ALL SYSTEMS HAVE THE SAME ENERGY. 125 



Let us imagine an ensemble of systems distributed in phase 

 according to the index of probability 



(e - c'V 



where e f is any constant which is a possible value of the 

 energy, except only the least value which is consistent with 

 the values of the external coordinates, and c and o> are other 

 constants. We have therefore 



all 



c 



e, w dp l . . . dq n 1, (403) 



phases 



or e =...e d Pl . . . dq n , (404) 



phases 



_ c | g 



or again e = C e ^ de. (405) 



From (404) we have 



all 



phases 



= 00 



, j 



^ (406) 



where H7i e denotes the average value of A 1 in those systems 

 of the ensemble which have any same energy e. (This 

 is the same thing as the average value of A l in a microcanoni- 

 cal ensemble of energy e.) The validity of the transformation 

 is evident, if we consider separately the part of each integral 

 which lies between two infimtesimally differing limits of 

 energy. Integrating by parts, we get 



