average value of ${\displaystyle u}$ in an ensemble in which the whole system is microcanonically distributed in phase, viz.,

${\displaystyle {\overline {u}}|_{\epsilon }=e^{-\phi }\int \limits _{V_{2}=0}^{\epsilon _{2}=\epsilon }u\,e^{\phi _{1}}\,dV_{2},}$

(387)

where ${\displaystyle \phi _{1}}$ and ${\displaystyle V_{2}}$ are connected by the equation

${\displaystyle \epsilon _{1}+\epsilon _{2}=\mathrm {constant} =\epsilon ,}$

(388)

and ${\displaystyle u}$, if given as function of ${\displaystyle \epsilon _{1}}$, or of ${\displaystyle \epsilon _{1}}$ and ${\displaystyle \epsilon _{2}}$, becomes in virtue of the same equation a function of ${\displaystyle \epsilon _{2}}$ alone.^[1] Thus

${\displaystyle {\overline {e^{-\phi _{1}}V_{1}}}|_{\epsilon }=e^{-\phi }\int \limits _{V_{2}=0}^{\epsilon _{2}=\epsilon }V_{1}\,dV_{2},}$

(389)

${\displaystyle e^{-\phi }V={\overline {e^{-\phi _{1}}V_{1}}}|_{\epsilon }={\overline {e^{-\phi _{2}}V_{2}}}|_{\epsilon }.}$

(390)

This requires a similar relation for canonical averages

${\displaystyle \Theta ={\overline {e^{-\phi }V}}|_{\Theta }={\overline {e^{-\phi _{1}}V_{1}}}|_{\Theta }={\overline {e^{-\phi _{2}}V_{2}}}|_{\Theta }.}$

(391)

Again

${\displaystyle {\overline {\frac {d\phi _{1}}{d\epsilon _{1}}}}{\bigg |}_{\epsilon }=e^{-\phi }\int \limits _{V_{2}=0}^{\epsilon _{2}=\epsilon }{\frac {d\phi _{1}}{d\epsilon _{1}}}e^{\phi _{1}}\,dV_{2}.}$

(392)

But if ${\displaystyle n_{1}>2}$, ${\displaystyle e^{\phi _{1}}}$ vanishes for ${\displaystyle V_{1}=0}$,^[2] and

${\displaystyle {\frac {d}{d\epsilon }}e^{\phi }={\frac {d}{d\epsilon }}\int \limits _{V_{2}=0}^{\epsilon _{2}=\epsilon }e^{\phi _{1}}\,dV_{2}=\int \limits _{V_{2}=0}^{\epsilon _{2}=\epsilon }{\frac {d\phi _{1}}{d\epsilon _{1}}}\,e^{\phi _{1}}\,dV_{2}.}$

(393)

Hence, if ${\displaystyle n_{1}>2}$, and ${\displaystyle n_{2}>2}$,

${\displaystyle {\frac {d\phi }{d\epsilon }}={\overline {\frac {d\phi _{1}}{d\epsilon _{1}}}}{\bigg |}_{\epsilon }={\overline {\frac {d\phi _{2}}{d\epsilon _{2}}}}{\bigg |}_{\epsilon },}$

(394)

1. ↑ In the applications of the equation (387), we cannot obtain all the results corresponding to those which we have obtained from equation (374), because ${\displaystyle \phi _{p}}$ is a known function of ${\displaystyle \epsilon _{p}}$, while ${\displaystyle \phi _{1}}$ must be treated as an arbitrary function of ${\displaystyle \epsilon _{1}}$, or nearly so.
2. ↑ See Chapter VIII, equations (306) and (316).