110 THE FUNCTION AND 



The impossibility of a canonical distribution occurs when 

 the equation 



e e 



e = e 



s* l-j-0 



=J e ' de (361) 



F=0 



fails to determine a finite value for ^. Evidently the equation 

 cannot make ty an infinite positive quantity, the impossibility 

 therefore occurs when the equation makes ty = oo . Now 

 we get easily from (191) 



If the canonical distribution is possible for any values of , 

 we can apply this equation so long as the canonical distribu- 

 tion is possible. The equation shows that as is increased 

 (without becoming infinite) ty cannot become infinite unless 

 6 simultaneously becomes infinite, and that as O is decreased 

 (without becoming zero) ^ cannot become infinite unless 

 simultaneously e becomes an infinite negative quantity. The 

 corresponding cases in thermodynamics would be bodies which 

 could absorb or give out an infinite amount of heat without 

 passing certain limits of temperature, when no external work 

 is done in the positive or negative sense. Such infinite values 

 present no analytical difficulties, and do not contradict the 

 general laws of mechanics or of thermodynamics, but they 

 are quite foreign to our ordinary experience of nature. In 

 excluding such cases (which are certainly not entirely devoid 

 of interest) we do not exclude any which are analogous to 

 any actual cases in thermodynamics. 



We assume then that for any finite value of the second 

 member of (361) has a finite value. 



When this condition is fulfilled, the second member of 

 (359) will vanish for u = e~+ V. For, if we set 6' = 26, 



? ___! _ f _ ^ 



F = V = 



