Page:Elementary Principles in Statistical Mechanics (1902).djvu/104

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mined. So also if ${\displaystyle \psi _{q}}$ or ${\displaystyle {\overline {\epsilon }}_{q}}$ is given as a function of ${\displaystyle \Theta }$, all averages of the form ${\displaystyle {\overline {\epsilon _{q}{}^{h}}}}$ or ${\displaystyle {\overline {(\epsilon _{q}-{\overline {\epsilon }}_{q})^{h})}}}$ are determined. But

Therefore if any one of the quantities ${\displaystyle \psi }$, ${\displaystyle \psi _{q}}$, ${\displaystyle {\overline {\epsilon }}}$, ${\displaystyle {\overline {\epsilon }}_{q}}$ is known as a function of ${\displaystyle \Theta }$, and ${\displaystyle n}$ is also known, all averages of any of the forms mentioned are thereby determined as functions of the same variable. In any case all averages of the form are known in terms of ${\displaystyle n}$ alone, and have the same value whether taken for the whole ensemble or limited to any particular configuration.

If we differentiate the equation

with respect to ${\displaystyle a_{1}}$, and multiply by ${\displaystyle \Theta }$, we have Differentiating again, with respect to ${\displaystyle a_{1}}$, with respect to ${\displaystyle a_{2}}$, and with respect to ${\displaystyle \Theta }$, we have