Page:Elementary Principles in Statistical Mechanics (1902).djvu/223

Equation (505), which we repeat in a slightly different form, viz.,

$e^{-{\frac {\Omega }{\Theta }}}=\Sigma _{\nu _{1}}\ldots \Sigma _{\nu _{h}}\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}{\frac {e^{\frac {\mu _{1}\nu _{1}\ldots +\mu _{h}\nu _{h}-\epsilon }{\Theta }}}{{\nu _{1}}!\ldots {\nu _{h}}!}}\,dp_{1}\ldots dq_{n},,$

(522)

shows that $\Omega$ is a function of $\Theta$ and $\mu _{1},\ldots \mu _{h}$; also of the external coördinates $a_{1}$, $a_{2}$, etc., which are involved implicitly in $\epsilon$. If we differentiate the equation regarding all these quantities as variable, we have

${\begin{aligned}&e^{-{\frac {\Omega }{\Theta }}}{\bigg (}-{\frac {d\Omega }{\Theta }}+{\frac {\Omega }{\Theta ^{2}}}d\Theta {\bigg )}=\\&\quad {}-{\frac {d\Theta }{\Theta ^{2}}}\Sigma _{\nu _{1}}\ldots \Sigma _{\nu _{h}}\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}{\frac {(\mu _{1}\nu _{1}\ldots +\mu _{h}\nu _{h}-\epsilon )e^{\frac {\mu _{1}\nu _{1}\ldots +\mu _{h}\nu _{h}-\epsilon }{\Theta }}}{{\nu _{1}}!\ldots {\nu _{h}}!}}\,dp_{1}\ldots dq_{n}\\&\quad {}+{\frac {d\mu _{1}}{\Theta }}\Sigma _{\nu _{1}}\ldots \Sigma _{\nu _{h}}\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}{\frac {\nu _{1}e^{\frac {\mu _{1}\nu _{1}\ldots +\mu _{h}\nu _{h}-\epsilon }{\Theta }}}{{\nu _{1}}!\ldots {\nu _{h}}!}}\,dp_{1}\ldots dq_{n}\\&\quad {}+\mathrm {etc.} \\&\quad {}-{\frac {da_{1}}{\Theta }}\Sigma _{\nu _{1}}\ldots \Sigma _{\nu _{h}}\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}{\frac {d\epsilon }{da_{1}}}\,{\frac {e^{\frac {\mu _{1}\nu _{1}\ldots +\mu _{h}\nu _{h}-\epsilon }{\Theta }}}{{\nu _{1}}!\ldots {\nu _{h}}!}}\,dp_{1}\ldots dq_{n}\\&\quad {}-\mathrm {etc.} \\\end{aligned}}$

(523)

If we multiply this equation by $e^{\frac {\Omega }{\Theta }}$, and set as usual $A_{1}$, $A_{2}$, etc., for $-d\epsilon /da_{1}$, $-d\epsilon /da_{2}$, etc., we get in virtue of the law expressed by equation (519),

${\begin{aligned}-{\frac {d\Omega }{\Theta }}+{\frac {\Omega }{\Theta ^{2}}}d\Theta &=-{\frac {d\Theta }{\Theta ^{2}}}(\mu _{1}{\overline {\nu }}_{1}\ldots \mu _{h}{\overline {\nu }}_{h}-{\overline {\epsilon }})\\&\quad {}+{\frac {d\mu _{1}}{\Theta }}{\overline {\nu }}_{1}+{\frac {d\mu _{2}}{\Theta }}{\overline {\nu }}_{2}+\mathrm {etc.} \\&\quad {}+{\frac {da_{1}}{\Theta }}{\overline {A}}_{1}+{\frac {da_{2}}{\Theta }}{\overline {A}}_{2}+\mathrm {etc.} ;\end{aligned}}$

(524)