ENSEMBLE OF SYSTEMS. 55 



When an ensemble of systems is distributed in configura- 

 tion in the manner indicated in this formula, i. e., when its 

 distribution in configuration is the same as that of an en- 

 semble canonically distributed in phase, we shall say, without 

 any reference to its velocities, that it is canonically distributed 

 in configuration. 



For any given configuration, the fractional part of the 

 systems which lie within any given limits of velocity is 

 represented by the quotient of the multiple integral 



d Pl ...dp n , 

 or its equivalent 



- 



l-- 





taken within those limits divided by the value of the same 

 integral for the limits oo. But the value, of the second 

 multiple integral for the limits oo is evidently 



We may therefore write 



~~~ du^ . . . du n , (143) 



The evaluation of this expression is similar to that of 



+00 +00 _!? 



...s f e & dp l ...dp n 



+00 +00 _ C JL 

 f...fe & d Pl ...dp n 



- 00 - CO 



which expresses the average value of the kinetic energy, and for which we 

 have found the value $ n 6. We have accordingly 



4-a = 2 na 

 Adding the equation 



*i> = 2 ne > 

 we have I e a = n e. 



