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

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

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

(295)

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

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If ${\displaystyle n=2}$, ${\displaystyle e^{\phi _{p}}=2\pi }$, and ${\displaystyle d\phi _{p}/d\epsilon _{p}=0}$, for any value of ${\displaystyle \epsilon _{p}}$.

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

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

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where the integrations cover all phases for which the energy is less than the limit ${\displaystyle \epsilon }$, for which the value of ${\displaystyle V}$ is sought. This gives

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

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and

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

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

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

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

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

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We may also write

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

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

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