THE CANONICAL DISTRIBUTION. Ill 



where ty r denotes the value of ^ for the modulus '. Since 

 the last member of this formula vanishes for e = oo , the 

 less value represented by the first member must also vanish 

 for the same value of e. Therefore the second member of 

 (359), which differs only by a constant factor, vanishes at 

 the upper limit. The case of the lower limit remains to be 

 considered. Now 



The second member of this formula evidently vanishes for 

 the value of e, which gives V 0, whether this be finite or 

 negative infinity. Therefore, the second member of (359) 

 vanishes at the lower limit also, and we have 



V 



or e V=. (362) 



This equation, which is subject to no restriction in regard to 

 the value of n, suggests a connection or analogy between the 

 function of the energy of a system which is represented by 

 iT^ V and the notion of temperature in thermodynamics. We 

 shall return to this subject in Chapter XIV. 



If n > 2, the second member of (359) may easily be shown 

 to vanish for any of the following values of u viz. : </>, e^, e, 

 e"*, where m denotes any positive number. It will also 

 vanish, when n > 4, for u = dfyde, and when n > 2 h for 

 u = e-* d h V/d^. When the second member of (359) van- 

 ishes, and n > 2, we may write 



We thus obtain the following equations : 

 If n > 2, 



(364) 



