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

For $h=1$, this gives

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

(218)

which agrees with (191).
From (215) we have also

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

(219)

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

(220)

In like manner from the identical equation

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

(221)

we get

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

(222)

and

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

(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

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

the equation

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

(224)

reduces to

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

(225)