118 A PERMANENT DISTRIBUTION IN WHICH 



de\ue Vp dV q 



v q =o 



But by (299) the value of the integral in the denominator 

 is e^. We have therefore 



(374) 



where e^ p and V q are connected by equation (373), and w, if 

 given as function of e p , or of e p and e q , becomes in virtue of 

 the same equation a function of e q alone. 



We shall assume that e^ has a finite value. If n > 1, it is 

 evident from equation (305) that e^ is an increasing function 

 of e, and therefore cannot be infinite for one value of e without 

 being infinite for all greater values of e, which would make 

 ty infinite.* When n > 1, therefore, if we assume that e^ 

 is finite, we only exclude such cases as we found necessary 

 to exclude in the study of the canonical distribution. But 

 when n = 1, cases may occur in which the canonical distribu- 

 tion is perfectly applicable, but in which the formulae for the 

 microcanonical distribution become illusory, for particular val- 

 ues of e, on account of the infinite value of e^. Such failing 

 cases of the microcanonical distribution for particular values 

 of the energy will not prevent us from regarding the canon- 

 ical ensemble as consisting of an infinity of microcanonical 

 ensembles, f 



* See equation (322). 



t An example of the failing case of the microcanonical distribution is 

 afforded by a material point, under the influence of gravity, and constrained 

 to remain in a vertical circle. The failing case occurs when the energy is 

 just sufficient to carry the material point to the highest point of the circle. 



It will be observed that the difficulty is inherent in the nature of the case, 

 and is quite independent of the mathematical formulae. The nature of the 

 difficulty is at once apparent if we try to distribute a finite number of 



