88 CERTAIN IMPORTANT FUNCTIONS 



function of 6, which becomes infinite with e, and vanishes 

 for the smallest possible value of e, or f or e = oo, if the 

 energy may be diminished without limit. 

 Let us also set 



dV 

 <f> = log (266) 



The extension in phase between any two limits of energy, ^ 

 and e", will be represented by the integral 



/ de. (267) 



And in general, we may substitute e* de for dp l . . . dq n in a 

 2tt-fold integral, reducing it to a simple integral, whenever 

 the limits can be expressed by the energy alone, and the other 

 factor under the integral sign is a function of the energy alone, 

 or with quantities which are constant in the integration. 



In particular we observe that the probability that the energy 

 of an unspecified system of a canonical ensemble lies between 

 the limits e' and e" will be represented by the integral * 



* ffe, (268) 



and that the average value in the ensemble of any quantity 

 which only varies with the energy is given by the equation j 



(269) 



where we may regard the constant *fy as determined by the 

 equation $ 



^ 

 =l 



6=00 





 & 



e de, (270) 



F=0 



In regard to the lower limit in these integrals, it will be ob- 

 served that V= is equivalent to the condition that the 

 value of e is the least possible. 



* Compare equation (93). t Compare equation (108). 



J Compare equation (92). 



