Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/205

For a given medium and light of a given period, the coefficients $a,b,$ etc., are constant.

This relation between the velocity of the waves and the direction of oscillation is capable of a very simple geometrical expression. Let $r$ be the radius vector of the ellipsoid

$ax^{2}+by^{2}+cz^{2}+eyz+fzx+gxy=1.$

(14)

Then

${\frac {1}{r^{2}}}={\frac {ax^{2}+by^{2}+cz^{2}+eyz+fzx+gxy}{r^{2}}}\cdot$

If this radius is drawn parallel to the electrical oscillations, we shall have

${\frac {x}{r}}={\frac {\alpha }{\rho }},$${\frac {y}{r}}={\frac {\beta }{\rho }},$${\frac {z}{r}}={\frac {\gamma }{\rho }},$

and

${\text{V}}^{2}={\frac {1}{r^{2}}}\cdot$

(15)

That is, the wave-velocity for any particular direction of oscillation is represented in the ellipsoid by the reciprocal of the radius vector which is parallel to that direction.

11. This relation between the wave-length, the period, and the direction of vibration, must hold true not only of such vibrations as actually occur, but also of such as we may imagine to occur under the influence of constraints determining the direction of vibration in the wave-plane. The directions of the natural or unconstrained vibrations in any wave-plane may be determined by the general mechanical principle that if the type of a natural vibration is infinitesimally altered by the application of a constraint, the value of the period will be stationary.^[1] Hence, in a system of stationary waves such as we have been considering, if the direction of an unconstrained vibration is infinitesimally varied in its wave-plane by a constraint while the wave-length remains constant, the period will be stationary. Therefore, if the direction of the unconstrained vibration is infinitesimally varied by constraint, and the period remains rigorously constant, the wave-length will be stationary. Hence, if we make a central section of the above described ellipsoid parallel to any wave-plane, the directions of natural vibration for that wave-plane will be parallel to the radii vectores of stationary value in that section, viz., to the axes of the ellipse, when the section is elliptical, or to all radii, when the section is circular.

12. For light of a single period, our hypothesis has led to a perfectly definite result, our equations expressing the fundamental laws of double refraction as enunciated by Fresnel. But if we ask how the velocity of light varies with the period, that is, if we seek

1. See Rayleigh's Theory of Sound, voll. i, p. 84.