are also identical with those given by Clausius for the corresponding quantities.

Equations (112) and (181) show that if ${\displaystyle \psi }$ or ${\displaystyle \psi _{q}}$ is known as function of ${\displaystyle \Theta }$ and ${\displaystyle a_{1}}$, ${\displaystyle a_{2}}$, etc., we can obtain by differentiation ${\displaystyle {\overline {\epsilon }}}$ or ${\displaystyle {\overline {\epsilon }}_{q}}$, and ${\displaystyle {\overline {A}}_{1}}$, ${\displaystyle {\overline {A}}_{2}}$ etc. as functions of the same variables. We have in fact

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

(191)

${\displaystyle {\overline {\epsilon }}_{q}=\psi _{q}-\Theta {\overline {\eta }}_{q}=\psi _{q}-\Theta {\frac {d\psi _{q}}{d\Theta }}.}$

(192)

The corresponding equation relating to kinetic energy,

${\displaystyle {\overline {\epsilon }}_{p}=\psi _{p}-\Theta \eta _{p}=\psi _{p}-\Theta {\frac {d\psi _{p}}{d\Theta }}.}$

(193)

which may be obtained in the same way, may be verified by the known relations (186), (187), and (188) between the variables. We have also

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

(194)

etc., so that the average values of the external forces may be derived alike from ${\displaystyle \psi }$ or from ${\displaystyle \psi _{q}}$.

The average values of the squares or higher powers of the energies (total, potential, or kinetic) may easily be obtained by repeated differentiations of ${\displaystyle \psi }$, ${\displaystyle \psi _{q}}$, ${\displaystyle \psi _{p}}$, or ${\displaystyle {\overline {\epsilon }}}$, ${\displaystyle {\overline {\epsilon }}_{q}}$, ${\displaystyle {\overline {\epsilon }}_{p}}$, with respect to ${\displaystyle \Theta }$. By equation (108) we have

${\displaystyle {\overline {\epsilon }}=\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}\epsilon e^{\frac {\psi -\epsilon }{\Theta }}\,dp_{1}\ldots dq_{n},}$

(195)

and differentiating with respect to ${\displaystyle \Theta }$,

${\displaystyle {\frac {d{\overline {\epsilon }}}{d\Theta }}=\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}{\bigg (}{\frac {\epsilon ^{2}-\psi \epsilon }{\Theta ^{2}}}+{\frac {\epsilon }{\Theta }}{\frac {d\psi }{d\Theta }}{\bigg )}e^{\frac {\psi -\epsilon }{\Theta }}\,dp_{1}\ldots dq_{n},}$

(196)

whence, again by (108),

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