Page:Elementary Principles in Statistical Mechanics (1902).djvu/137

Rh If $$V_q$$ is a continuous increasing function of $$\epsilon_q$$, commencing with $$V_q = 0$$, the average value in a canonical ensemble of any function of $$\epsilon_q$$, either alone or with the modulus and the external coördinates, is given by equation (275), which is identical with (357) except that $$\epsilon$$, $$\phi$$, and $$\psi$$ have the suffix $$(~)_q$$. The equation may be transformed so as to give an equation identical with (359) except for the suffixes. If we add the same suffixes to equation (361), the finite value of its members will determine the possibility of the canonical distribution.

From these data, it is easy to derive equations similar to (360), (362)-(372), except that the conditions of their validity must be differently stated. The equation requires only the condition already mentioned with respect to $$V_q$$. This equation corresponds to (362), which is subject to no restriction with respect to the value of $$n$$. We may observe, however, that $$V$$ will always satisfy a condition similar to that mentioned with respect to $$V_q$$.

If $$V_q$$ satisfies the condition mentioned, and $$e^{\phi_q}$$ a similar condition, i. e., if $$e^{\phi_q}$$ is a continuous increasing function of $$\epsilon_q$$, commencing with the value $$e^{\phi_q} = 0$$, equations will hold similar to those given for the case when $$n > 2$$, viz., similar to (360), (364)-(368). Especially important is If $$V_q$$, $$e^{\phi_q}$$ (or $$dV_q/d\epsilon_q$$), $$d^2 V_q/d\epsilon_q{}^2$$ all satisfy similar conditions, we shall have an equation similar to (369), which was subject to the condition $$n > 4$$. And if $$d^3 V_q/d\epsilon_q{}^3$$ also satisfies a similar condition, we shall have an equation similar to (372), for which the condition was $$n > 6$$. Finally, if $$V_q$$ and $$h$$ successive differential coefficients satisfy conditions of the kind mentioned, we shall have equations like (370) and (371) for which the condition was $$n > 2h$$.