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

206 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 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. 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 whence  and  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