Page:Elementary Principles in Statistical Mechanics (1902).djvu/100

For ${\displaystyle h=1}$, this gives

${\displaystyle {\overline {\epsilon }}=-\Theta ^{2}{\frac {d}{d\Theta }}{\bigg (}{\frac {\psi }{\Theta }}{\bigg )}}$

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${\displaystyle {\overline {\epsilon ^{h}}}={\overline {\epsilon }}{\overline {\epsilon ^{h-1}}}+\Theta ^{2}{\frac {d{\overline {\epsilon ^{h-1}}}}{d\Theta }}={\bigg (}{\overline {\epsilon }}+\Theta ^{2}{\frac {d}{d\Theta }}{\bigg )}\epsilon ^{h-1},}$

(219)

${\displaystyle {\overline {\epsilon ^{h}}}={\bigg (}{\overline {\epsilon }}+\Theta ^{2}{\frac {d}{d\Theta }}{\bigg )}^{h-1}{\overline {\epsilon }}.}$

(220)

In like manner from the identical equation

${\displaystyle \mathop {\int \ldots \int } _{\rm {config.}}^{\rm {all}}\epsilon _{q}{}^{h}e^{-{\frac {\epsilon _{q}}{\Theta }}}\Delta _{\dot {q}}{}^{\frac {1}{2}}\,dq_{1}\ldots dq_{n}=\Theta ^{2}{\frac {d}{d\Theta }}\mathop {\int \ldots \int } _{\rm {config.}}^{\rm {all}}\epsilon _{q}{}^{h-1}e^{-{\frac {\epsilon _{q}}{\Theta }}}\Delta _{\dot {q}}{}^{\frac {1}{2}}\,dq_{1}\ldots dq_{n},}$

(221)

${\displaystyle {\overline {\epsilon _{q}{}^{h}}}=e^{\frac {\psi _{q}}{\Theta }}{\bigg (}\Theta ^{2}{\frac {d}{d\Theta }}{\bigg )}^{h}e^{-{\frac {\psi _{q}}{\Theta }}}.,}$

(222)

${\displaystyle {\overline {\epsilon _{q}{}^{h}}}={\bigg (}{\overline {\epsilon }}_{q}+\Theta ^{2}{\frac {d}{d\Theta }}{\bigg )}^{h-1}{\overline {\epsilon }}_{q}.}$

(223)

With respect to the kinetic energy similar equations will hold for averages taken for any particular configuration, or for the whole ensemble. But since

${\displaystyle {\overline {\epsilon }}_{p}={\frac {n}{2}}\Theta ,}$

${\displaystyle {\overline {\epsilon _{p}{}^{h}}}={\bigg (}{\overline {\epsilon }}_{p}+\Theta ^{2}{\frac {d}{d\Theta }}{\bigg )}{\overline {\epsilon _{p}{}^{h-1}}}}$

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${\displaystyle {\overline {\epsilon _{p}{}^{h}}}={\bigg (}{\frac {n}{2}}\Theta +{}^{2}\Theta {\frac {d}{d\Theta }}{\bigg )}{\overline {\epsilon _{p}{}^{h-1}}}={\frac {n}{2}}{\bigg (}{\frac {n}{2}}\Theta +\Theta ^{2}{\frac {d}{d\Theta }}{\bigg )}^{h-1}\Theta .}$

(225)