Page:Elementary Principles in Statistical Mechanics (1902).djvu/135

where $\psi '$ denotes the value of $\psi$ for the modulus $\Theta '$. Since the last member of this formula vanishes for $\epsilon =\infty$, the less value represented by the first member must also vanish for the same value of $\epsilon$. Therefore the second member of (359), which differs only by a constant factor, vanishes at the upper limit. The case of the lower limit remains to be considered. Now

$e^{-{\frac {\epsilon }{\Theta }}}V\leq \int \limits _{V=0}^{\epsilon }e^{-{\frac {\epsilon }{\Theta }}+\phi }\,d\epsilon .$

The second member of this formula evidently vanishes for the value of $\epsilon$, which gives $V=0$, whether this be finite or negative infinity. Therefore, the second member of (359) vanishes at the lower limit also, and we have

$1-{\overline {e^{-\phi }V{\frac {d\phi }{d\epsilon }}}}-{\overline {e^{-\phi }{\frac {V}{\Theta }}}}+{\overline {e^{-\phi }V{\frac {d\phi }{d\epsilon }}}}=0,$

or

${\overline {e^{-\phi }V}}=\Theta .$

(362)

This equation, which is subject to no restriction in regard to the value of $n$, suggests a connection or analogy between the function of the energy of a system which is represented by $e^{-\phi }V$ and the notion of temperature in thermodynamics. We shall return to this subject in Chapter XIV.

If $n>2$, the second member of (359) may easily be shown to vanish for any of the following values of $u$ viz.: $\phi$, $e^{\phi }$, $\epsilon$, $\epsilon ^{m}$, where $m$ denotes any positive number. It will also vanish, when $n>4$, for $u=d\phi /d\epsilon$, and when $n>2h$ for $u=e^{-\phi }\,d^{h}V/d\epsilon ^{h}$. When the second member of (359) vanishes, and $n>2$, we may write

${\overline {(u-{\overline {u}}){\bigg (}{\frac {d\phi }{d\epsilon }}-{\frac {1}{\Theta }}{\bigg )}}}={\overline {u{\frac {d\phi }{d\epsilon }}}}-{\frac {\overline {u}}{\Theta }}=-{\overline {\frac {du}{d\epsilon }}}.$

(363)

We thus obtain the following equations:

If $n>2$,

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

(364)