136 MAXIMUM AND MINIMUM PROPERTIES. 



of the probability-coefficients of the original ensembles, the 

 average index of probability of the resulting ensemble cannot 

 be greater than the same linear function of the average indices 

 of the original ensembles. It can be equal to it only when 

 the original ensembles are similarly distributed in phase. 



Let PijP%, etc. be the probability-coefficients of the original 

 ensembles, and P that of the ensemble formed by combining 

 them ; and let N^ , -ZV^ , etc. be the numbers of systems in the 

 original ensembles. It is evident that we shall have 



P = e l P l + c 2 P 2 + etc. = 2 (cjPi), (445) 



where Ci = =-^V> c 2 = ^, etc. (446) 



The main proposition to be proved is that 



all all 



/ ./P log Pd Pl . . . <*? ^ s pi/ -/P, log P, ^ . . . dfcTI 



phases L phases - 



(447) 



all 



f . . . f [2 (c l P l log PO - P log P] d Pl ... dq n > 0. (448) 

 J J 



or 



J 



phases 



If we set 



ft = P! log P! - P! log P - P! + P 



Q 1 will be positive, except when it vanishes for P 1 = P. To 

 prove this, we may regard P l and P as any positive quantities. 

 Then 



\dPi*J P PI ' 



Since Q 1 and dQ 1 /dP 1 vanish for P l P, and the second 

 differential coefficient is always positive, Q 1 must be positive 

 except when P 1 = P. Therefore, if # 2 , etc. have similar 

 definitions, 



2 fa ft) ^ 0. (449) 



