Page:Elementary Principles in Statistical Mechanics (1902).djvu/108

or

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

(251)

Setting ${\displaystyle u=1}$ in this equation, we get

${\displaystyle {\frac {d\psi }{d\Theta }}={\frac {\psi -{\overline {\epsilon }}}{\Theta }},}$

and substituting this value, we have

${\displaystyle {\frac {\overline {u}}{d\Theta }}={\overline {\frac {du}{d\Theta }}}+{\frac {\overline {u\epsilon }}{\Theta ^{2}}}-{\frac {{\overline {u}}\,{\overline {\epsilon }}}{\Theta ^{2}}}}$

or

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

(252)

If we differentiate equation (250) with respect to ${\displaystyle a}$ (which may represent any of the external coördinates), and write ${\displaystyle A}$ for the force ${\displaystyle -{\frac {d\epsilon }{dA}}}$, we get

${\displaystyle {\frac {\overline {u}}{da}}=\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}{\bigg (}{\frac {du}{da}}+{\frac {u}{\Theta }}{\frac {d\psi }{da}}+{\frac {u}{\Theta }}A{\bigg )}e^{\frac {\psi -\epsilon }{\Theta }}\,dp_{1}\ldots dq_{n}}$

or

${\displaystyle {\frac {\overline {u}}{da}}={\overline {\frac {du}{da}}}+{\frac {\overline {u}}{\Theta }}{\frac {d\psi }{da}}+{\frac {\overline {uA}}{\Theta }}}$

(253)

Setting ${\displaystyle u=1}$ in this equation, we get

${\displaystyle {\frac {d\psi }{da}}=-{\overline {A}}.}$

Substituting this value, we have

${\displaystyle {\frac {\overline {u}}{da}}={\overline {\frac {du}{da}}}+{\frac {\overline {uA}}{\Theta }}-{\frac {{\overline {u}}\,{\overline {A}}}{\Theta }},}$

(254)

or

${\displaystyle \Theta {\frac {\overline {u}}{da}}-\Theta {\overline {\frac {du}{da}}}={\overline {uA}}-{\overline {u}}\,{\overline {A}}={\overline {(u-{\overline {u}})(A-{\overline {A}})}}.}$

(255)

Repeated applications of the principles expressed by equations (252) and (255) are perhaps best made in the particular cases. Yet we may write (252) in this form