Page:Elementary Principles in Statistical Mechanics (1902).djvu/103

But since identically

${\frac {\overline {(\epsilon _{p}-{\overline {\epsilon }}_{p})^{0}}}{{\overline {\epsilon }}_{p}{}^{0}}}=1,\quad {\frac {\overline {(\epsilon _{p}-{\overline {\epsilon }}_{p})^{1}}}{{\overline {\epsilon }}_{p}{}^{1}}}=0$

the value of the corresponding expression for any index will be independent of $\Theta$ and the formula reduces to

${\overline {{\bigg (}{\frac {\epsilon _{p}-{\overline {\epsilon }}_{p}}{{\overline {\epsilon }}_{p}}}{\bigg )}^{h+1}}}={\frac {2h}{n}}{\overline {{\bigg (}{\frac {\epsilon _{p}-{\overline {\epsilon }}_{p}}{{\overline {\epsilon }}_{p}}}{\bigg )}^{h}}}+{\frac {2h}{n}}{\overline {{\bigg (}{\frac {\epsilon _{p}-{\overline {\epsilon }}_{p}}{{\overline {\epsilon }}_{p}}}{\bigg )}^{h-1}}}$

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we have therefore

${\overline {{\bigg (}{\frac {\epsilon _{p}-{\overline {\epsilon }}_{p}}{{\overline {\epsilon }}_{p}}}{\bigg )}^{0}}}=1,\quad {\overline {{\bigg (}{\frac {\epsilon _{p}-{\overline {\epsilon }}_{p}}{{\overline {\epsilon }}_{p}}}{\bigg )}^{3}}}={\frac {8}{n^{2}}},$

${\overline {{\bigg (}{\frac {\epsilon _{p}-{\overline {\epsilon }}_{p}}{{\overline {\epsilon }}_{p}}}{\bigg )}^{1}}}=0,\quad {\overline {{\bigg (}{\frac {\epsilon _{p}-{\overline {\epsilon }}_{p}}{{\overline {\epsilon }}_{p}}}{\bigg )}^{4}}}={\frac {48}{n^{3}}}+{\frac {12}{n^{2}}},$

${\overline {{\bigg (}{\frac {\epsilon _{p}-{\overline {\epsilon }}_{p}}{{\overline {\epsilon }}_{p}}}{\bigg )}^{2}}}={\frac {2}{n}},\quad \mathrm {etc.}$[1]

It will be observed that when $\psi$ or ${\overline {\epsilon }}$ is given as function of $\Theta$, all averages of the form ${\overline {\epsilon ^{h}}}$ or ${\overline {(\epsilon -{\overline {\epsilon }})^{h}}}$ are thereby deter-

1. In the case discussed in the preceding foot-notes we get easily

   ${\overline {(\epsilon _{q}-{\overline {\epsilon }}_{q})^{h}}}={\overline {(\epsilon _{p}-{\overline {\epsilon }}_{p})^{h}}},$

   and

   ${\overline {{\bigg (}{\frac {\epsilon _{q}-{\overline {\epsilon }}_{q}}{{\overline {\epsilon }}_{q}-\epsilon _{a}}}{\bigg )}^{h}}}={\overline {{\bigg (}{\frac {\epsilon _{p}-{\overline {\epsilon }}_{p}}{{\overline {\epsilon }}_{p}}}{\bigg )}^{h}}}.$

   For the total energy we have in this case

   ${\overline {{\bigg (}{\frac {\epsilon -{\overline {\epsilon }}}{{\overline {\epsilon }}-\epsilon _{a}}}{\bigg )}^{h+1}}}={\frac {h}{n}}{\overline {{\bigg (}{\frac {\epsilon -{\overline {\epsilon }}}{{\overline {\epsilon }}-\epsilon _{a}}}{\bigg )}^{h}}}+{\overline {{\frac {h}{n}}{\bigg (}{\frac {\epsilon -{\overline {\epsilon }}}{{\overline {\epsilon }}-\epsilon _{a}}}{\bigg )}^{h-1}}}.$

   ${\overline {{\bigg (}{\frac {\epsilon -{\overline {\epsilon }}}{{\overline {\epsilon }}-\epsilon _{a}}}{\bigg )}^{2}}}={\frac {1}{n}}.\quad {\overline {{\bigg (}{\frac {\epsilon -{\overline {\epsilon }}}{{\overline {\epsilon }}-\epsilon _{a}}}{\bigg )}^{4}}}={\frac {3}{n^{2}}}+{\frac {6}{n^{3}}},$

   ${\overline {{\bigg (}{\frac {\epsilon -{\overline {\epsilon }}}{{\overline {\epsilon }}-\epsilon _{a}}}{\bigg )}^{3}}}={\frac {2}{n^{2}}},\quad \mathrm {etc.}$