Page:Elementary Principles in Statistical Mechanics (1902).djvu/33

Rh would alter the values of the $$\dot p$$'s as determined by equations (3), and thus disturb the relation expressed in the last equation.

If we write equation (19) in the form it will be seen to express a theorem of remarkable simplicity. Since $$D$$ is a function of $$t$$, $$p_1, \ldots p_n$$, $$q_1 \ldots q_n$$, its complete differential will consist of parts due to the variations of all these quantities. Now the first term of the equation represents the increment of $$D$$ due to an increment of $$t$$ (with constant values of the $$p$$'s and $$q$$'s), and the rest of the first member represents the increments of $$D$$ due to increments of the $$p$$'s and $$q$$'s, expressed by $$\dot p_1 \, dt$$, $$\dot q_1 \, dt$$, etc. But these are precisely the increments which the $$p$$'s and $$q$$'s receive in the movement of a system in the time $$dt$$. The whole expression represents the total increment of $$D$$ for the varying phase of a moving system. We have therefore the theorem:—

In an ensemble of mechanical systems identical in nature and subject to forces determined by identical laws, but distributed in phase in any continuous manner, the density-in-phase is constant in time for the varying phases of a moving system; provided, that the forces of a system are functions of its coördinates, either alone or with the time.

This may be called the principle of conservation of density-in-phase. It may also be written where $$a,\ldots h$$ represent the arbitrary constants of the integral equations of motion, and are suffixed to the differential