kinetic energy due to ${\displaystyle V_{1}}$ and ${\displaystyle V_{2}{}''}$ combined is the sum of the kinetic energies due to these velocities taken separately. And the velocity ${\displaystyle V_{3}}$ may be regarded as compounded of three, ${\displaystyle V_{3}{}'}$, ${\displaystyle V_{3}{}''}$, ${\displaystyle V_{3}{}'''}$, of which ${\displaystyle V_{3}{}'}$ is of the same nature as ${\displaystyle V_{1}}$, ${\displaystyle V_{3}{}''}$ of the same nature as ${\displaystyle V_{2}{}''}$, while ${\displaystyle V_{3}{}'''}$ satisfies the relations that if combined either with ${\displaystyle V_{1}}$ or ${\displaystyle V_{2}{}''}$ the kinetic energy of the combined velocities is the sum of the kinetic energies of the velocities taken separately. When all the velocities ${\displaystyle V_{2},\ldots V_{n}}$ have been thus decomposed, the square root of the product of the doubled kinetic energies of the several velocities ${\displaystyle V_{1}}$, ${\displaystyle V_{2}{}''}$, ${\displaystyle V_{3}{}'''}$, 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 ${\displaystyle \epsilon _{12}}$ for the kinetic energy of the velocities ${\displaystyle V_{1}}$ and ${\displaystyle V_{2}}$ 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 ${\displaystyle V_{1}}$ and ${\displaystyle V_{2}}$. Let the mutual energy of every pair of the velocities ${\displaystyle V_{1},\ldots V_{n}}$ be expressed in the same way. Analogy would make ${\displaystyle \epsilon _{11}}$ represent the energy of twice ${\displaystyle V_{1}}$ diminished by twice the energy of ${\displaystyle V_{1}}$, i. e., ${\displaystyle \epsilon _{11}}$ would represent twice the energy of ${\displaystyle V_{1}}$, although the term mutual energy is hardly appropriate to this case. At all events, let ${\displaystyle \epsilon _{11}}$ have this signification, and ${\displaystyle \epsilon _{22}}$ represent twice the energy of ${\displaystyle V_{2}}$, etc. The square root of the determinant

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

represents the value of the extension-in-velocity determined as above described by the velocities ${\displaystyle V_{1},\ldots V_{n}}$.

The statements of the preceding paragraph may be readily proved from the expression (157) on page 60, viz.,

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

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