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

Rh 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 $$\Theta$$:  The two last equations give

If $$\psi$$ or $$\overline\epsilon$$ is known as function of $$\Theta$$, $$a_1$$, $$a_2$$, etc., $$\overline{(\epsilon-\overline\epsilon)^2}$$ may be obtained by differentiation as function of the same variables. And if $$\psi$$, or $$\overline A_1$$, or $$\overline\eta$$ is known as function of $$\Theta$$, $$a_1$$, etc., $$\overline{(A_1 - \overline A_1)(\epsilon - \overline\epsilon)}$$ may be obtained by differentiation. But $$\overline{(A_1 - \overline A_1)^2}$$ and $$\overline{(A_1 - \overline A_1)(A_2 - \overline A_2)}$$ cannot be obtained in any similar manner. We have seen that $$\overline{(\epsilon - \overline\epsilon)^2}$$ is in general a vanishing quantity for very great values of $$n$$, which we may regard as contained implicitly in $$\Theta$$ as a divisor. The same is true of $$\overline{(A_1 - \overline A_1)(\epsilon - \overline\epsilon)}$$. It does not appear that we can assert the same of $$\overline{(A_1 - \overline A_1)^2}$$ or $$\overline{(A_1 - \overline A_1)(A_2 - \overline A_2)}$$, since