The exact accordance of this construction with observation was proved by the care ful experiments of Wollaston. We have only to add, that the two rays A/3, and A/3, are, in all cases, completely polarized in planes at right angles to each other.
The experiments of Brewster showed that in by far the greater number of minerals and artificial crystals, both rays are extraordinary—i. e., neither of them can be accounted for by disturbances propagated spherically in the crystal. But no tentative process could lead to the form of the wave-surface in this most general case. Here Fresnel's genius supplied the necessary construction.
He assumes that the ether in a crystallized body is possessed of different rigidity, of different inertia, in different directions; a supposition in itself extremely probable, from the mechanical and other properties of crystals. In the general case there are shown to be three principal directions in a crystal, in any one of which, if the ether be displaced, the resulting elastic force is in the direction of the displacement. Each of these is, in all cases, perpendicular to the others. Any given displacement of the ether corresponds to partial calculable displacements parallel to each of these lines, and thus the elastic force consequent on any displacement whatever is known if we know those for the three rectangular directions. All the calculations are thus dependent on three numbers only, for each substance.
To find the form in which a disturbance will spread, Fresnel proceeds as follows: Let the plane of the paper represent the front of a wave in the crystal, and suppose a particle of ether to be displaced in it from A to B (fig. 8). This displacement may be resolved (by the law of the parallelogram of velocities, forces, etc.) into two components in any two directions in the plade of the paper. Assume AP to be one of these, and let PQ be the force produced by disturbing the particle of ether from A to P. In general PQ will not lie in the plane of the paper. Let fall a perpendicular, QR, upon the plane of the paper. In general the point R will not lie in AP. The portion RQ, of the elastic force of the ether, Fresnel neglects; because it would produce vibrations perpendicular to the wave-front, i.e., similar to those of sound, and he assumes that such normal vibrations do not produce visible light. We shall recur to this point. Fresnel now assumes that the vibrations which will be propagated continuously in the crystal are such as have PR coincident in direction with AP; and then the rate of their propagation will depend upon the ratio of PR to PA. He shows by mathematical reasoning that there are two such directions in every wave-front, and that they are always perpendicular to each other. This, of course, at once accounts for double refraction, the complete polarization of each of the two rays, and their being polarized in planes perpendicular to each other. The original plane wave is now broken into two, both parallel to the first, but in general moving at different rates. He next considers a disturbance at any point in a crystal as i equivalent to waves having fronts in every plane passing through that point, and nvesti gates mathematically the form of the surface which is touched by the planes of all the pairs of polarized rays which- have (in any given time) proceeded from each of those wave-fronts. The form of this surface is very remarkable. It is symmetrical with reference to three planes at right angles to each other. These, of course together, cut it into eight parts, one of which is figured below (fig. 9). From this it appears, though Fresncl did not perceive it, that the surface lias four conical cusps, as they are called, the inner portion seeming to he drawn through a hole, as it were, and then spreading out again to form the, outer portion. The external appearance of these points very much resembles the portion of an apple round the point of attachment of the stalk. Fresnel showed that, in particular cases, when two of the three principal elasticities are equal, this surface degenerates into the sphere and spheroid of Huyghens already described for Iceland spar; and that, when all three are equal, it becomes a single sphere, as in glass, water, and other singly refracting bodies. All this, of course, is in complete accord with experiment. But there is vastly more. If we use the wave-surface of Fresnel to con struct the refracted rays, just as we employed the sphere for simple refraction, or the sphere and spheroid for Iceland spar, we rind generally two definite refracted rays (both usually out of the plane of incidence) for one incident ray. But Hamilton (q.v.), who
was the first to perceive the existence of the cusps already described, saw that they indicated the existence of a very remarkable phenomenon, to which he gave the name of conical refraction (q. v.). The ray which, in the crystal, passes from A to C (the cusp, see last figure), has not, like other rays such as ApP, to definite wave-fronts. For if at p and P, where the line ApP meets the inner and outer portions of the wave-surface, we draw tangent planes, these are the definite fronts of the corresponding waves; so that such a ray will split into two only, on leaving the crystal. But AC intersects the surface at C, where it is conical, and has an infinite number of tangent planes, so that when it leaves the crystal it will split into an infinite number, forming a hollow cone. Hamil ton's prediction then was: If a single ray of light be made to pass through a plate of a biaxal crystal in the direction AC (limiting it, for instance, by sheets of tin-foil with small holes in them properly fixed on each side), it will enter and emerge as a hollow cone. Also the plane of polarization will differ for different rays in this cone. Lloyd com pletely verified this wonderful prediction by experiments made with a plate of arrayon ite (q.v.). But more, Hamilton observed that (see last figure) the wave surface can be touched by a tangent plane in a circle surrounding the cusp. If, then, we make the construction of fig. 7 with Fresnel's wave instead of the sphere and spheroid, there will be a definite direction of the incident ray aA, for which the tangent planes and in that figure will coincide, and will touch the wave-surface in the circle about the cusp. Any line drawn from A to a point in that circle will be a direction for a refracted ray. Hence the ray aA will be broken up into a hollow cone of rays, the vertex of the cone being A, and its base this circle. If the crystal be cut into a plate each ray will of course emerge parallel to czA, and the ensemble of them will form a hollow cylinder. The prediction, then, is that a single definite_ray-,--falling in a given direction on such a plate of crystal, will emerge as a hollow cylinder. This and the predicted laws of the polarization of the light of the cylinder were also verified by Lloyd.
"The formulm which led to such triumphantly successful predictions may have been deduced from incomplete or even erroneous premises; but they represent a truth, and must in time conduct us step by step back to ultimate proof of the truth of Frcsuel's assumptions, and of the undulatory theory of light, as now understood, or show us what modifications may be required in the original conceptions." It would unduly lengthen this article, and besides would lead us into discussions far too recondite for a work like this, to enter upon the question of whether the vibrations in polarized light are perpendicular to or in the plane of polarization, a subject which has recently been well investigated by Stokes (q.v.); or to consider the production of elliptically polarized light by reflection at the surface of metals, diamond, etc.; and various other most important points of the theory. We can only mention that Green, Cauchy, Stokes, and others, who have entered deeply into the mechanical question of luminiferous vibrations, have found themselves obliged to take into account the normal wave, which, as we have seen, Fresnel neglected.
Fluorescence (see PHOSPHORESCENCE), spectrum analysis (see SPECTRUM), and various other important recent additions to the theory, must be merely mentioned; as also the very remarkable observation of Maxwell, which appears to connect light and electricity, and was derived from a theory which assumes the ether to be the vehicle of electricity and magnetism as well as of light and heat, and by which it appears that the velocity of light is expressible in terms of the static and kinetic units of electricity.
For further information, we refer the reader to Lloyd's an excellent elementary treatise; while to the more advanced mathematician we may commend Airy's Tract oft the Undulatory Theory, and Herschel's article " Light" in the Encyclopcedia .31etropolitana.