Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/222

In equations (29), we are to read $+$ or $-$ in the second members, according as the ray is right-handed or left-handed. (See § 16.) It follows that if the value of $\phi$ is positive, the greater velocity will belong to a right-handed ray, and the smaller to a left-handed, but if the value of $\phi$ is negative, the opposite is the case. Except when $\phi =0,$ and the polarization is linear, there will be one right-handed and one left-handed ray for any given wave-normal and period.

18. When ${\text{U}}_{1}={\text{U}}_{2},$ equations (29) give

$\rho _{1}=\rho _{2},$⁠${\text{V}}^{2}={\text{U}}^{2}\pm {\frac {\phi }{\text{V}}},$

where ${\text{U}}$ represents the common value of ${\text{U}}_{1}$ and ${\text{U}}_{2}.$ The polarization is therefore circular. The converse is also evident from equations (29), viz., that a ray can be circularly polarized only when the direction of its wave-normal is such that ${\text{U}}_{1}={\text{U}}_{2}.$ Such a direction, which is determined by a circular section of the ellipsoid (24) precisely as an optic axis of a crystal which conforms to Fresnel's law of double refraction, may be called an optic axis, although its physical properties are not the same as in the more ordinary case.^[1] If we write ${\text{V}}_{\text{R}}$ and ${\text{V}}_{\text{L}},$ respectively, for the wave- velocities of the right-handed and left-handed rays, we have

${\text{V}}_{\text{R}}^{2}={\text{U}}^{2}+{\frac {\phi }{{\text{V}}_{\text{R}}}},$⁠${\text{V}}_{\text{L}}^{2}={\text{U}}^{2}-{\frac {\phi }{{\text{V}}_{\text{L}}}};$

(33)

whence

${\text{V}}_{\text{R}}^{2}-{\text{V}}_{\text{L}}^{2}=\phi \left({\frac {1}{{\text{V}}_{\text{R}}}}+{\frac {1}{{\text{V}}_{\text{L}}}}\right)=\phi {\frac {{\text{V}}_{\text{R}}+{\text{V}}_{\text{L}}}{{\text{V}}_{\text{R}}{\text{V}}_{\text{L}}}},$

and

${\text{V}}_{\text{R}}-{\text{V}}_{\text{L}}={\frac {\phi }{{\text{V}}_{\text{R}}{\text{V}}_{\text{L}}}}\cdot$

(34)

The phenomenon best observed with respect to an optic axis is the rotation of the plane of linearly polarized light. If we denote by $\theta$ the amount of this rotation per unit of the distance traversed by the wave-plane, regarding it as positive when it appears clockwise to the

1. Our experimental knowledge of circularly or elliptically polarizing media is confined to such as are optically either isotropic or uniaxial. The general theory of such media, embracing the case of two optic axes, has however been discussed by Professor von Lang ("Theorie der Circularpolarization," Sitz. Ber. Wiener Akad., vol. lxxv, p. 719). The general results of the present paper, although derived from physical hypotheees of an entirely difierent nature, are quite similar to those of the memoir cited. They would become identical, the writer believes, by the substitution of a constant for ${\frac {p\phi }{l}}$ or ${\frac {\phi }{\text{V}}}$ in the equations of this paper. (See especiaUy equations (18), (20), (28).)
   That a complete discussion of the subject on any theory must include the case of biaxial media having the property of circular or elliptical polarization, is evident from the consideration that it must at least be possible to produce examples of such media artificially. An isotropic or uniaxial ciystal may be made biaxial by pressure. If it has the property of circular and elliptic polarization, that property cannot be wholly destroyed by the application of small pressures.