Page:Elementary Principles in Statistical Mechanics (1902).djvu/131

whence

${\overline {{\bigg (}{\frac {d\phi }{d\epsilon }}-{\frac {1}{\Theta }}{\bigg )}(\epsilon -\epsilon _{0})}}={\bigg (}{\frac {d^{2}\phi }{d\epsilon ^{2}}}{\bigg )}_{0}{\overline {(\epsilon -\epsilon _{0})^{2}}}=-1,$

(349)

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

(350)

This is of the order of magnitude of $n$.^[1]

It should be observed that the approximate distribution of the ensemble in energy according to the 'law of errors' is not dependent on the particular form of the function of the energy which we have assumed for the index of probability ($\eta$). In any case, we must have

$\int \limits _{V=0}^{\epsilon =\infty }e^{\eta +\phi }\,d\epsilon =1,$

(351)

where $e^{\eta +\phi }$ is necessarily positive. This requires that it shall vanish for $\epsilon =\infty$, and also for $\epsilon =-\infty$, if this is a possible value. It has been shown in the last chapter that if $\epsilon$ has a (finite) least possible value (which is the usual case) and $n>2$, $e^{\phi }$ will vanish for that least value of $\epsilon$. In general therefore $\eta +\phi$ will have a maximum, which determines the most probable value of the energy. If we denote this value by $\epsilon _{0}$ and distinguish the corresponding values of the functions of the energy by the same suffix, we shall have

${\bigg (}{\frac {d\eta }{d\epsilon }}{\bigg )}_{0}+{\bigg (}{\frac {d\phi }{d\epsilon }}{\bigg )}_{0}=0.$

(352)

The probability that an unspecified system of the ensemble

1. We shall find hereafter that the equation

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

   is exact for any value of $n$ greater than 2, and that the equation

   ${\overline {{\bigg (}{\frac {d\phi }{d\epsilon }}-{\frac {1}{\Theta }}{\bigg )}^{2}}}=-{\overline {\frac {d^{2}\phi }{d\epsilon ^{2}}}}$

   is exact for any value of $n$ greater than 4.