Page:Elementary Principles in Statistical Mechanics (1902).djvu/88

In the general case, the notions of extension-in-configuration and extension-in-velocity may be connected as follows.

If an ensemble of similar systems of $n$ degrees of freedom have the same configuration at a given instant, but are distributed throughout any finite extension-in-velocity, the same ensemble after an infinitesimal interval of time $\delta t$ will be distributed throughout an extension in configuration equal to its original extension-in-velocity multiplied by $\delta t^{n}$.

In demonstrating this theorem, we shall write $q_{1}{}',\ldots q_{n}{}'$ for the initial values of the coördinates. The final values will evidently be connected with the initial by the equations

$q_{1}-q_{1}{}'={\dot {q}}_{1}\delta t,\ldots q_{n}-q_{n}{}'={\dot {q}}_{n}\delta t.$

(170)

Now the original extension-in-velocity is by definition represented by the integral

$\int \ldots \int \Delta _{\dot {q}}{}^{\frac {1}{2}}\,d{\dot {q}}_{1}\ldots d{\dot {q}}_{n},$

(171)

where the limits may be expressed by an equation of the form

$F({\dot {q}}_{1},\ldots {\dot {q}}_{n})=0.$

(172)

The same integral multiplied by the constant $\delta t^{n}$ may be written

$\int \ldots \int \Delta _{\dot {q}}{}^{\frac {1}{2}}\,d({\dot {q}}_{1}\delta t),\ldots d({\dot {q}}_{n}\delta t),$

(173)

and the limits may be written

$F({\dot {q}}_{1},\ldots {\dot {q}}_{n})=f({\dot {q}}_{1}\delta t,\ldots {\dot {q}}_{n}\delta t)=0.$

(174)

(It will be observed that $\delta t$ as well as $\Delta _{\dot {q}}$ is constant in the integrations.) Now this integral is identically equal to

$\int \ldots \int \Delta _{q}{}^{\frac {1}{2}}\,d(q_{1}-q_{1}{}')\ldots d(q_{n}-q_{n}{}'),$

(175)

or its equivalent

$\int \ldots \int \Delta _{\dot {q}}{}^{\frac {1}{2}}\,dq_{1}\ldots dq_{n},$

(176)

with limits expressed by the equation

$f(q_{1}-q_{1}{}',\ldots q_{n}-q_{n}{}')=0.$

(177)