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

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

(72)

Moreover, since ${\displaystyle b,\ldots h}$ and ${\displaystyle t-a}$ are independent functions of ${\displaystyle r_{1},\ldots r_{2n}}$, the latter variables are functions of the former. The Jacobian in (71) is therefore function of ${\displaystyle b,\ldots h}$, and ${\displaystyle t-a}$, and since it does not vary with ${\displaystyle t}$ it cannot vary with ${\displaystyle a}$. We have therefore in the case considered, viz., where the forces are functions of the coördinates alone,

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

(73)

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

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

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

whence

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

(74)

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

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