Page:Elementary Principles in Statistical Mechanics (1902).djvu/113

In like manner, let us denote by ${\displaystyle V_{q}}$ the extension-in-configuration below a certain limit of potential energy which we may call ${\displaystyle \epsilon _{q}}$. That is, let

${\displaystyle V_{q}=\int \ldots \int \Delta _{\dot {q}}^{\frac {1}{2}}dq_{1}\ldots dq_{n},}$

(271)

the integration being extended (with constant values of the external coördinates) over all configurations for which the potential energy is less than ${\displaystyle \epsilon _{q}}$. ${\displaystyle V_{q}}$ will be a function of ${\displaystyle \epsilon _{q}}$ with the external coördinates, an increasing function of ${\displaystyle \epsilon _{q}}$, which does not become infinite (in such cases as we shall consider^[1]) for any finite value of ${\displaystyle \epsilon _{q}}$. It vanishes for the least possible value of ${\displaystyle \epsilon _{q}}$, or for ${\displaystyle \epsilon _{q}=-\infty }$, if ${\displaystyle \epsilon _{q}}$ can be diminished without limit. It is not always a continuous function of ${\displaystyle \epsilon _{q}}$. In fact, if there is a finite extension-in-configuration of constant potential energy, the corresponding value of ${\displaystyle V_{q}}$ will not include that extension-in-configuration, but if ${\displaystyle \epsilon _{q}}$ be increased infinitesimally, the corresponding value of ${\displaystyle V_{q}}$ will be increased by that finite extension-in-configuration.

Let us also set

${\displaystyle \phi _{q}=\log {\frac {dV_{q}}{d\epsilon _{q}}}.}$

(272)

The extension-in-configuration between any two limits of potential energy ${\displaystyle \epsilon _{q}'}$ and ${\displaystyle \epsilon _{q}''}$ may be represented by the integral

${\displaystyle \int _{\epsilon _{q}'}^{\epsilon _{q}''}e^{\phi _{q}}\,d\epsilon _{q}}$

(273)

whenever there is no discontinuity in the value of ${\displaystyle V_{q}}$ as function of ${\displaystyle \epsilon _{q}}$ between or at those limits, that is, whenever there is no finite extension-in-configuration of constant potential energy between or at the limits. And in general, with the restriction mentioned, we may substitute ${\displaystyle e^{\phi _{q}}\,d\epsilon _{q}}$ for ${\displaystyle \Delta _{\dot {q}}^{\frac {1}{2}}dq_{1}\ldots dq_{n},}$ in an ${\displaystyle n}$-fold integral, reducing it to a simple integral, when the limits are expressed by the potential energy, and the other factor under the integral sign is a function of

1. ↑ If ${\displaystyle V_{q}}$ were infinite for finite values of ${\displaystyle \epsilon _{q}}$, ${\displaystyle V}$ would evidently be infinite for finite values of ${\displaystyle \epsilon }$.