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

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

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

(295)

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

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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

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

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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

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

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and

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

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where $V_{p}$ and $e^{\phi _{p}}$ are connected with $V_{q}$ by the equation

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

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

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

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We may also write

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

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

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