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

80 . So also if $$\psi_q$$ or $$\overline\epsilon_q$$ is given as a function of $$\Theta$$, all averages of the form $$\overline{\epsilon_q{}^h}$$ or $$\overline{(\epsilon_q - \overline\epsilon_q)^h)}$$ are determined. But Therefore if any one of the quantities $$\psi$$, $$\psi_q$$, $$\overline\epsilon$$, $$\overline\epsilon_q$$ is known as a function of $$\Theta$$, and $$n$$ 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 $$n$$ 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 $$a_1$$, and multiply by $$\Theta$$, we have  Differentiating again, with respect to $$a_1$$, with respect to $$a_2$$, and with respect to $$\Theta$$, we have