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)