Page:Elementary Principles in Statistical Mechanics (1902).djvu/36

12 second configuration will coincide with the third system of coördinates. This will give $$D_2' = D_3''$$. We have therefore $$D_1' = D_2'$$.

It follows, or it may be proved in the same way, that the value of an extension-in-phase is independent of the system of coördinates which is used in its evaluation. This may easily be verified directly. If $$q_1, \ldots q_n$$, $$Q_1, \ldots Q_n$$ are two systems of coördinates, and $$p_1, \ldots p_n$$, $$P_1, \ldots P_n$$ the corresponding momenta, we have to prove that when the multiple integrals are taken within limits consisting of the same phases. And this will be evident from the principle on which we change the variables in a multiple integral, if we prove that where the first member of the equation represents a Jacobian or functional determinant. Since all its elements of the form $$dQ/dp$$ are equal to zero, the determinant reduces to a product of two, and we have to prove that We may transform any element of the first of these determinants as follows. By equations (2) and (3), and in view of the fact that the $$\dot Q$$'s are linear functions of the $$\dot q$$'s and therefore of the $$p$$'s, with coefficients involving the $$q$$'s, so that a differential coefficient of the form $$d\dot Q_r/dp_y$$ is function of the $$q$$'s alone, we get