Page:Elementary Principles in Statistical Mechanics (1902).djvu/119

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OF THE ENERGIES OF A SYSTEM.

95

{\overline {e^{\phi _{p}}}}={\frac {\Gamma (n-1)}{[\Gamma ({\tfrac {1}{2}}n)]^{2}}}(2\pi )^{\tfrac {n}{2}}\Theta ^{{\tfrac {n}{2}}-1},\quad \mathrm {if} \quad n>1;

(294)

{\overline {\frac {d\phi _{p}}{d\epsilon _{p}}}}={\frac {1}{\Theta }},\quad \mathrm {if} \quad n>2;

(295)

{\overline {e^{-\phi _{p}}V_{p}}}=\Theta ,.

(296)

If

n=2

,

e^{\phi _{p}}=2\pi

, and

d\phi _{p}/d\epsilon _{p}=0

, for any value of

\epsilon _{p}

.

The definitions of $V$ , $V_{q}$ , and $V_{p}$ give

V=\int \int dV_{p}\,dV_{q}

(297)

where the integrations cover all phases for which the energy is less than the limit

\epsilon

, for which the value of

V

is sought. This gives

V=\int \limits _{V_{q}=0}^{\epsilon _{q}=\epsilon }V_{p}\,dV_{q},

(298)

and

e^{\phi }={\frac {dV}{d\epsilon }}=\int \limits _{V_{q}=0}^{\epsilon _{q}=\epsilon }e^{\phi _{p}}\,dV_{q},

(299)

where

V_{p}

and

e^{\phi _{p}}

are connected with

V_{q}

by the equation

\epsilon _{p}+\epsilon _{q}=\mathrm {constant} =\epsilon .

(300)

If $n>2$ , $e^{\phi _{p}}$ vanishes at the upper limit, i. e., for $\epsilon _{p}=0$ , and we get by another differentiation

e^{\phi }{\frac {d\phi }{d\epsilon }}=\int \limits _{V_{q}=0}^{\epsilon _{q}=\epsilon }e^{\phi _{p}}{\frac {d\phi _{p}}{d\epsilon _{p}}}\,dV_{q}.

(301)

We may also write

V=\int \limits _{V_{q}=0}^{\epsilon _{q}=\epsilon }V_{p}e^{\phi _{q}}\,d\epsilon _{q},

(302)

e^{\phi }=\int \limits _{V_{q}=0}^{\epsilon _{q}=\epsilon }e^{\phi _{p}+\phi _{q}}\,d\epsilon _{q},

(303)