Page:Elementary Principles in Statistical Mechanics (1902).djvu/90

66 kinetic energy due to $$V_1$$ and $$V_2{}''$$ combined is the sum of the kinetic energies due to these velocities taken separately. And the velocity $$V_3$$ may be regarded as compounded of three, $$V_3{}'$$, $$V_3{}$$, $$V_3{}$$, of which $$V_3{}'$$ is of the same nature as $$V_1$$, $$V_3{}$$ of the same nature as $$V_2{}$$, while $$V_3{}$$ satisfies the relations that if combined either with $$V_1$$ or $$V_2{}$$ the kinetic energy of the combined velocities is the sum of the kinetic energies of the velocities taken separately. When all the velocities $$V_2,\ldots V_n$$ have been thus decomposed, the square root of the product of the doubled kinetic energies of the several velocities $$V_1$$, $$V_2{}$$, $$V_3{}'$$, etc., will be the value of the extension-in-velocity which is sought.

This method of evaluation of the extension-in- velocity which we are considering is perhaps the most simple and natural, but the result may be expressed in a more symmetrical form. Let us write $$\epsilon_{12}$$ for the kinetic energy of the velocities $$V_1$$ and $$V_2$$ combined, diminished by the sum of the kinetic energies due to the same velocities taken separately. This may be called the mutual energy of the velocities $$V_1$$ and $$V_2$$. Let the mutual energy of every pair of the velocities $$V_1,\ldots V_n$$ be expressed in the same way. Analogy would make $$\epsilon_{11}$$ represent the energy of twice $$V_1$$ diminished by twice the energy of $$V_1$$, i. e., $$\epsilon_{11}$$ would represent twice the energy of $$V_1$$, although the term mutual energy is hardly appropriate to this case. At all events, let $$\epsilon_{11}$$ have this signification, and $$\epsilon_{22}$$ represent twice the energy of $$V_2$$, etc. The square root of the determinant represents the value of the extension-in-velocity determined as above described by the velocities $$V_1,\ldots V_n$$.

The statements of the preceding paragraph may be readily proved from the expression (157) on page 60, viz., by which the notion of an element of extension-in-velocity was