Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/223

Rh observer, who looks in the direction opposite to that of the propagation of the light, we have By the preceding equation, this reduces to  Without any appreciable error, we may substitute $$\text{U}^4$$ for $$\text{V}_{\text{R}}^2 \text{V}_{\text{L}}^2,$$ which will give  19. Since these equations involve unknown functions of the period they will not serve for an exact determination of the relation between and the period. For a rough approximation, however, we may assume that the manner in which the general displacement in any small part of the medium distributes itself among the molecules and intermolecular spaces is independent of the period, being determined entirely by the values of $$\xi, \eta, \zeta,$$ and their differential coefficients with respect to the coordinates. For a fixed direction of the wave-normal, $$\Phi$$ and $$\Phi '$$ will then be constant. Now equations (15) and (36) give To express this result in terms of the quantities directly observed, we may use the equations  where $$k$$ denotes the velocity of light in vacuo, $$\lambda$$ the wave-length in vacuo of the light employed, $$n_{\text{R}}, n_{\text{L}}$$ the absolute indices of refraction of the two rays, and $$n$$ the index for the optic axis as derived from the ellipsoid (24) by Fresnel's law. We thus obtain