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

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

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

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

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 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-