Page:Elementary Principles in Statistical Mechanics (1902).djvu/151

be true for any value of $\omega$. If we diminish $\omega$, the average value of the parenthesis at the limit when $\omega$ vanishes becomes identical with the value for $\epsilon =\epsilon '$. But this may be any value of the energy, except the least possible. We have therefore

${\frac {d{\overline {A_{1}}}|_{\epsilon }}{d\epsilon }}+{\overline {A_{1}}}|_{\epsilon }\,{\frac {d\phi }{d\epsilon }}-{\frac {d\phi }{da_{1}}}=0,$

(410)

unless it be for the least value of the energy consistent with the external coördinates, or for particular values of the external coördinates. But the value of any term of this equation as determined for particular values of the energy and of the external coördinates is not distinguishable from its value as determined for values of the energy and external coördinates indefinitely near those particular values. The equation therefore holds without limitation. Multiplying by $e^{\phi }$, we get

$e^{\phi }{\frac {d{\overline {A_{1}}}|_{\epsilon }}{d\epsilon }}+{\overline {A_{1}}}|_{\epsilon }\,e^{\phi }\,{\frac {d\phi }{d\epsilon }}=e^{\phi }\,{\frac {d\phi }{da_{1}}}={\frac {de^{\phi }}{da_{1}}}={\frac {d^{2}V}{da_{1}\,d\epsilon }}.$

(411)

The integral of this equation is

${\overline {A_{1}}}|_{\epsilon }e^{\phi }={\frac {dV}{da_{1}}}+F_{1},,$

(412)

where $F_{1}$ is a function of the external coördinates. We have an equation of this form for each of the external coördinates. This gives, with (266), for the complete value of the differential of $V$

$dV=e^{\phi }\,d\epsilon +(e^{\phi }{\overline {A_{1}}}|_{\epsilon }-F_{1})da_{1}+(e^{\phi }{\overline {A_{2}}}|_{\epsilon }-F_{2})da_{2}+\mathrm {etc.} ,$

(413)

or

$dV=e^{\phi }(d\epsilon +{\overline {A_{1}}}|_{\epsilon }\,da_{1}+{\overline {A_{2}}}|_{\epsilon }\,da_{2}+\mathrm {etc.} )-F_{1}\,da_{1}-F_{2}\,da_{2}-\mathrm {etc.}$

(414)

To determine the values of the functions $F_{1}$, $F_{2}$, etc., let us suppose $a_{1}$, $a_{2}$, etc. to vary arbitrarily, while $\epsilon$ varies so as always to have the least value consistent with the values of the external coördinates. This will make $V=0$, and $dV=0$. If $n<2$, we shall have also $e^{\phi }=0$, which will give

$F_{1}=0,\quad F_{2}=0,\quad \mathrm {etc.}$

(415)