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

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

${\frac {dr_{1}}{da}}=-{\dot {r}}_{1},\quad {\frac {dr_{2}}{da}}=-{\dot {r}}_{2},\quad \mathrm {etc.}$

(72)

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,

${\frac {d(r_{1},\ldots r_{2n})}{d(a,\ldots h)}}=\operatorname {func.} (b,\ldots h).$

(73)

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 $c,\ldots h$.

$dr_{1}={\frac {dr_{1}}{da}}\,da+{\frac {dr_{1}}{db}}\,db,$

$dr_{2}={\frac {dr_{2}}{da}}\,da+{\frac {dr_{2}}{db}}\,db,$

whence

${\frac {d(r_{1},r_{2})}{d(a,b)}}\,db=-{\frac {dr_{2}}{da}}\,dr_{1}+{\frac {dr_{1}}{da}}\,dr_{2}.$

(74)

Now, by the ordinary formula for the change of variables,

${\begin{aligned}&\int \ldots \int {\frac {d(r_{1},r_{2})}{d(a,b)}}\,da\,db\,dr_{3}\ldots dr_{2n}=\int \ldots \int dr_{1}\ldots dr_{2n}\\&\qquad \qquad {}=\int \ldots \int {\frac {d(r_{1},\ldots r_{2n})}{d(a,\ldots h)}}\,da\ldots dh\\&{}=\int \ldots \int {\frac {d(r_{1},\ldots r_{2n})}{d(a,\ldots h)}}{\frac {d(c,\ldots h)}{d(r_{3},\ldots r_{2n})}}\,da\,db\,dr_{3}\ldots dr_{2n},\end{aligned}}$