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

${\overline {u}}|_{\epsilon }={\frac {d\epsilon \int \limits _{V_{q}=0}^{\epsilon _{q}=\epsilon }u\,e^{\phi _{p}}\,dV_{q}}{d\epsilon \int \limits _{V_{q}=0}^{\epsilon _{q}=\epsilon }e^{\phi _{p}}\,dV_{q}}}$

But by (299) the value of the integral in the denominator is $e^{\phi }$. We have therefore

${\overline {u}}|_{\epsilon }=e^{-\phi }\int \limits _{V_{q}=0}^{\epsilon _{q}=\epsilon }u\,e^{\phi _{p}}\,dV_{q}$

(374)

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.^[1] 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.^[2]

1. See equation (322).
2. 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