Page:Elementary Principles in Statistical Mechanics (1902).djvu/76

52 the $$u$$'s. The integrals may always be taken from a less to a greater value of a $$u$$.

The general integral which expresses the fractional part of the ensemble which falls within any given limits of phase is thus reduced to the form

For the average value of the part of the kinetic energy which is represented by $$\tfrac 12 u_1{}^2$$ whether the average is taken for the whole ensemble, or for a given configuration, we have therefore and for the average of the whole kinetic energy, $$\tfrac 12 n\Theta$$, as before.

The fractional part of the ensemble which lies within any given limits of configuration, is found by integrating (184) with respect to the $$u$$'s from $$-\infty$$ to $$+\infty$$. This gives which shows that the value of the Jacobian is independent of the manner in which $$2\epsilon_p$$ is divided into a sum of squares. We may verify this directly, and at the same time obtain a more convenient expression for the Jacobian, as follows.

It will be observed that since the $$u$$'s are linear functions of the $$p$$'s, and the $$p$$'s linear functions of the $$\dot q$$'s, the $$u$$'s will be linear functions of the $$\dot q$$'s, so that a differential coefficient of the form $$du/d\dot q$$ will be independent of the $$\dot q$$'s, and function of the $$q$$'s alone. Let us write $$dp_x/du_y$$ for the general element of the Jacobian determinant. We have