Page:Elementary Principles in Statistical Mechanics (1902).djvu/105

The multiple integrals in the last four equations represent the average values of the expressions In the brackets, which we may therefore set equal to zero. The first gives

as already obtained. With this relation and (191) we get from the other equations We may add for comparison equation (205), which might be derived from (236) by differentiating twice with respect to ${\displaystyle \Theta }$: The two last equations give If ${\displaystyle \psi }$ or ${\displaystyle {\overline {\epsilon }}}$ is known as function of ${\displaystyle \Theta }$, ${\displaystyle a_{1}}$, ${\displaystyle a_{2}}$, etc., ${\displaystyle {\overline {(\epsilon -{\overline {\epsilon }})^{2}}}}$ may be obtained by differentiation as function of the same variables. And if ${\displaystyle \psi }$, or ${\displaystyle {\overline {A}}_{1}}$, or ${\displaystyle {\overline {\eta }}}$ is known as function of ${\displaystyle \Theta }$, ${\displaystyle a_{1}}$, etc., ${\displaystyle {\overline {(A_{1}-{\overline {A}}_{1})(\epsilon -{\overline {\epsilon }})}}}$ may be obtained by differentiation. But ${\displaystyle {\overline {(A_{1}-{\overline {A}}_{1})^{2}}}}$ and ${\displaystyle {\overline {(A_{1}-{\overline {A}}_{1})(A_{2}-{\overline {A}}_{2})}}}$ cannot be obtained in any similar manner. We have seen that ${\displaystyle {\overline {(\epsilon -{\overline {\epsilon }})^{2}}}}$ is in general a vanishing quantity for very great values of ${\displaystyle n}$, which we may regard as contained implicitly in ${\displaystyle \Theta }$ as a divisor. The same is true of ${\displaystyle {\overline {(A_{1}-{\overline {A}}_{1})(\epsilon -{\overline {\epsilon }})}}}$. It does not appear that we can assert the same of ${\displaystyle {\overline {(A_{1}-{\overline {A}}_{1})^{2}}}}$ or ${\displaystyle {\overline {(A_{1}-{\overline {A}}_{1})(A_{2}-{\overline {A}}_{2})}}}$, since