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

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

${\Bigg (}{\frac {dD}{dt}}{\Bigg )}_{p,q}dt+\Sigma {\Bigg (}{\frac {dD}{dp_{1}}}{\dot {p}}_{1}dt+{\frac {dD}{dq_{1}}}{\dot {q}}_{1}dt{\Bigg )}=0$

(21)

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.^[1]

This may be called the principle of conservation of density-in-phase. It may also be written

${\Bigg (}{\frac {dD}{dt}}{\Bigg )}_{a,\ldots h}=0,$

(22)

where $a,\ldots h$ represent the arbitrary constants of the integral equations of motion, and are suffixed to the differential co-

1. The condition that the forces $F_{1},\ldots F_{n}$ are functions of $q_{1},\ldots q_{n}$ and $a_{1}$, $a_{2}$, etc., which last are functions of the time, is analytically equivalent to the condition that $F_{1},\ldots F_{n}$ are functions of $q_{1},\ldots q_{n}$ and the time. Explicit mention of the external coördinates, $a_{1}$, $a_{2}$, etc., has been made in the preceding pages, because our purpose will require us hereafter to consider these coördinates and the connected forces, $A_{1}$, $A_{2}$, etc., which represent the action of the systems on external bodies.