132 MAXIMUM AND MINIMUM PROPERTIES. 



In the canonical ensemble rj + e / has the constant value 

 -|r / <s) ; in the other ensemble it has the value A/T / -f- A?/. 

 The proposition to be proved may therefore be written 



all 



phases 



where 



r/- ^ 

 d Pl ...dq n =J...Je e d Pl ...d<i, = l. (429) 



phases phases 



In virtue of this condition, since i/r / is constant, the propo- 

 sition to be proved reduces to 



all j-t 



// ^r + A f 

 ...J A^6 < cZq l ...dp n , (430) 



phases 



where the demonstration may be concluded as in the last 

 theorem. 



If we should substitute for the energy in the preceding 

 theorems any other function of the phase, the theorems, mu- 

 tatis mutandis, would still hold. On account of the unique 

 importance of the energy as a function of the phase, the theo- 

 rems as given are especially worthy of notice. When the case 

 is such that other functions of the phase have important 

 properties relating to statistical equilibrium, as described 

 in Chapter IV,* the three following theorems, which are 

 generalizations of the preceding, may be useful. It will be 

 sufficient to give them without demonstration, as the principles 

 involved are in no respect different. 



Theorem IV. If an ensemble of systems is so distributed in 

 phase that the index of probability is any function of F v JP 2 , 

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

 value of the index is less than for any other distribution in 

 phase in which the distribution with respect to the functions 

 F v F v etc. is unchanged. 



* See pages 37-41. 



