Page:Elementary Principles in Statistical Mechanics (1902).djvu/35

which we have called the extension-in-phase, is also constant in time.^[1]

Since the system of coördinates employed in the foregoing discussion is entirely arbitrary, the values of the coördinates relating to any configuration and its immediate vicinity do not impose any restriction upon the values relating to other configurations. The fact that the quantity which we have called density-in-phase is constant in time for any given system, implies therefore that its value is independent of the coördinates which are used in its evaluation. For let the density-in-phase as evaluated for the same time and phase by one system of coördinates be $D_{1}'$, and by another system $D_{2}'$. A system which at that time has that phase will at another time have another phase. Let the density as calculated for this second time and phase by a third system of coördinates be $D_{3}''$. Now we may imagine a system of coördinates which at and near the first configuration will coincide with the first system of coördinates, and at and near the second configuration will coincide with the third system of coördinates. This will give $D_{1}'=D_{3}''$. Again we may imagine a system of coördinates which at and near the first configuration will coincide with the second system of coördinates, and at and near the

1. If we regard a phase as represented by a point in space of $2n$ dimensions, the changes which take place in the course of time in our ensemble of systems will be represented by a current in such space. This current will be steady so long as the external coördinates are not varied. In any case the current will satisfy a law which in its various expressions is analogous to the hydrodynamic law which may be expressed by the phrases conservation of volumes or conservation of density about a moving point, or by the equation

   ${\frac {d{\dot {x}}}{dx}}+{\frac {d{\dot {y}}}{dy}}+{\frac {d{\dot {z}}}{dz}}=0.$

   The analogue in statistical mechanics of this equation, viz.,

   ${\frac {d{\dot {p}}_{1}}{dp_{1}}}+{\frac {d{\dot {q}}_{1}}{dq_{1}}}+{\frac {d{\dot {p}}_{2}}{dp_{2}}}+{\frac {d{\dot {q}}_{2}}{dq_{2}}}+\mathrm {etc.} =0$

   may be derived directly from equations (3) or (6), and may suggest such theorems as have been enunciated, if indeed it is not regarded as making them intuitively evident. The somewhat lengthy demonstrations given above will at least serve to give precision to the notions involved, and familiarity with their use.