THE CANONICAL DISTRIBUTION. 107 



whence 



This is of the order of magnitude of n.* 



It should be observed that the approximate distribution of 

 the ensemble in energy according to the 'law of errors' is 

 not dependent on the particular form of the function of the 

 energy which we have assumed for the index of probability 

 (77). In any case, we must have 



(351) 



where e^+t is necessarily positive. This requires that it 

 shall vanish for e = oo , and also for e = oo , if this is a possi- 

 ble value. It has been shown in the last chapter that if e has 

 a (finite) least possible value (which is the usual case) and 

 n > 2, e* will vanish for that least value of e. In general 

 therefore 77 + < will have a maximum, which determines the 

 most probable value of the energy. If we denote this value 

 by e > and distinguish the corresponding values of the func- 

 tions of the energy by the same suffix, we shall have 



- a 



The probability that an unspecified system of the ensemble 



* We shall find hereafter that the equation 



is exact for any value of n greater than 2, and that the equation 



fd(f> IV __ <^0 

 \d* ) ' rf? 

 is exact for any value of n greater than 4. 



