Page:Elementary Principles in Statistical Mechanics (1902).djvu/115

the integration being extended, with constant values of the coördinates, both internal and external, over all values of the momenta for which the kinetic energy is less than the limit ${\displaystyle \epsilon _{p}}$. ${\displaystyle V_{p}}$ will evidently be a continuous increasing function of ${\displaystyle \epsilon _{p}}$ which vanishes and becomes infinite with ${\displaystyle \epsilon _{p}}$. Let us set

${\displaystyle \phi _{p}=\log {\frac {V_{p}}{d\epsilon _{p}}}.}$

(277)

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

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

(278)

And in general, we may substitute ${\displaystyle e^{\phi _{p}}\,d\epsilon _{p}}$ for ${\displaystyle \Delta _{p}^{\frac {1}{2}}dp_{1}\ldots dp_{n}}$ or ${\displaystyle \Delta _{\dot {q}}^{\frac {1}{2}}d{\dot {q}}_{1}\ldots d{\dot {q}}_{n}}$ in an ${\displaystyle n}$-fold integral in which the coördinates are constant, reducing it to a simple integral, when the limits are expressed by the kinetic energy, and the other factor under the integral sign is a function of the kinetic energy, either alone or with quantities which are constant in the integration.

It is easy to express ${\displaystyle V_{p}}$ and ${\displaystyle \phi _{p}}$ in terms of ${\displaystyle \epsilon _{p}}$. Since ${\displaystyle \Delta _{p}}$ is function of the coördinates alone, we have by definition

${\displaystyle V_{p}=\Delta _{p}^{\frac {1}{2}}\int \ldots \int dp_{1}\ldots dp_{n}}$

(279)

the limits of the integral being given by ${\displaystyle \epsilon _{p}}$. That is, if

${\displaystyle \epsilon _{p}=F(p_{1},\ldots p_{n}),}$

(280)

the limits of the integral for ${\displaystyle \epsilon _{p}=1}$, are given by the equation

${\displaystyle F(p_{1},\ldots p_{n})=1,}$

(281)

and the limits of the integral for ${\displaystyle \epsilon _{p}=a^{2}}$, are given by the equation

${\displaystyle F(p_{1},\ldots p_{n})=a^{2}.}$

(282)

But since ${\displaystyle F}$ represents a quadratic function, this equation may be written

${\displaystyle F{\bigg (}{\frac {p_{1}}{a}},\ldots {\frac {p_{n}}{a}}{\bigg )}=1}$

(283)