Page:Elementary Principles in Statistical Mechanics (1902).djvu/78

54 the fractional part of the ensemble which lies within any given limits of configuration (136) may be written where the constant $$\psi_q$$ may be determined by the condition that the integral extended over all configurations has the value unity. In the simple but important case in which $$\Delta_{\dot q}$$ is independent of the $$q$$'s, and $$\epsilon_q$$ a quadratic function of the $$q$$'s, if we write $$\epsilon_a$$ for the least value of $$\epsilon_q$$ (or of $$\epsilon$$) consistent with the given values of the external coördinates, the equation determining $$\psi_q$$ may be written If we denote by $$q_1{}',\ldots q_n{}'$$ the values of $$q_1,\ldots q_n$$ which give $$\epsilon_q$$ its least value $$\epsilon_a$$, it is evident that $$\epsilon_q - \epsilon_a$$ is a homogenous quadratic function of the differences $$q_1 - q_1{}'$$, etc., and that $$dq_1,\ldots dq_n$$ may be regarded as the differentials of these differences. The evaluation of this integral is therefore analytically similar to that of the integral for which we have found the value $$\Delta_p{}^{-\frac 12} (2\pi\Theta)^\frac n2$$. By the same method, or by analogy, we get where $$\Delta_q$$ is the Hessian of the potential energy as function of the $$q$$'s. It will be observed that $$\Delta_q$$ depends on the forces of the system and is independent of the masses, while $$\Delta_{\dot q}$$ or its reciprocal $$\Delta_p$$ depends on the masses and is independent of the forces. While each Hessian depends on the system of coördinates employed, the ratio $$\Delta_q/\Delta_{\dot q}$$ is the same for all systems.

Multiplying the last equation by (140), we have

For the average value of the potential energy, we have