94 CERTAIN IMPORTANT FUNCTIONS 



falls within given limits is independent also of the value of 

 the modulus, being determined entirely by the number of 

 degrees of freedom of the system and the limiting values 

 of the ratio. 



The average value of any function of the kinetic energy, 

 either for the whole ensemble, or for any particular configura- 

 tion, is given by 



p 



?-i 

 ue e,2 de p *(291) 



Thus: 



^"' if m + ^>> t(292) 



* The corresponding equation for the average value of any function of 

 the potential energy, when this is a quadratic function of the ^'s, and A is 

 independent of the q's, is 



In the same case, the average value of any function of the (total) energy is 

 given by the equation 



Hence in this case 





j . f m + n>0- 



and = , if 



ii f vy 



If n = 1, e* = 2 ir and d^jde = for any value of e. If n = 2, the case is 

 the same with respect to 2 . 



t This equation has already been proved for positive integral powers of 

 the kinetic energy. See page 77. 



