Page:Elementary Principles in Statistical Mechanics (1902).djvu/94

70 are also identical with those given by Clausius for the corresponding quantities.

Equations (112) and (181) show that if $$\psi$$ or $$\psi_q$$ is known as function of $$\Theta$$ and $$a_1$$, $$a_2$$, etc., we can obtain by differentiation $$\overline\epsilon$$ or $$\overline\epsilon_q$$, and $$\overline A_1$$, $$\overline A_2$$ etc. as functions of the same variables. We have in fact  The corresponding equation relating to kinetic energy,  which may be obtained in the same way, may be verified by the known relations (186), (187), and (188) between the variables. We have also etc., so that the average values of the external forces may be derived alike from $$\psi$$ or from $$\psi_q$$.

The average values of the squares or higher powers of the energies (total, potential, or kinetic) may easily be obtained by repeated differentiations of $$\psi$$, $$\psi_q$$, $$\psi_p$$, or $$\overline\epsilon$$, $$\overline\epsilon_q$$, $$\overline\epsilon_p$$, with respect to $$\Theta$$. By equation (108) we have and differentiating with respect to $$\Theta$$,  whence, again by (108),