Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/257

The sum of these may be equated to the kinetic energy, giving an equation of the form

${\frac {\pi ^{2}{\text{B}}h^{2}}{l^{2}}}+{\frac {bh^{2}}{4}}={\frac {\pi ^{2}{\text{A}}'h^{2}}{p^{2}}}\cdot$

(24)

${\text{B}}$ is an absolute constant (the rigidity of the ether, previously represented by the same letter), ${\text{A}}'$ and $b$ will be constant (for the same medium and the same direction of the wave-normal) except so far as the type of the motion changes, i.e., except so far as the manner in which the motion of the ether distributes itself between the ponderable molecules, and the degree in which these take part in the motion, may undergo a change. When the period of vibration varies, the type of motion will vary more or less, and ${\text{A}}'$ and ${\text{b}}$ will vary more or less.

In a manner entirely analogous,^[1] the kinetic energy, on the electrical theory, may be divided into two parts, of which one is due to those general fluxes which are represented by the equations of wave-motions, and the other to those irregularities in the fluxes which are caused by the presence of the ponderable molecules, as well as to such motions of the ponderable particles themselves as may sometimes occur. These parts of the kinetic energy may be represented respectively by

${\frac {\pi {\text{F}}l^{2}h^{2}}{p^{2}}}$and${\frac {\pi ^{2}fh^{2}}{p^{2}}}\cdot$

Their sum equated to the potential energy gives

${\frac {\pi {\text{F}}l^{2}h^{2}}{p^{2}}}+{\frac {\pi ^{2}fh^{2}}{p^{2}}}={\frac {{\text{G}}h^{2}}{4}}\cdot$

(25)

Here ${\text{F}}$ is the constant of electrodynamic induction, which is unity if we use the electromagnetic system of units, $f$ and ${\text{G}}$ (like ${\text{A}}'$ and $b$) vary only so far as the type of motion varies.

We have the means of forming a very exact numerical estimate of the ratio of the two parts into which the statical energy is thus divided on the elastic theory, or the kinetic energy on the electric theory. The means for this estimate is afforded by the principle that the period of a natural vibration is stationary when its type is infinitesimally altered by any constraint.^[2] Let us consider a case of simple wave-motion, and suppose the period to be infinitesimally varied: the wave-length will also vary, and presumably to some extent the type of vibration. But, by the principle just stated, if the ether or the electricity could be constrained to vibrate in the original type, the variations of $l$ and $p$ would be the same as in the actual

1. See page 182 of this volume.
2. See Lord Rayleigh's Theory of Sound, vol. i, p. 84. The application of the principle is most simple in the case of stationary waves.