Page:Elementary Principles in Statistical Mechanics (1902).djvu/148

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

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

Accordingly

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

(397)

and we may write

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

(398)

and

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

(399)

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

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

(400)

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

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

(401)

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

(402)

We cannot apply the methods employed in the preceding pages to the microcanonical averages of the (generalized) forces $A_{1}$, $A_{2}$, 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.