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

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

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

(75)

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

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

(76)

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

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

(77)

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

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

(78)

which may be integrated by quadratures and gives ${\displaystyle b'}$ as functions of ${\displaystyle r_{1},r_{2},\ldots c,\ldots h}$, and thus as function of ${\displaystyle r_{1},\ldots r_{2n}}$. 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-