Page:Elementary Principles in Statistical Mechanics (1902).djvu/52

where the limits of the multiple integrals are formed by the same phases. Hence

${\frac {d(r_{1},r_{2})}{d(a,b)}}={\frac {d(r_{1},\ldots r_{2n})}{d(a,\ldots h)}}{\frac {d(c,\ldots h)}{d(r_{3},\ldots r_{2n})}},$

(75)

With the aid of this equation, which is an identity, and (72), we may write equation (74) in the form

${\frac {d(r_{1},\ldots r_{2n})}{d(a,\ldots h)}}{\frac {d(c,\ldots h)}{d(r_{3},\ldots r_{2n})}}\,db={\dot {r}}_{2}\,dr_{1}-{\dot {r}}_{1}\,dr_{2}.$

(76)

The separation of the variables is now easy. The differential equations of motion give ${\dot {r}}_{1}$ and ${\dot {r}}_{2}$ in terms of $r_{1},\ldots r_{2n}$. The integral equations already obtained give $c,\ldots h$ and therefore the Jacobian $d(c,\ldots h)/d(r_{3},\ldots r_{2n})$, in terms of the same variables. But in virtue of these same integral equations, we may regard functions of $r_{1},\ldots r_{2n}$ as functions of $r_{1}$ and $r_{2}$ with the constants $c,\ldots h$. If therefore we write the equation in the form

${\frac {d(r_{1},\ldots r_{2n})}{d(a,\ldots h)}}\,db={\frac {{\dot {r}}_{2}}{\frac {d(c,\ldots h)}{d(r_{3},\ldots r_{2n})}}}\,dr_{1}-{\frac {{\dot {r}}_{1}}{\frac {d(c,\ldots h)}{d(r_{3},\ldots r_{2n})}}}\,dr_{2},$

(77)

the coefficients of $dr_{1}$ and $dr_{2}$ may be regarded as known functions of $r_{1}$ and $r_{2}$ with the constants $c,\ldots h$. The coefficient of $db$ is by (73) a function of $b,\ldots h$. It is not indeed a known function of these quantities, but since $c,\ldots h$ are regarded as constant in the equation, we know that the first member must represent the differential of some function of $b,\ldots h$, for which we may write $b'$. We have thus

$db'={\frac {{\dot {r}}_{2}}{\frac {d(c,\ldots h)}{d(r_{3},\ldots r_{2n})}}}\,dr_{1}-{\frac {{\dot {r}}_{1}}{\frac {d(c,\ldots h)}{d(r_{3},\ldots r_{2n})}}}\,dr_{2},$

(78)

which may be integrated by quadratures and gives $b'$ as functions of $r_{1},r_{2},\ldots c,\ldots h$, and thus as function of $r_{1},\ldots r_{2n}$. This integration gives us the last of the arbitrary constants which are functions of the coördinates and momenta without the time. The final integration, which introduces the remain-