MAXIMUM AND MINIMUM PROPERTIES. 131 



all itr f a 11 J' 



/(* + AIJ r r 



. . .J e* d Pl . . . dq n =J . . .J e & d Pl . . . dq n = 1, 



phases phases 



(424) 

 and to that relating to the constant average energy, that 



all f all 



J. . . Je e"^" 4 * 4,, . . . <*? =J . . . Je e~e~ fe . . . <*?.. (425) 



phases phases 



It is to be proved that 



phases 



all 



phases 



Now in virtue of the first condition (424) we may cancel the 

 constant term ^r / in the parentheses in (426), and in virtue 

 of the second condition (425) we may cancel the term e/O. 

 The proposition to be proved is thus reduced to 



all ty~ 



I A>7 e & dpi . . . dq n > 0, 



phases 



which may be written, in virtue of the condition (424), 



all if/ e 



f. . . f (Ar; e Al? + 1 - /") e~ dpi... dq n > 0. (427) 

 J J 



phases 



In this form its truth is evident for the same reasons which 

 applied to (423). 



Theorem III. If is any positive constant, the average 

 value in an ensemble of the expression 77 -|- e / (77 denoting 

 as usual the index of probability and e the energy) is less when 

 the ensemble is distributed canonically with modulus , than 

 for any other distribution whatever. 



In accordance with our usual notation let us write 

 (i/r e) / for the index of the canonical distribution. In any 

 other distribution let the index be (>/r e)/ + AT;. 



