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

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

${\begin{vmatrix}\epsilon _{11}&\epsilon _{12}&\ldots &\epsilon _{1n}\\\epsilon _{21}&\epsilon _{22}&\ldots &\epsilon _{2n}\\\ldots &\ldots &\ldots &\ldots \\\epsilon _{n1}&\epsilon _{n2}&\ldots &\epsilon _{nn}\end{vmatrix}}$

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.,

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

by which the notion of an element of extension-in-velocity was