Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/200

reducible to this. Then, with the origin of coordinates and the zero of time suitably chosen, the regular part of the displacement may be represented by the equations

$\xi$	$=\alpha \cos 2\pi {\frac {u}{l}}\cos 2\pi {\frac {t}{p}},$	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,}$ (1)
$\eta$	$=\beta \cos 2\pi {\frac {u}{l}}\cos 2\pi {\frac {t}{p}},$
$\zeta$	$=\gamma \cos 2\pi {\frac {u}{l}}\cos 2\pi {\frac {t}{p}},$

where $l$ denotes the wave-length, $p$ the period of vibration, $\alpha ,\beta ,\gamma$ the maximum amplitudes of the displacements $\xi ,\eta ,\zeta ,$ and $u$ the distance of the point considered from the wave-plane which passes through the origin. Since $u$ is a linear function of $x,y,$ and $z,$ we may regard these equations as giving the values of $\xi ,\eta ,\zeta ,$ for a given system of waves, in terms of $x,y,z,$ and $t.$

4. The components of the irregular displacement, $\xi ',\eta ',\zeta ',$ at any given point, will evidently be simple harmonic functions of the time, having the same period as the regular part of the displacement. That they will also have the same phase is not quite so evident, and would not be the case in a medium in which there were any absorption or dispersion of light. It will however appear from the following considerations that in perfectly transparent media the irregular oscillations are synchronous with the regular. For if they are not synchronous, we may resolve the irregular oscillations into two parts, of which one shall be synchronous with the regular oscillations, and the other shall have a difference of phase of one-fourth of a complete oscillation. Now if the mediimi is one in which there is no absorption or dispersion of light, we may assume that the same electrical configurations may also be passed through in the inverse order, which would be represented analytically by writing $-t$ for $t$ in the equations which give $\xi ,\eta ,\zeta ,\xi ',\eta ',\zeta ',$ as functions of $x,y,z,$ and $t.$ But this change would not affect the regular oscillations, nor the synchronous part of the irregular oscillations, which depends on the cosine of the time, while the non-synchronous part of the irregular oscillations, which depends on the sine of the time, would simply have its direction reversed. Hence, by taking first one-half the sum, and secondly one-half the difference, of the original motion and that obtained by substitution of $-t$ for $t,$ we may separate the non-synchronous part of the irregular oscillations from the rest of the motion. Therefore, the supposed non-synchronous part of the irregular displacement, if capable of existence, is at least wholly independent of the wave-motion and need not be considered by us.

We may go farther in the determination of the quantities $\xi ',\eta ',\zeta '.$ For in view of the very fine-grained structure of the medium, it will