Let us imagine an ensemble of systems distributed in phase according to the index of probability

${\displaystyle c-{\frac {(\epsilon -\epsilon ')^{2}}{\omega ^{2}}},}$

where ${\displaystyle \epsilon '}$ is any constant which is a possible value of the energy, except only the least value which is consistent with the values of the external coördinates, and ${\displaystyle c}$ and ${\displaystyle \omega }$ are other constants. We have therefore

${\displaystyle \mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}e^{c-{\frac {(\epsilon -\epsilon ')^{2}}{\omega ^{2}}}}\,dp_{1}\ldots dq_{n}=1,}$

(403)

or

${\displaystyle e^{-c}=\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}e^{-{\frac {(\epsilon -\epsilon ')^{2}}{\omega ^{2}}}}\,dp_{1}\ldots dq_{n},}$

(404)

or again

${\displaystyle e^{-c}=\int \limits _{V=0}^{\epsilon =\infty }e^{-{\frac {(\epsilon -\epsilon ')^{2}}{\omega ^{2}}}+\phi }\,d\epsilon .}$

(405)

From (404) we have

${\displaystyle {\begin{aligned}{\frac {de^{-c}}{da_{1}}}&=\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}2{\frac {\epsilon -\epsilon '}{\omega ^{2}}}A_{1}\,e^{-{\frac {(\epsilon -\epsilon ')^{2}}{\omega ^{2}}}}\,dp_{1}\ldots dq_{n}\\&=\int \limits _{V=0}^{\epsilon =\infty }2{\frac {\epsilon -\epsilon '}{\omega ^{2}}}{\overline {A_{1}}}|_{\epsilon }\,e^{-{\frac {(\epsilon -\epsilon ')^{2}}{\omega ^{2}}}+\phi }\,d\epsilon ,\end{aligned}}}$

(406)

where ${\displaystyle {\overline {A_{1}}}|_{\epsilon }}$ denotes the average value of ${\displaystyle A_{1}}$ in those systems of the ensemble which have any same energy ${\displaystyle \epsilon }$. (This is the same thing as the average value of ${\displaystyle A_{1}}$ in a microcanonical ensemble of energy ${\displaystyle \epsilon }$.) The validity of the transformation is evident, if we consider separately the part of each integral which lies between two infinitesimally differing limits of energy. Integrating by parts, we get