Page:Elementary Principles in Statistical Mechanics (1902).djvu/51

Rh remaining constant ($$a$$) will then be introduced in the final integration, (viz., that of an equation containing $$dt$$,) and will be added to or subtracted from $$t$$ in the integral equation. Let us have it subtracted from $$t$$. It is evident then that

Moreover, since $$b,\ldots h$$ and $$t-a$$ are independent functions of $$r_1,\ldots r_{2n}$$, the latter variables are functions of the former. The Jacobian in (71) is therefore function of $$b,\ldots h$$, and $$t-a$$, and since it does not vary with $$t$$ it cannot vary with $$a$$. We have therefore in the case considered, viz., where the forces are functions of the coördinates alone,

Now let us suppose that of the first $$2n-1$$ integrations we have accomplished all but one, determining $$2n-2$$ arbitrary constants (say $$c,\ldots h$$) as functions of $$r_1,\ldots r_{2n}$$, leaving $$b$$ as well as $$a$$ to be determined. Our $$2n-2$$ finite equations enable us to regard all the variables $$r_1,\ldots r_{2n}$$, and all functions of these variables as functions of two of them, (say $$r_1$$ and $$r_2$$,) with the arbitrary constants $$c,\ldots h$$. To determine $$b$$, we have the following equations for constant values of $$c,\ldots h$$. whence Now, by the ordinary formula for the change of variables,