Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/246

of the common perpendicular to both will be linear fimctions of the direction-cosines of each, and a quadratic function of the direction-cosines of the common perpendicular will be a quadratic function of the direction-cosines of each. We may thus reconcile the theory with the law of double refraction, in a certain sense, by supposing that ${\text{A}}_{\text{D}}$ and $b_{\text{D}}$ are independent of the direction of displacement, and that ${\text{B}}_{\text{ND}}$ and therefore ${\text{V}}^{2}$ is a quadratic function of the direction-cosines of the common perpendicular to the wave-normal and the displacement. But this supposition, besides its intrinsic improbability so far as ${\text{A}}_{\text{D}}$ and $b_{\text{D}}$ are concerned, involves a direction of the displacement which is certainly or almost certainly wrong.

We are thus driven to suppose that the undisturbed medium is in a state of stress, which, moreover, is not a simple hydraulic stress. In this case, by attributing certain definite physical properties to the medium, we may make the function ${\text{B}}_{\text{ND}}$ become independent of the direction of the wave-normal, and reduce to a quadratic function of the direction-cosines of the displacement.^[1] This entirely satisfies Fresnel's Law, including the direction of displacement, if we can suppose ${\text{A}}_{\text{D}}$ and $f_{\text{D}}$ independent of the direction of displacement. But this supposition, in any case difficult for aeolotropic bodies, seems quite irreconcilable with that of a permanent (not hydrostatic) stress. For this stress can only be kept up by the action of the ponderable molecules, and by a sort of action which hinders the passage of the ether past the molecules. Now the phenomena of reflection and refraction would be very different from what they are, if the optical homogeneity of a crystal did not extend up very close to the surface. This implies that the stress is produced by the ponderable particles in a very thin lamina at the surface of the crystal, much less in thickness, it would seem probable, than a wave-length of yellow light. And this again implies that the power of the ponderable particles to pin down the ether, as it were, to a particular position is very great, and that the term in the energy relating to the motion of the ether relative to the ponderable particles is very important. This is the term containing the factor $b_{\text{D}},$ which it is difficult to suppose independent of the direction of displacement because the dimensions and arrangement of the particles are different in different directions. But our present hypothesis has brought in a new reason for supposing $b_{\text{D}}$ depend on the direction of displacement, viz., on account of the stress of the medium. A general displacement of the medium midway between two nodal planes, when it is restrained at innumerable points by the ponderable particles, will

1. See note on page 224.