MAXIMUM AND MINIMUM PROPERTIES. 135 



It appears from the definitions of ^ and 7? 2 that (436) may 

 also be written 



f . . . Cru e n dp l ...dq n + J". . . J ^ e^dp l ...dq n < 



f... fa <&..<<%., (440) 



or f . . . f 0? - >?i - in)***! . . . dq n > 0, 



where the integrations cover all phases. Adding the equation 



... <?<?* = 1, (441) 



f . 



a 



f. . . C 



which we get by multiplying (438) and (439), and subtract- 

 ing (437), we have for the proposition to be proved 



all 



J. . .J[(, - % - Tfc) J + ** - e"] <$* . . . dq n > 0. (442) 



phases 



Let 



U = r 1 r }1 r ]2 . (443) 



The main proposition to be proved may be written 



all 



n > 0. (444) 



phases 



This is evidently true since the quantity in the parenthesis is 

 incapable of a negative value.* Moreover the sign = can 

 hold only when the quantity in the parenthesis vanishes for 

 all phases, i. e., when u = for all phases. This makes 

 i) = tj l + ?7 2 for all phases, which is the analytical condition 

 which expresses that the distributions in phase of the two 

 parts of the system are independent. 



Theorem VIII. If two or more ensembles of systems which 

 are identical in nature, but may be distributed differently in 

 phase, are united to form a single ensemble, so that the prob- 

 ability-coefficient of the resulting ensemble is a linear function 



* See Theorem I, where this is proved of a similar expression. 



