Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/211

XII.

ON DOUBLE REFRACTION IN PERFECTLY TRANSPARENT MEDIA WHICH EXHIBIT THE PHENOMENA OF CIRCULAR POLARIZATION.

[American Journal of Science, ser. 3, vol. xxiii, pp. 460–476, June, 1882.]

1. In the April number of this Journal,^[1] the velocity of propagation of a system of plane waves of light, regarded as oscillating electrical fluxes, was discussed with such a degree of approximation as would account for the dispersion of colors and give Fresnel's laws of double refraction. It is the object of this paper to supplement that discussion by carrying the approximation so much further as is necessary in order to embrace the phenomena of circularly polarizing media.

2. If we imagine all the velocities in any progressive system of plane waves to be reversed at a given instant without affecting the displacements, and the system of wave-motion thus obtained to be superposed upon the original system, we obtain a system of stationary waves having the same wave-length and period of oscillation as the original progressive system. If we then reduce the magnitude of the displacements in the uniform ratio of two to one, they will be identical, at an instant of maximum displacement, with those of the original system at the same instant.

Following the same method as in the paper cited, let us especially consider the system of stationary waves, and divide the whole displacement into the regular part, represented by $\xi ,\eta ,\zeta$, and the irregular part, represented by $\xi ',\eta ',\zeta '$, in accordance with the definitions of §2 of that paper.

3. The regular part of the displacement is subject to the equations of wave-motion, which may be written (in the most general case of plane stationary waves)

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

1. See page 182 of this volume.