and

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

(395)

We have compared certain functions of the energy of the whole system with average values of similar functions of the kinetic energy of the whole system, and with average values of similar functions of the whole energy of a part of the system. We may also compare the same functions with average values of the kinetic energy of a part of the system.

We shall express the total, kinetic, and potential energies of the whole system by ${\displaystyle \epsilon }$, ${\displaystyle \epsilon _{p}}$, and ${\displaystyle \epsilon _{q}}$, and the kinetic energies of the parts by ${\displaystyle \epsilon _{1p}}$, and ${\displaystyle \epsilon _{2p}}$. These kinetic energies are necessarily separate: we need not make any supposition concerning potential energies. The extension-in-phase within any limits which can be expressed in terms of ${\displaystyle \epsilon _{q}}$, ${\displaystyle \epsilon _{1p}}$, ${\displaystyle \epsilon _{2p}}$ may be represented in the notations of Chapter VIII by the triple integral

${\displaystyle \iiint dV_{1p}\,dV_{2p}\,dV_{q}}$

taken within those limits. And if an ensemble of systems is distributed with a uniform density within those limits, the average value of any function of ${\displaystyle \epsilon _{q}}$, ${\displaystyle \epsilon _{1p}}$, ${\displaystyle \epsilon _{2p}}$ will be expressed by the quotient

${\displaystyle {\frac {\iiint u\,dV_{1p}\,dV_{2p}\,dV_{q}}{\iiint dV_{1p}\,dV_{2p}\,dV_{q}}}}$

or

${\displaystyle {\frac {\iiint u\,e^{\phi _{1p}}\,d\epsilon \,dV_{2p}\,dV_{q}}{\iiint e^{\phi _{1p}}\,d\epsilon \,dV_{2p}\,dV_{q}}}}$

To get the average value of ${\displaystyle u}$ for a microcanonical distribution, we must make the limits ${\displaystyle \epsilon }$ and ${\displaystyle \epsilon +d\epsilon }$. The denominator in this case becomes ${\displaystyle \epsilon ^{\phi }\,d\epsilon }$, and we have

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

(396)