Now we have identically

${\displaystyle A_{1}-{\overline {A}}_{1}=(A_{1}-{\overline {A_{1}}}|_{\epsilon })+({\overline {A_{1}}}|_{\epsilon }-{\overline {A}}_{1}),}$

where ${\displaystyle A_{1}-{\overline {A_{1}}}|_{\epsilon }}$ denotes the excess of the force (tending to increase ${\displaystyle a_{1}}$ exerted by any system above the average of such forces for systems of the same energy. Accordingly,

${\displaystyle {\overline {(A_{1}-{\overline {A}}_{1})^{2}}}={\overline {(A_{1}-{\overline {A_{1}}}|_{\epsilon })^{2}}}+2{\overline {(A_{1}-{\overline {A_{1}}}|_{\epsilon })({\overline {A_{1}}}|_{\epsilon }-{\overline {A}}_{1})}}+{\overline {({\overline {A_{1}}}|_{\epsilon }-{\overline {A}}_{1})^{2}}}.}$

But the average value of ${\displaystyle (A_{1}-{\overline {A_{1}}}|_{\epsilon })({\overline {A_{1}}}|_{\epsilon }-{\overline {A}}_{1})}$ for systems of the ensemble which have the same energy is zero, since for such systems the second factor is constant. Therefore the average for the whole ensemble is zero, and

${\displaystyle {\overline {(A_{1}-{\overline {A}}_{1})^{2}}}={\overline {(A_{1}-{\overline {A_{1}}}|_{\epsilon })^{2}}}+{\overline {({\overline {A_{1}}}|_{\epsilon }-{\overline {A}}_{1})^{2}}}.}$

(248)

In the same way it may be shown that

${\displaystyle {\overline {(A_{1}-{\overline {A}}_{1})(\epsilon -{\overline {\epsilon }})}}={\overline {({\overline {A_{1}}}|_{\epsilon }-{\overline {A}}_{1})(\epsilon -{\overline {\epsilon }})}}.}$

(249)

It is evident that in ensembles in which the anomalies of energy ${\displaystyle \epsilon -{\overline {\epsilon }}}$ may be regarded as insensible the same will be true of the quantities represented by ${\displaystyle {\overline {A_{1}}}|_{\epsilon }-{\overline {A}}_{1}}$.

The properties of quantities of the form ${\displaystyle {\overline {A_{1}}}|_{\epsilon }}$ will be farther considered in Chapter X, which will be devoted to ensembles of constant energy.

It may not be without interest to consider some general formulae relating to averages in a canonical ensemble, which embrace many of the results which have been given in this chapter.

Let ${\displaystyle u}$ be any function of the internal and external coördinates with the momenta and modulus. We have by definition

${\displaystyle {\overline {u}}=\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}ue^{\frac {\psi -\epsilon }{\Theta }}\,dp_{1}\ldots dq_{n}}$

(250)

If we differentiate with respect to ${\displaystyle \Theta }$, we have

${\displaystyle {\frac {d{\overline {u}}}{d\Theta }}=\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}{\bigg (}{\frac {du}{d\Theta }}-{\frac {u}{\Theta ^{2}}}(\psi -\epsilon )+{\frac {u}{\Theta }}{\frac {d\psi }{d\Theta }}{\bigg )}e^{\frac {\psi -\epsilon }{\Theta }}\,dp_{1}\ldots dq_{n},}$