be 



