158 On the Law of Distribution of Energy. 



parameters v 1} t? a , &c, on which a l9 6 12 , &c, depend. The 

 second law may from one point of view be regarded as the 

 law of the variation of B when T and the parameters vary 

 very slowly, so that stationary motion is always attained. On 

 this assumption the proofs of the second law depend. 



We have seen that S = T and is independent of the para- 



7 7(^( 



meters, or — S =0 for each v. But -j- is not generally zero. 



It may, therefore, be the case that work has to be done on 

 variation of any parameter v. This work, will be denoted by 



— -=- "dv. It will include the work done against all external 



forces. The energy imparted during any small variation of 

 T and the parameters will be denoted by dQ. Then the 



second law requires that -jr shall be a complete differential. 



(18) It will be sufficient to prove the law for any para- 

 meter v on which a ly b 12 , &c. depend. So far as this proof is 

 concerned, there may be many such. 



We have in this case 



BQ = BT-^-^ 



~° ° V n \chV~ + ~dv~ ~W + '"J 

 "° ° V n 2\dv Ddcti dv T)db 12 + •••) 



^ T .. Tl dT> 



n udv 



c)Q_^i_. rr 1 



n 



Now in this case 



^ = BlogT-^logD. 



9B = blog|x/D(^J. 





