${\displaystyle \mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}e^{{\frac {\psi -\epsilon }{\Theta }}+\Delta \eta }\,dp_{1}\ldots dq_{n}=\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}e^{\frac {\psi -\epsilon }{\Theta }}\,dp_{1}\ldots dq_{n}=1,}$

(424)

and to that relating to the constant average energy, that

${\displaystyle \mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}\epsilon e^{{\frac {\psi -\epsilon }{\Theta }}+\Delta \eta }\,dp_{1}\ldots dq_{n}=\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}\epsilon e^{\frac {\psi -\epsilon }{\Theta }}\,dp_{1}\ldots dq_{n}.}$

(425)

It is to be proved that

${\displaystyle {\begin{aligned}&\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}{\bigg (}{\frac {\psi }{\Theta }}-{\frac {\epsilon }{\Theta }}+\Delta \eta {\bigg )}e^{{\frac {\psi -\epsilon }{\Theta }}+\Delta \eta }\,dp_{1}\ldots dq_{n}>\\&\qquad \qquad \qquad \mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}{\bigg (}{\frac {\psi }{\Theta }}-{\frac {\epsilon }{\Theta }}{\bigg )}e^{\frac {\psi -\epsilon }{\Theta }}\,dp_{1}\ldots dq_{n}.\end{aligned}}}$

(426)

Now in virtue of the first condition (424) we may cancel the constant term ${\displaystyle \psi /\Theta }$ in the parentheses in (426), and in virtue of the second condition (425) we may cancel the term ${\displaystyle \epsilon /\Theta }$. The proposition to be proved is thus reduced to

${\displaystyle \mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}\Delta \eta e^{{\frac {\psi -\epsilon }{\Theta }}+\Delta \eta }\,dp_{1}\ldots dq_{n}>0,}$

which may be written, in virtue of the condition (424),

${\displaystyle \mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}(\Delta \eta e^{\Delta \eta }+1-e^{\Delta \eta })e^{\frac {\psi -\epsilon }{\Theta }}\,dp_{1}\ldots dq_{n}>0.}$

(427)

In this form its truth is evident for the same reasons which applied to (423).

Theorem III. If ${\displaystyle \Theta }$ is any positive constant, the average value in an ensemble of the expression ${\displaystyle \eta +\epsilon /\Theta }$ (${\displaystyle \eta }$ denoting as usual the index of probability and ${\displaystyle \epsilon }$ the energy) is less when the ensemble is distributed canonically with modulus ${\displaystyle \Theta }$, than for any other distribution whatever.

In accordance with our usual notation let us write ${\displaystyle (\psi -\epsilon )/\Theta }$ for the index of the canonical distribution. In any other distribution let the index be ${\displaystyle (\psi -\epsilon )/\Theta +\Delta \eta }$.