Page:Elementary Principles in Statistical Mechanics (1902).djvu/115

Rh the integration being extended, with constant values of the coördinates, both internal and external, over all values of the momenta for which the kinetic energy is less than the limit $$\epsilon_p$$. $$V_p$$ will evidently be a continuous increasing function of $$\epsilon_p$$ which vanishes and becomes infinite with $$\epsilon_p$$. Let us set The extension-in-velocity between any two limits of kinetic energy $$\epsilon_p'$$ and $$\epsilon_p''$$ may be represented by the integral  And in general, we may substitute $$e^{\phi_p}\,d\epsilon_p$$ for $$\Delta_p^{\frac 12} dp_1\ldots dp_n$$ or $$\Delta_{\dot q}^{\frac 12} d\dot q_1\ldots d\dot q_n$$ in an $$n$$-fold integral in which the coördinates are constant, reducing it to a simple integral, when the limits are expressed by the kinetic energy, and the other factor under the integral sign is a function of the kinetic energy, either alone or with quantities which are constant in the integration.

It is easy to express $$V_p$$ and $$\phi_p$$ in terms of $$\epsilon_p$$. Since $$\Delta_p$$ is function of the coördinates alone, we have by definition the limits of the integral being given by $$\epsilon_p$$. That is, if the limits of the integral for $$\epsilon_p=1$$, are given by the equation  and the limits of the integral for $$\epsilon_p = a^2$$, are given by the equation  But since $$F$$ represents a quadratic function, this equation may be written