falls within any given limits of energy (${\displaystyle \epsilon '}$ and ${\displaystyle \epsilon ''}$) is represented by

${\displaystyle \int \limits _{\epsilon '}^{\epsilon ''}e^{\eta +\phi }\,d\epsilon .}$

If we expand ${\displaystyle \eta }$ and ${\displaystyle \phi }$ in ascending powers of ${\displaystyle \epsilon -\epsilon _{0}}$, without going beyond the squares, the probability that the energy falls within the given limits takes the form of the 'law of errors'—

${\displaystyle e^{\eta _{0}+\phi _{0}}\int \limits _{\epsilon '}^{\epsilon ''}e^{{\big [}{\big (}{\frac {d^{2}\eta }{d\epsilon ^{2}}}{\big )}_{0}+{\big (}{\frac {d^{2}\phi }{d\epsilon ^{2}}}{\big )}_{0}{\big ]}{\frac {(\epsilon -\epsilon _{0})^{2}}{2}}}\,d\epsilon .}$

(353)

This gives

${\displaystyle \eta _{0}+\phi _{0}={\tfrac {1}{2}}\log {\bigg [}{\frac {-1}{2\pi }}{\bigg (}{\frac {d^{2}\eta }{d\epsilon ^{2}}}{\bigg )}_{0}-{\frac {1}{2\pi }}{\bigg (}{\frac {d^{2}\phi }{d\epsilon ^{2}}}{\bigg )}_{0}{\bigg ]},}$

(354)

and

${\displaystyle {\overline {(\epsilon -\epsilon _{0})^{2}}}={\bigg [}-{\bigg (}{\frac {d^{2}\eta }{d\epsilon ^{2}}}{\bigg )}_{0}-{\bigg (}{\frac {d^{2}\phi }{d\epsilon ^{2}}}{\bigg )}_{0}{\bigg ]}^{-1}.}$

(355)

We shall have a close approximation in general when the quantities equated in (355) are very small, i. e., when

${\displaystyle -{\bigg (}{\frac {d^{2}\eta }{d\epsilon ^{2}}}{\bigg )}_{0}-{\bigg (}{\frac {d^{2}\phi }{d\epsilon ^{2}}}{\bigg )}_{0}}$

(356)

is very great. Now when ${\displaystyle n}$ is very great, ${\displaystyle -d^{2}\phi /d\epsilon ^{2}}$ is of the same order of magnitude, and the condition that (356) shall be very great does not restrict very much the nature of the function ${\displaystyle \eta }$.

We may obtain other properties pertaining to average values in a canonical ensemble by the method used for the average of ${\displaystyle d\phi /d\epsilon }$. Let ${\displaystyle u}$ be any function of the energy, either alone or with ${\displaystyle \Theta }$ and the external coördinates. The average value of ${\displaystyle u}$ in the ensemble is determined by the equation

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

(357)