578 The Second Law of Thermodynamics. 



And now we obtain a solution in the form 



f =*0tt} 



•u 



/-♦$> 



where <£ and ty denote any functions. And since F and/ must 

 vanish for all infinite values of the variables, we are led to 



— \ 



U+r 



F = Ce t , 



where X is some positive numerical quantity, and C, C are 

 found from the conditions 



G$Fd<rd(r' = l, 



G'$fd<r =1. 



13. But these are the same conditions which make the 

 motion stationary withT, V\ . . . v r constant. This has been satis- 

 factorily proved for the case where the mutual action between 

 the molecules is restricted to u binary encounters." (Watson's 

 Kinetic Theory of Gases, second edition, p. 36.) 



In the January number of this Magazine I have endeavoured 

 to show that the same form of F gives stationary motion if 

 we remove this restriction on mutual action. 



14. If the necessity for making -=- a complete differential 



can be established as a substantive law by independent evi- 

 dence, the above investigation affords an independent proof 

 of the distribution 



U + T 



for such a system as above described, and that whether the 

 mutual action is restricted to binary encounters or not. 



15. I have assumed that the continued products of differen- 

 tials do- and da do not vary. If they do vary, that is in effect 



if the limits of integration vary, the assumption F = </>( — ™ — ] 

 will still make -^ a complete differential. 



