Page:Elementary Principles in Statistical Mechanics (1902).djvu/126

If their values are varied, we shall have by differentiation, if $n>2$,

${\begin{aligned}&e^{-{\frac {\psi }{\Theta }}}{\bigg (}-{\frac {1}{\Theta }}\,d\psi +{\frac {\psi }{\Theta ^{2}}}\,d\Theta {\bigg )}={\frac {1}{\Theta ^{2}}}\,d\Theta \int \limits _{V=0}^{\epsilon =\infty }\epsilon e^{-{\frac {\epsilon }{\Theta }}+\phi }\,d\epsilon \\&\qquad {}+da_{1}\int \limits _{V=0}^{\epsilon =\infty }{\frac {d\phi }{da_{1}}}e^{-{\frac {\epsilon }{\Theta }}+\phi }\,d\epsilon +da_{2}\int \limits _{V=0}^{\epsilon =\infty }{\frac {d\phi }{da_{2}}}e^{-{\frac {\epsilon }{\Theta }}+\phi }\,d\epsilon +\mathrm {etc.} \end{aligned}}$

(323)

(Since $e^{\phi }$ vanishes with $V$, when $n>2$, there are no terms due to the variations of the limits.) Hence by (269)

$-{\frac {1}{\Theta }}\,d\psi +{\frac {\psi }{\Theta ^{2}}}\,d\Theta ={\frac {\overline {\epsilon }}{\Theta ^{2}}}\,d\Theta +{\overline {\frac {d\phi }{da_{1}}}}\,da_{1}+{\overline {\frac {d\phi }{da_{2}}}}\,da_{2}+\mathrm {etc.} ,$

(324)

or, since

${\frac {\psi +{\overline {\epsilon }}}{\Theta }}={\overline {\eta }},$

(325)

$d\psi ={\overline {\eta }}\,d\Theta -\Theta {\overline {\frac {d\phi }{da_{1}}}}\,da_{1}-\Theta {\overline {\frac {d\phi }{da_{2}}}}\,da_{2}-\mathrm {etc.} .$

(326)

Comparing this with (112), we get

${\overline {\frac {d\phi }{da_{1}}}}={\frac {\overline {A_{1}}}{\Theta }},\quad {\overline {\frac {d\phi }{da_{2}}}}={\frac {\overline {A_{2}}}{\Theta }},\quad \mathrm {etc.}$

(327)

The first of these equations might be written^[1]

${\overline {{\bigg (}{\frac {d\phi }{da_{1}}}{\bigg )}}}_{\epsilon ,a}=-{\overline {{\bigg (}{\frac {d\phi }{d\epsilon }}{\bigg )}}}_{a}{\overline {{\bigg (}{\frac {d\epsilon }{da_{1}}}{\bigg )}}}_{a,q}$

(328)

but must not be confounded with the equation

${\overline {{\bigg (}{\frac {d\phi }{da_{1}}}{\bigg )}}}_{\epsilon ,a}=-{\overline {{\bigg (}{\frac {d\phi }{d\epsilon }}{\bigg )}_{a}{\bigg (}{\frac {d\epsilon }{da_{1}}}{\bigg )}_{\phi ,a}}}$

(329)

which is derived immediately from the identity

${\bigg (}{\frac {d\phi }{da_{1}}}{\bigg )}_{\epsilon ,a}=-{\bigg (}{\frac {d\phi }{d\epsilon }}{\bigg )}_{a}{\bigg (}{\frac {d\epsilon }{da_{1}}}{\bigg )}_{\phi ,a}$

(330)

1. See equations (321) and (104). Suffixes are here added to the differential coefficients, to make the meaning perfectly distinct, although the same quantities may be written elsewhere without the suffixes, when it is believed that there is no danger of misapprehension. The suffixes indicate the quantities which are constant in the differentiation, the single letter $a$ standing for all the letters $a_{1}$, $a_{2}$, etc., or all except the one which is explicitly varied.