Page:Elementary Principles in Statistical Mechanics (1902).djvu/87

Comparing (160) and (162) with (40), we get

$e^{\eta _{q}}e^{\eta _{p}}=P=e^{\eta }$

(XXX)

or

$\eta _{q}+\eta _{p}=\eta .$

(XXX)

That is: the product of the coefficients of probability of configuration and of velocity is equal to the coefficient of probability of phase; the sum of the indices of probability of configuration and of velocity is equal to the index of probability of phase.

It is evident that $e^{\eta _{q}}$ and $e^{\eta _{p}}$ have the dimensions of the reciprocals of extension-in-configuration and extension-in-velocity respectively, i. e., the dimensions of $t^{-n}\epsilon ^{-{\frac {n}{2}}}$ and $\epsilon ^{-{\frac {n}{2}}}$, where $t$ represent any tine, and $\epsilon$ any energy. If, therefore, the unit of time is multiplied by $c_{t}$, and the unit of energy by $c_{\epsilon }$, every $\eta _{q}$ will be increased by the addition of

$n\log c_{t}+{\tfrac {1}{2}}n\log c_{\epsilon },$

(168)

and every $\eta _{p}$ by the addition of

${\tfrac {1}{2}}n\log c_{\epsilon }.$[1]

(169)

It should be observed that the quantities which have been called extension-in-configuration and extension-in-velocity are not, as the terms might seem to imply, purely geometrical or kinematical conceptions. To express their nature more fully, they might appropriately have been called, respectively, the dynamical measure of the extension in configuration, and the dynamical measure of the extension in velocity. They depend upon the masses, although not upon the forces of the system. In the simple case of material points, where each point is limited to a given space, the extension-in-configuration is the product of the volumes within which the several points are confined (these may be the same or different), multiplied by the square root of the cube of the product of the masses of the several points. The extension-in-velocity for such systems is most easily defined as the extension-in-configuration of systems which have moved from the same configuration for the unit of time with the given velocities.

1. Compare (47) in Chapter I.