Page:Elementary Principles in Statistical Mechanics (1902).djvu/113

Rh In like manner, let us denote by $$V_q$$ the extension-in-configuration below a certain limit of potential energy which we may call $$\epsilon_q$$. That is, let the integration being extended (with constant values of the external coördinates) over all configurations for which the potential energy is less than $$\epsilon_q$$. $$V_q$$ will be a function of $$\epsilon_q$$ with the external coördinates, an increasing function of $$\epsilon_q$$, which does not become infinite (in such cases as we shall consider ) for any finite value of $$\epsilon_q$$. It vanishes for the least possible value of $$\epsilon_q$$, or for $$\epsilon_q = -\infty$$, if $$\epsilon_q$$ can be diminished without limit. It is not always a continuous function of $$\epsilon_q$$. In fact, if there is a finite extension-in-configuration of constant potential energy, the corresponding value of $$V_q$$ will not include that extension-in-configuration, but if $$\epsilon_q$$ be increased infinitesimally, the corresponding value of $$V_q$$ will be increased by that finite extension-in-configuration.

Let us also set The extension-in-configuration between any two limits of potential energy $$\epsilon_q'$$ and $$\epsilon_q''$$ may be represented by the integral  whenever there is no discontinuity in the value of $$V_q$$ as function of $$\epsilon_q$$ between or at those limits, that is, whenever there is no finite extension-in-configuration of constant potential energy between or at the limits. And in general, with the restriction mentioned, we may substitute $$e^{\phi_q}\,d\epsilon_q$$ for $$\Delta_{\dot q} dq_1 \ldots dq_n,$$ in an $$n$$-fold integral, reducing it to a simple integral, when the limits are expressed by the potential energy, and the other factor under the integral sign is a function of