134 MAXIMUM AND MINIMUM PROPERTIES. 



where the integrations cover all phases of the first system, 

 the integral (431) will reduce to the form 



f . . . 



dp!... dp m d^ . . . dq m) (434) 



when the limits can be expressed in terms of the coordinates 

 and momenta of the first part of the system. The same integral 

 will reduce to 



J . . . J (?* dp m+l ...dp n dq m+1 ...dq r 



(435) 



when the limits can be expressed in terms of the coordinates 

 and momenta of the second part of the system. It is evident 

 that rj 1 and r) 2 are the indices of probability for the two parts 

 of the system taken separately. 



The main proposition to be proved may be written 



f 



(436) 



where the first integral is to be taken over all phases of the first 

 part of the system, the second integral over all phases of the 

 second part of the system, and the last integral over all phases 

 of the whole system. Now we have 



..%. = !, (437) 



..dq m = l t (438) 



and * 



where the limits cover in each case all the phases to which the 

 variables relate. The two last equations, which are in them- 

 selves evident, may be derived by partial integration from the 

 first. 



J*. . .Je^dp m+l ...dq n = l, (439) 



