130 MAXIMUM AND MINIMUM PROPERTIES. 



Since 77 is a function of the energy, and may therefore be re- 

 garded as a constant within the limits of integration of (421), 

 we may multiply by T; under the integral sign hi both mem- 

 bers, which gives 



C 



J. 



71 dp^ . . . dq n . 



U U \J 



=' e' 



Since this is true within the limits indicated, and for every 

 value of e', it will be true if the integrals are taken for all 

 phases. We may therefore cancel the corresponding parts of 

 (419), which gives 



all 



f A r, e 1 ** 11 d Pl ... dq n > 0. (422) 



J 



phases 



But by (420) this is equivalent to 



all 



/. . . / (Ar;e Al7 + 1 e^e'dpi . . . dq n > 0. (423) 

 tj 



phases 



Now AT; e^ + 1 e^ is a decreasing function of AT; for nega- 

 tive values of AT;, and an increasing function of AT; for positive 

 values of AT;. It vanishes for AT; = 0. The expression is 

 therefore incapable of a negative value, and can have the value 

 only for AT; = 0. The inequality (423) will hold therefore 

 unless AT; = for all phases. The theorem is therefore 

 proved. 



Theorem II. If an ensemble of systems is canonically dis- 

 tributed in phase, the average index of probability is less than 

 in any other distribution of the ensemble having the same 

 average energy. 



For the canonical distribution let the index be (^ e) / , 

 and for another having the same average energy let the index 

 be (t/r e)/0 + AT;, where AT; is an arbitrary function of the 

 phase subject only to the limitation involved in the notion of 

 the index, that 



