54 AVERAGE VALUES IN A CANONICAL 



the fractional part of the ensemble which lies within any 

 given limits of configuration (136) may be written 



dq l . . . dq n , (142) 



where the constant ty q may be determined by the condition 

 that the integral extended over all configurations has the value 

 unity.* 



* In the simple but important case in which Aj is independent of the ^'s, 

 and j a quadratic function of the q's, if we write e a for the least value of q 

 (or of e) consistent with the given values of the external coordinates, the 

 equation determining \l/ q may be written 



00 00 



If we denote by q t . . . q n ' the values of qi , . . . q n which give f q its least value 

 e a , it is evident that e g e a is a homogenous quadratic function of the differ- 

 ences ?! qi, etc., and that dq t , . . . dq n may be regarded as the differentials 

 of these differences. The evaluation of this integral is therefore analytically 

 similar to that of the integral 



+00 +00_J 



J. . .fe & dp! . . . dp n , 



00 CO 



for which we have found the value A p * (2 TT 9) 3. By the same method, or 

 by analogy, we get 



where A 9 is the Hessian of the potential energy as function of the q's. It 

 will be observed that A ? depends on the forces of the system and is independ- 

 ent of the masses, while A^ or its reciprocal A p depends on the masses and 

 is independent of the forces. While each Hessian depends on the system of 

 coordinates employed, the ratio A^/A^ is the same for all systems. 

 Multiplying the last equation by (140), we have 



For the average value of the potential energy, we have 



+00 +00 *g~ e a 



J ' ' -f ( Q f a)e dq l . . . dq n 



00 00 



+00 +eo * a 



J . . .J e dqi . . . dq n 



