Page:Elementary Principles in Statistical Mechanics (1902).djvu/104

mined. So also if $\psi _{q}$ or ${\overline {\epsilon }}_{q}$ is given as a function of $\Theta$, all averages of the form ${\overline {\epsilon _{q}{}^{h}}}$ or ${\overline {(\epsilon _{q}-{\overline {\epsilon }}_{q})^{h})}}$ are determined. But

${\overline {\epsilon }}_{q}={\overline {\epsilon }}-{\tfrac {1}{2}}n\Theta .$

Therefore if any one of the quantities $\psi$, $\psi _{q}$, ${\overline {\epsilon }}$, ${\overline {\epsilon }}_{q}$ is known as a function of $\Theta$, and $n$ is also known, all averages of any of the forms mentioned are thereby determined as functions of the same variable. In any case all averages of the form

${\overline {{\bigg (}{\frac {\epsilon _{p}-{\overline {\epsilon }}_{p}}{{\overline {\epsilon }}_{p}}}{\bigg )}^{h}}}$

are known in terms of $n$ alone, and have the same value whether taken for the whole ensemble or limited to any particular configuration.

If we differentiate the equation

$\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}e^{\frac {\psi -\epsilon }{\Theta }}\,dp_{1}\ldots dq_{n}=1$

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with respect to $a_{1}$, and multiply by $\Theta$, we have

$\int \ldots \int {\bigg [}{\frac {d\psi }{da_{1}}}-{\frac {d\epsilon }{da_{1}}}{\bigg ]}e^{\frac {\psi -\epsilon }{\Theta }}\,dp_{1}\ldots dq_{n}=0.$

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Differentiating again, with respect to $a_{1}$, with respect to $a_{2}$, and with respect to $\Theta$, we have

$\int \ldots \int {\bigg [}{\frac {d^{2}\psi }{da_{1}{}^{2}}}-{\frac {d^{2}\epsilon }{da_{1}{}^{2}}}+{\frac {1}{\Theta }}{\bigg (}{\frac {d\psi }{da_{1}}}-{\frac {d\epsilon }{da_{1}}}{\bigg )}^{2}{\bigg ]}e^{\frac {\psi -\epsilon }{\Theta }}\,dp_{1}\ldots dq_{n}=0,$

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${\begin{aligned}\int \ldots \int {\bigg [}{\frac {d^{2}\psi }{da_{1}\,da_{2}}}-{\frac {d^{2}\epsilon }{da_{1}\,da_{2}}}&+{\frac {1}{\Theta }}{\bigg (}{\frac {d\psi }{da_{1}}}-{\frac {d\epsilon }{da_{1}}}{\bigg )}{\bigg (}{\frac {d\psi }{da_{2}}}-{\frac {d\epsilon }{da_{2}}}{\bigg )}{\bigg ]}\\&\qquad \qquad \qquad e^{\frac {\psi -\epsilon }{\Theta }}\,dp_{1}\ldots dq_{n}=0,\end{aligned}}$

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${\begin{aligned}\int \ldots \int {\bigg [}{\frac {d^{2}\psi }{da_{1}\,d\Theta }}+{\bigg (}{\frac {d\psi }{da_{1}}}-{\frac {d\epsilon }{da_{1}}}{\bigg )}&{\bigg (}{\frac {1}{\Theta }}{\frac {d\psi }{d\Theta }}-{\frac {\psi -\epsilon }{\Theta ^{2}}}{\bigg )}{\bigg ]}\\&\qquad e^{\frac {\psi -\epsilon }{\Theta }}\,dp_{1}\ldots dq_{n}=0.\end{aligned}}$

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