${\displaystyle \phi _{1p}}$

${\displaystyle V_{2p}}$

${\displaystyle V_{q}}$

${\displaystyle \epsilon _{1p}+\epsilon _{2p}+\epsilon _{q}=\mathrm {constant} =\epsilon .}$

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

(397)

${\displaystyle e^{-\phi }V={\overline {e^{-\phi _{1p}}V_{1p}}}|_{\epsilon }={\overline {e^{-\phi _{2p}}V_{2p}}}|_{\epsilon }={\frac {2}{n_{1}}}{\overline {\epsilon _{1p}}}|_{\epsilon }={\frac {2}{n_{2}}}{\overline {\epsilon _{2p}}}|_{\epsilon },}$

(398)

${\displaystyle \Theta ={\overline {e^{-\phi _{p}}V_{p}}}|_{\Theta }={\overline {e^{-\phi _{1p}}V_{1p}}}|_{\Theta }={\overline {e^{-\phi _{2p}}V_{2p}}}|_{\Theta }={\frac {2}{n_{1}}}{\overline {\epsilon _{1p}}}|_{\Theta }={\frac {2}{n_{2}}}{\overline {\epsilon _{2p}}}|_{\Theta }.}$

(399)

Again, if ${\displaystyle n_{1}>2}$,

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

(400)

${\displaystyle n_{1}>2}$

${\displaystyle n_{2}>2}$

${\displaystyle {\frac {d\phi }{d\epsilon }}={\overline {\frac {d\phi _{1p}}{d\epsilon _{1p}}}}{\bigg |}_{\epsilon }={\overline {\frac {d\phi _{2p}}{d\epsilon _{2p}}}}{\bigg |}_{\epsilon }=({\tfrac {1}{2}}n_{1}-1){\overline {{\epsilon _{1p}}^{-1}}}|_{\epsilon }=({\tfrac {1}{2}}n_{2}-1){\overline {{\epsilon _{2p}}^{-1}}}|_{\epsilon },}$

(401)

${\displaystyle {\frac {1}{\Theta }}={\overline {\frac {d\phi _{p}}{d\epsilon _{p}}}}{\bigg |}_{\Theta }={\overline {\frac {d\phi _{1p}}{d\epsilon _{1p}}}}{\bigg |}_{\Theta }={\overline {\frac {d\phi _{2p}}{d\epsilon _{2p}}}}{\bigg |}_{\Theta }=({\tfrac {1}{2}}n_{1}-1){\overline {{\epsilon _{1p}}^{-1}}}|_{\Theta }=({\tfrac {1}{2}}n_{2}-1){\overline {{\epsilon _{2p}}^{-1}}}|_{\Theta }.}$

(402)

We cannot apply the methods employed in the preceding pages to the microcanonical averages of the (generalized) forces ${\displaystyle A_{1}}$, ${\displaystyle A_{2}}$, etc., exerted by a system on external bodies, since these quantities are not functions of the energies, either kinetic or potential, of the whole or any part of the system. We may however use the method described on page 116.