Page:Elementary Principles in Statistical Mechanics (1902).djvu/142

118 But by (299) the value of the integral in the denominator is $$e^\phi$$. We have therefore where $$e^{\phi_p}$$ and $$V_q$$ are connected by equation (373), and $$u$$, if given as function of $$\epsilon_p$$, or of $$\epsilon_p$$ and $$\epsilon_q$$, becomes in virtue of the same equation a function of $$\epsilon_q$$ alone.

We shall assume that $$e^\phi$$ has a finite value. If $$n>1$$, it is evident from equation (305) that $$e^\phi$$ is an increasing function of $$\epsilon$$, and therefore cannot be infinite for one value of $$\epsilon$$ without being infinite for all greater values of $$\epsilon$$, which would make $$-\psi$$ infinite. When $$n>1$$, therefore, if we assume that $$e^\phi$$ 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 distribution is perfectly applicable, but in which the formulae for the microcanonical distribution become illusory, for particular values of $$\epsilon$$, on account of the infinite value of $$e^\phi$$. Such failing cases of the microcanonical distribution for particular values of the energy will not prevent us from regarding the canonical ensemble as consisting of an infinity of microcanonical ensembles.