MAXIMUM AND MINIMUM PROPERTIES. 137 



But since . 2 (c x P x ) = P 



and 2 <?i = 1, 



2 fa ft) = 2 fa P! log P x ) - P log P. (450) 



This proves (448), and shows that the sign = will hold only 



when 



P 1 = P, P 2 = P, etc. 



for all phases, i. e., only when the distribution in phase of the 

 original ensembles are all identical. 



Theorem IX. A uniform distribution of a given number of 

 systems within given limits of phase gives a less average index 

 of probability of phase than any other distribution. 



Let 77 be the constant index of the uniform distribution, and 

 T? + A?; the index of some other distribution. Since the num- 

 ber of systems within the given limits is the same in the two 

 distributions we have 



J. . . Je"+ A * dp,... dq n = J. . . J> dp, . . . dq n , (451) 



where the integrations, like those which follow, are to be 

 taken within the given limits. The proposition to be proved 

 may be written 



Pl ... dq n > . . . ,; Jd Pl . . . dq n , (452) 



or, since 77 is constant, 



l ...dq n >. . .rjdp! . . . dq n . (453) 



In (451) also we may cancel the constant factor e^, and multiply 

 by the constant factor (rj + 1). This gives 



f. . . 



The subtraction of this equation will not alter the inequality 

 to be proved, which may therefore be written 



/. . ./(A, - 1) /" d Pl ... dj. >/. . ./- cfc . . . dj. 



