576 Mr. S. H. Burbury on the 



7. With this notation, if the bar now denotes mean values 

 for the whole system, 



U=jJ UFda d<r'=§ Ufda If do* 



Similarly, 



Also 



dv J J dv 

 dv JJ< 



dr 



t F da do'. 



dv 



And if we denote by T the mean kinetic energy of a molecule 

 for the whole system, 



T=JJtF da-da". 



8. We may now suppose v ± . . . v r to vary in any manner 

 for all the molecules simultaneously. Further we may, with 

 or without variation of i\. . . v r , make T vary. Now r is a 

 quadratic function of d\ . . . x n , with coefficients which may 

 be functions of v lt ..v r as well as of x x . . . x n . And there- 

 fore when v 1 . . . v r vary, T, or \ \ t F da da', will generally 

 vary. But if v x . . . v r be constant, T can only vary by the 

 variation of F. We might suppose F a function of a variable 

 6 not contained in t or U. Then with v x . . . v 7 constant, 



^T^eWT^dada'. 



And when v 1 . . . v r also vary, the whole variation of T is 

 BT= {{$^-dvFdada'+ ^r^-dvdada' ' 



+ 1 1 t -jg'bO da da' ', 

 = t~-dv+^r'dFdada f , 



or 



d 



t^-dv=-dT-$T-dFdada'. . . . (I.) 



