Now we have identically

${\displaystyle \int \limits _{V=0}^{\epsilon =\infty }{\bigg (}{\frac {du}{d\epsilon }}-{\frac {u}{\Theta }}+u{\frac {d\phi }{d\epsilon }}{\bigg )}e^{{\frac {\psi -\epsilon }{\Theta }}+\phi }\,d\epsilon ={\bigg [}u\,e^{{\frac {\psi -\epsilon }{\Theta }}+\phi }{\bigg ]}_{V=0}^{\epsilon =\infty }}$

(358)

Therefore, by the preceding equation

${\displaystyle {\overline {\frac {du}{d\epsilon }}}-{\frac {\overline {u}}{\Theta }}+{\overline {u{\frac {d\phi }{d\epsilon }}}}={\bigg [}u\,e^{{\frac {\psi -\epsilon }{\Theta }}+\phi }{\bigg ]}_{V=0}^{\epsilon =\infty }}$

[1](359)

If we set ${\displaystyle u=1}$, (a value which need not be excluded,) the second member of this equation vanishes, as shown on page 101, if ${\displaystyle n>2}$, and we get

${\displaystyle {\overline {\frac {d\phi }{d\epsilon }}}={\frac {1}{\Theta }}}$

(360)

as before. It is evident from the same considerations that the second member of (359) will always vanish if ${\displaystyle n>2}$, unless ${\displaystyle u}$ becomes infinite at one of the limits, in which case a more careful examination of the value of the expression will be necessary. To facilitate the discussion of such cases, it will be convenient to introduce a certain limitation in regard to the nature of the system considered. We have necessarily supposed, in all our treatment of systems canonically distributed, that the system considered was such as to be capable of the canonical distribution with the given value of the modulus. We shall now suppose that the system is such as to be capable of a canonical distribution with any (finite)^[2] modulus. Let us see what cases we exclude by this last limitation.

^[1]

1. ↑ A more general equation, which is not limited to ensembles canonically distributed, is

   ${\displaystyle {\overline {\frac {du}{d\epsilon }}}+{\overline {u{\frac {d\eta }{d\epsilon }}}}+{\overline {u{\frac {d\phi }{d\epsilon }}}}={\bigg [}u\,e^{\eta +\phi }{\bigg ]}_{V=0}^{\epsilon =\infty }}$

   where ${\displaystyle \eta }$ denotes, as usual, the index of probability of phase.
2. ↑ The term finite applied to the modulus is intended to exclude the value zero as well as infinity.