Page:Elementary Principles in Statistical Mechanics (1902).djvu/91

originally defined. Since $\Delta _{\dot {q}}$ in this expression represents the determinant of which the general element is

${\frac {d^{2}\epsilon }{d{\dot {q}}_{i}\,d{\dot {q}}_{j}}}$

the square of the preceding expression represents the determinant of which the general element is

${\frac {d^{2}\epsilon }{d{\dot {q}}_{i}\,d{\dot {q}}_{j}}}\,d{\dot {q}}_{i}\,d{\dot {q}}_{j}.$

Now we may regard the differentials of velocity $d{\dot {q}}_{i}$, $d{\dot {q}}_{j}$ as themselves infinitesimal velocities. Then the last expression represents the mutual energy of these velocities, and

${\frac {d^{2}\epsilon }{d{\dot {q}}_{i}{}^{2}}}\,d{\dot {q}}_{i}{}^{2}$

represents twice the energy due to the velocity $d{\dot {q}}_{i}$.

The case which we have considered is an extension-in-velocity of the simplest form. All extensions-in-velocity do not have this form, but all may be regarded as composed of elementary extensions of this form, in the same manner as all volumes may be regarded as composed of elementary parallelepipeds.

Having thus a measure of extension-in-velocity founded, it will be observed, on the dynamical notion of kinetic energy, and not involving an explicit mention of coördinates, we may derive from it a measure of extension-in-configuration by the principle connecting these quantities which has been given in a preceding paragraph of this chapter.

The measure of extension-in-phase may be obtained from that of extension-in-configuration and of extension-in-velocity. For to every configuration in an extension-in-phase there will belong a certain extension-in-velocity, and the integral of the elements of extension-in-configuration within any extension-in-phase multiplied each by its extension-in-velocity is the measure of the extension-in-phase.