CHAPTER XI. 



MAXIMUM AND MINIMUM PROPERTIES OF VARIOUS DIS- 

 TRIBUTIONS IN PHASE. 



IN the following theorems we suppose, as always, that the 

 systems forming an ensemble are identical in nature and in 

 the values of the external coordinates, which are here regarded 

 as constants. 



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

 phase that the index of probability is a function of the energy, 

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

 bution in which the distribution in energy is unaltered. 



Let us write TJ for the index which is a function of the 

 energy, and 77 + A?? for any other which gives the same dis- 

 tribution in energy. It is to be proved that 



all all 



J*. . . J* (i, + Ar,) e"** 1 d Pl ... dq n >f. . . Jr? 6* dp,... dq n , (419) 



pliases phases 



where ?? is a function of the energy, and A?; a function of the 

 phase, which are subject to the conditions that 



all all 



J. . . Je^ 4 " dp,... dq n = f. . . J> d&... dy n = 1, (420) 



phases phases 



and that for any value of the energy (e') 



dp,... dq n =. . .fdpi ...dq n . (421) 



Equation (420) expresses the general relations which -77 and 

 77 + AT; must satisfy in order to be indices of any distributions, 

 and (421) expresses the condition that they give the same 

 distribution in energy. 



