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

64 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 Now the original extension-in-velocity is by definition represented by the integral  where the limits may be expressed by an equation of the form  The same integral multiplied by the constant $$\delta t^n$$ may be written  and the limits may be written  (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  or its equivalent  with limits expressed by the equation