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

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

(241)

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

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

(242)

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

(243)

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

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

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

(244)

The two last equations give

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

(245)

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