Page:Elementary Principles in Statistical Mechanics (1902).djvu/150

${\displaystyle {\begin{aligned}{\frac {de^{-c}}{da_{1}}}&=-{\Big [}{\overline {A_{1}}}|_{\epsilon }\,e^{-{\frac {(\epsilon -\epsilon ')^{2}}{\omega ^{2}}}+\phi }{\Big ]}_{V=0}^{\epsilon =\infty }\\&\qquad {}+\int \limits _{V=0}^{\epsilon =\infty }{\bigg (}{\frac {d{\overline {A_{1}}}|_{\epsilon }}{d\epsilon }}+{\overline {A_{1}}}|_{\epsilon }\,{\frac {d\phi }{d\epsilon }}{\bigg )}e^{-{\frac {(\epsilon -\epsilon ')^{2}}{\omega ^{2}}}+\phi }\,d\epsilon .\end{aligned}}}$

(407)

Differentiating (405), we get

${\displaystyle {\frac {de^{-c}}{da_{1}}}=\int \limits _{V=0}^{\epsilon =\infty }{\frac {d\phi }{da_{1}}}\,e^{-{\frac {(\epsilon -\epsilon ')^{2}}{\omega ^{2}}}+\phi }\,d\epsilon -{\bigg (}e^{-{\frac {(\epsilon -\epsilon ')^{2}}{\omega ^{2}}}+\phi }\,{\frac {d\epsilon _{a}}{da_{1}}}{\bigg )}_{V=0}}$

(408)

where ${\displaystyle \epsilon _{a}}$ denotes the least value of ${\displaystyle \epsilon }$ consistent with the external coördinates. The last term in this equation represents the part of ${\displaystyle de^{-c}/da_{1}}$ which is due to the variation of the lower limit of the integral. It is evident that the expression in the brackets will vanish at the upper limit. At the lower limit, at which ${\displaystyle \epsilon _{p}=0}$, and ${\displaystyle \epsilon _{q}}$ has the least value consistent with the external coördinates, the average sign on ${\displaystyle {\overline {A_{1}}}|_{\epsilon }}$ is superfluous, as there is but one value of ${\displaystyle A_{1}}$ which is represented by ${\displaystyle -d\epsilon _{a}/da_{1}}$. Exceptions may indeed occur for particular values of the external coördinates, at which ${\displaystyle d\epsilon _{a}/da_{1}}$ receive a finite increment, and the formula becomes illusory. Such particular values we may for the moment leave out of account. The last term of (408) is therefore equal to the first term of the second member of (407). (We may observe that both vanish when ${\displaystyle n>2}$ on account of the factor ${\displaystyle e^{\phi }}$.)

We have therefore from these equations

${\displaystyle \int \limits _{V=0}^{\epsilon =\infty }{\bigg (}{\frac {d{\overline {A_{1}}}|_{\epsilon }}{d\epsilon }}+{\overline {A_{1}}}|_{\epsilon }\,{\frac {d\phi }{d\epsilon }}{\bigg )}e^{-{\frac {(\epsilon -\epsilon ')^{2}}{\omega ^{2}}}+\phi }\,d\epsilon =\int \limits _{V=0}^{\epsilon =\infty }{\frac {d\phi }{da_{1}}}\,e^{-{\frac {(\epsilon -\epsilon ')^{2}}{\omega ^{2}}}+\phi }\,d\epsilon ,}$

or

${\displaystyle \int \limits _{V=0}^{\epsilon =\infty }{\bigg (}{\frac {d{\overline {A_{1}}}|_{\epsilon }}{d\epsilon }}+{\overline {A_{1}}}|_{\epsilon }\,{\frac {d\phi }{d\epsilon }}-{\frac {d\phi }{da_{1}}}{\bigg )}e^{c-{\frac {(\epsilon -\epsilon ')^{2}}{\omega ^{2}}}+\phi }\,d\epsilon =0.}$

(409)

That is: the average value in the ensemble of the quantity represented by the principal parenthesis is zero. This must