Page:Elementary Principles in Statistical Mechanics (1902).djvu/39

${\frac {d}{dt}}{\frac {d(p_{1},\ldots q_{n})}{d(p_{1}'',\ldots q_{n}'')}}=0,$

which substituted in (34) will give

${\frac {d}{dt}}{\frac {d(p_{1},\ldots q_{n})}{d(p_{1}',\ldots q_{n}')}}=0.$

The determinant in this equation is therefore a constant, the value of which may be determined at the instant when $t=t'$, when it is evidently unity. Equation (33) is therefore demonstrated.

Again, if we write $a,\ldots h$ for a system of $2n$ arbitrary constants of the integral equations of motion, $p_{1}$, $q_{1}$, etc. will be functions of $a,\ldots h$, and $t$, and we may express an extension-in-phase in the form

$\int \ldots \int {\frac {d(p_{1},\ldots q_{n})}{d(a,\ldots h)}}da\ldots dh.$

(35)

If we suppose the limits specified by values of $a,\ldots h$, a system initially at the limits will remain at the limits. The principle of conservation of extension-in-phase requires that an extension thus bounded shall have a constant value. This requires that the determinant under the integral sign shall be constant, which may be written

${\frac {d}{dt}}{\frac {d(p_{1},\ldots q_{n})}{d(a,\ldots h)}}=0.$

(36)

This equation, which may be regarded as expressing the principle of conservation of extension-in-phase, may be derived directly from the identity

${\frac {d(p_{1},\ldots q_{n})}{d(a,\ldots h)}}={\frac {d(p_{1},\ldots q_{n})}{d(p_{1}',\ldots q_{n}')}}{\frac {d(p_{1}',\ldots q_{n}')}{d(a,\ldots h)}}$

in connection with equation (33).

Since the coördinates and momenta are functions of $a,\ldots h$, and $t$, the determinant in (36) must be a function of the same variables, and since it does not vary with the time, it must be a function of $a,\ldots h$ alone. We have therefore

${\frac {d(p_{1},\ldots q_{n})}{d(a,\ldots h)}}=\operatorname {func.} (a,\ldots h).$

(37)