We have therefore

${\displaystyle e^{-\phi }{\frac {d^{h}V}{d\epsilon ^{h}}}={\overline {e^{-\phi _{p}}{\frac {d^{h}V_{p}}{d\epsilon _{p}^{h}}}}}{\bigg |}_{\epsilon }={\frac {\Gamma ({\tfrac {1}{2}}n)}{\Gamma ({\tfrac {1}{2}}n-h+1)}}{\overline {\epsilon _{p}^{1-h}}}|_{\epsilon },}$

(380)

if ${\displaystyle h<{\tfrac {1}{2}}n+1}$. For example, when ${\displaystyle n}$ is even, we may make ${\displaystyle h={\tfrac {1}{2}}n}$, which gives, with (307),

${\displaystyle (2\pi )^{\tfrac {n}{2}}e^{-\phi }(V_{q})_{\epsilon _{q}=\epsilon }=\Gamma ({\tfrac {1}{2}}n){\overline {{\epsilon _{p}}^{1-{\tfrac {n}{2}}}}}{\Big |}_{\epsilon }.}$

(381)

Since any canonical ensemble of systems may be regarded as composed of microcanonical ensembles, if any quantities ${\displaystyle u}$ and ${\displaystyle v}$ have the same average values in every microcanonical ensemble, they will have the same values in every canonical ensemble. To bring equation (380) formally under this rule, we may observe that the first member being a function of ${\displaystyle \epsilon }$ is a constant value in a microcanonical ensemble, and therefore identical with its average value. We get thus the general equation

${\displaystyle {\overline {e^{-\phi }{\frac {d^{h}V}{d\epsilon ^{h}}}}}{\bigg |}_{\Theta }={\overline {e^{-\phi _{p}}{\frac {d^{h}V_{p}}{d\epsilon _{p}^{h}}}}}{\bigg |}_{\Theta }={\frac {\Gamma ({\tfrac {1}{2}}n)}{\Gamma ({\tfrac {1}{2}}n-h+1)}}{\overline {\epsilon _{p}^{1-h}}}|_{\Theta }=\Theta ^{1-h}}$

(382)

if ${\displaystyle h<{\frac {1}{2}}n+1}$.^[1] The equations

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

(383)

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

(384)

may be regarded as particular cases of the general equation. The last equation is subject to the condition that ${\displaystyle n>2}$.

The last two equations give for a canonical ensemble, if ${\displaystyle n>2}$,

${\displaystyle {\bigg (}1-{\frac {2}{n}}{\bigg )}{\overline {\epsilon _{p}}}|_{\Theta }\,{\overline {\epsilon _{p}^{-1}}}|_{\Theta }=1.}$

(385)

The corresponding equations for a microcanonical ensemble give, if ${\displaystyle n>2}$,

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

(386)

1. ↑ See equation (292).