Page:Elementary Principles in Statistical Mechanics (1902).djvu/54

taken within limits formed by phases regarded as contemporaneous represents the extension-in-phase within those limits.

The case is somewhat different when the forces are not determined by the coördinates alone, but are functions of the coördinates with the time. All the arbitrary constants of the integral equations must then be regarded in the general case as functions of $r_{1}\ldots r_{2n}$, and $t$. We cannot use the principle of conservation of extension-in-phase until we have made $2n-1$ integrations. Let us suppose that the constants $b,\ldots h$ have been determined by integration in terms of $r_{1}\ldots r_{2n}$, and $t$, leaving a single constant ($a$) to be thus determined. Our $2n-1$ finite equations enable us to regard all the variables $r_{1}\ldots r_{2n}$ as functions of a single one, say $r_{1}$.

For constant values of $b,\ldots h$, we have

$dr_{1}={\frac {dr_{1}}{da}}\,da+{\dot {r}}_{1}\,dt.$

(82)

Now

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

where the limits of the integrals are formed by the same phases. We have therefore

${\frac {dr_{1}}{da}}={\frac {d(r_{1},\ldots r_{2n})}{d(a,\ldots h)}}{\frac {d(b,\ldots h)}{d(r_{2},\ldots r_{2n})}},$

(83)

by which equation (82) may be reduced to the form

${\frac {d(r_{1},\ldots r_{2n})}{d(a,\ldots h)}}\,da={\frac {1}{\frac {d(b,\ldots h)}{d(r_{2},\ldots r_{2n})}}}\,dr_{1}-{\frac {{\dot {r}}_{1}}{\frac {d(b,\ldots h)}{d(r_{2},\ldots r_{2n})}}}\,dt.$

(84)

Now we know by (71) that the coefficient of $da$ is a function of $a,\ldots h$. Therefore, as $b,\ldots h$ are regarded as constant in the equation, the first number represents the differential