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

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

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

(522)

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

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

(523)

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

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

(524)