Page:Elementary Principles in Statistical Mechanics (1902).djvu/105

The multiple integrals in the last four equations represent the average values of the expressions In the brackets, which we may therefore set equal to zero. The first gives

${\frac {d\psi }{da_{1}}}={\overline {\frac {d\epsilon }{da_{1}}}}=-{\overline {A}}_{1},$

(241)

as already obtained. With this relation and (191) we get from the other equations

${\overline {(A_{1}-{\overline {A}}_{1})^{2}}}=\Theta {\bigg (}{\overline {\frac {d^{2}\epsilon }{da_{1}{}^{2}}}}-{\frac {d^{2}\psi }{da_{1}{}^{2}}}{\bigg )}=\Theta {\bigg (}{\frac {d{\overline {A}}_{1}}{da_{1}}}-{\overline {\frac {dA_{1}}{da_{1}}}}{\bigg )}$

(242)

${\begin{aligned}{\overline {(A_{1}-{\overline {A}}_{1})(A_{2}-{\overline {A}}_{2})}}&=\Theta {\bigg (}{\overline {\frac {d^{2}\epsilon }{da_{1}\,da_{2}}}}-{\frac {d^{2}\psi }{da_{1}\,da_{2}}}{\bigg )}\\&=\Theta {\bigg (}{\frac {d{\overline {A}}_{1}}{da_{2}}}-{\overline {\frac {dA_{1}}{da_{2}}}}{\bigg )}=\Theta {\bigg (}{\frac {d{\overline {A}}_{2}}{da_{1}}}-{\overline {\frac {dA_{2}}{da_{1}}}}{\bigg )}\end{aligned}}$

(243)

${\overline {(A_{1}-{\overline {A}}_{1})(\epsilon -{\overline {\epsilon }})}}=-\Theta ^{2}{\frac {d^{2}\psi }{da_{1}\,d\Theta }}=\Theta ^{2}{\frac {d{\overline {A}}_{1}}{d\Theta }}=-\Theta ^{2}{\frac {d{\overline {\eta }}}{da_{1}}}.$

We may add for comparison equation (205), which might be derived from (236) by differentiating twice with respect to $\Theta$:

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

(244)

The two last equations give

${\overline {(A_{1}-{\overline {A}}_{1})(\epsilon -{\overline {\epsilon }})}}={\frac {d{\overline {A}}}{d{\overline {\epsilon }}}}{\overline {(\epsilon -{\overline {\epsilon }})^{2}}}.$

(245)

If $\psi$ or ${\overline {\epsilon }}$ is known as function of $\Theta$, $a_{1}$, $a_{2}$, etc., ${\overline {(\epsilon -{\overline {\epsilon }})^{2}}}$ may be obtained by differentiation as function of the same variables. And if $\psi$, or ${\overline {A}}_{1}$, or ${\overline {\eta }}$ is known as function of $\Theta$, $a_{1}$, etc., ${\overline {(A_{1}-{\overline {A}}_{1})(\epsilon -{\overline {\epsilon }})}}$ may be obtained by differentiation. But ${\overline {(A_{1}-{\overline {A}}_{1})^{2}}}$ and ${\overline {(A_{1}-{\overline {A}}_{1})(A_{2}-{\overline {A}}_{2})}}$ cannot be obtained in any similar manner. We have seen that ${\overline {(\epsilon -{\overline {\epsilon }})^{2}}}$ is in general a vanishing quantity for very great values of $n$, which we may regard as contained implicitly in $\Theta$ as a divisor. The same is true of ${\overline {(A_{1}-{\overline {A}}_{1})(\epsilon -{\overline {\epsilon }})}}$. It does not appear that we can assert the same of ${\overline {(A_{1}-{\overline {A}}_{1})^{2}}}$ or ${\overline {(A_{1}-{\overline {A}}_{1})(A_{2}-{\overline {A}}_{2})}}$, since