where ${\displaystyle \psi '}$ denotes the value of ${\displaystyle \psi }$ for the modulus ${\displaystyle \Theta '}$. Since the last member of this formula vanishes for ${\displaystyle \epsilon =\infty }$, the less value represented by the first member must also vanish for the same value of ${\displaystyle \epsilon }$. 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

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

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

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

or

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

(362)

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

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

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

(363)

We thus obtain the following equations:

If ${\displaystyle n>2}$,

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

(364)