MAXIMUM AND MINIMUM PROPERTIES. 133 



Theorem V. If an ensemble of systems is so distributed 

 in phase that the index of probability is a linear function of 

 F v F v etc., (these letters denoting functions of the phase,) the 

 average value of the index is less than for any other distribu- 

 tion in which the functions F v F^ etc. have the same average 

 values. 



Theorem VI. The average value in an ensemble of systems 

 of 77 + F (where 77 denotes as usual the index of probability and 

 F any function of the phase) is less when the ensemble is so 

 distributed that 77 + F is constant than for any other distribu- 

 tion whatever. 



Theorem VII. If a system which in its different phases 

 constitutes an ensemble consists of two parts, and we consider 

 the average index of probability for the whole system, and 

 also the average indices for each of the parts taken separately, 

 the sum of the average indices for the parts will be either less 

 than the average index for the whole system, or equal to it, 

 but cannot be greater. The limiting case of equality occurs 

 when the distribution in phase of each part is independent of 

 that of the other, and only in this case. 



Let the coordinates and momenta of the whole system be 



2l ZifiPl > -Pni O f Wnicl1 ft ' <lm Pi , -Pm relate to ne 



part of the system, and q m+l ,...<?, p m+l , . . . p n to the other. 

 If the index of probability for the whole system is denoted by 

 77, the probability that the phase of an unspecified system lies 

 within any given limits is expressed by the integral 



f. . .fe*d Pl ...dq, (431) 



taken for those limits. If we set 



J . . .fa dp m+l . . . dp n dq^ ...dq n =. e\ (432) 



where the integrations cover all phases of the second system, 

 and 



J. . . JV d Pl . . . dp m d qi ... dq m = e^ (433) 



