THE CANONICAL DISTRIBUTION. 109 



Now we have identically 



Therefore, by the preceding equation 



If we set u = 1, (a value which need not be excluded,) the 

 second member of this equation vanishes, as shown on page 

 101, if n > 2, and we get 



^ = i, (360) 



as before. It is evident from the same considerations that the 

 second member of (359) will always vanish if n > 2, unless u 

 becomes infinite at one of the limits, in which case a more care- 

 ful examination of the value of the expression will be necessary. 

 To facilitate the discussion of such cases, it will be convenient 

 to introduce a certain limitation in regard to the nature of the 

 system considered. We have necessarily supposed, in all our 

 treatment of systems canonically distributed, that the system 

 considered was such as to be capable of the canonical distri- 

 bution with the given value of the modulus. We shall now 

 suppose that the system is such as to be capable of a canonical 

 distribution with any (finite) f modulus. Let us see what 

 cases we exclude by this last limitation. 



* A more general equation, which is not limited to ensembles canonically 

 distributed, is 



^ + M ^4. M ^- \ue f *~\* = * > 

 df U de U de ~ I"* J F=0 



where t\ denotes, as usual, the index of probability of phase. 



t The term finite applied to the modulus is intended to exclude the value 

 zero as well as infinity. 



