We shall commence by briefly adverting to the facts of double refraction, and to its laws so far as they were ascertained before Fresnel's researches on the subject. It. was in Iceland spar that the phenomenon was first discovered, and this crystal, from the great power of its double refraction was well suited for a study of the subject, especially at a time when the instrumental means of examination were far inferior to what we at present possess. Of the two rays into which Iceland spar divides iu general a single ray incident upon it, the more refracted was found to obey the ordinary law of refraction, but the less refracted was found to obey a more complicated law, not even lying in the plane of incidence, except in particular cases. On measuring the refractive index of the spar with respect to the hatter or extraordinary ray by methods applicable to ordinary media, different values were obtained, varying from a maximum equal to the refractive index for the ordinary ray, and obtained when the course of the ray within the crystal was parallel to the axis, to a minimum, which was obtained in any direction perpendicular to the axis, around which everything relating to the optical properties was symmetrical. Since there are here two rays, we must, if we adopt the undulatory theory at all, assume that a disturbance excited at any point of the crystal would give rise not to one wave, but to two waves, diverging from that point, or what comes to the same, to a wave represented by a surface of two sheets. As the more refracted ray in Iceland spar obeys the ordinary law of refraction, the inner sheet of this surface must be supposed to be a sphere. The facts which have been mentioned respecting the refraction of the extraordinary ray, show that the outer sheet must be a surface of revolution around the axis of the crystal, along which it touches the inner sheet, and must. be most protuberant at the equator. Apparently as being next in simplicity to a sphere, Huygens assumed this surface or sheet to be an oblate spheroid of revolution, and found the calculations thence resulting as to the course of the extraordinary ray confirmed by the result of his experiments.
The demonstration of the lawa of reflection and refraction in the case of ordinary media, require but a slight. modification to adapt them to the case of a crystal for which we are supposed to know the form of the wave surface. In fact, if we for simplicity, the incident waves and the surface of the crystal to be both plane, we have only to replace the hemispheres within the media by the corresponding wave surfaces, which will form a system of similar and similarly situated curved surfaces with two plane envelopes, one for each sheet. I knee results the following construction. Draw a line perpendicular to tho incident waves in air, and therefore representing the course of an inci dent ray. With the point of incidence for centre, describe a sphere representing the velocity of propagation in air, and likewise within the refracting medium draw the wave surface on a corresponding scale, so that its radius in any direction represents the velocity of a ray propa gated in that direction. Produce the incident ray to meet the sphere within the refracting medium which is a continuation of the hemi sphere described in air, and at the point of meeting draw a tangent plane to the sphere, or, iu other words, a plane perpendicular to the ray. Through the line of intersection of this tangent plane with the plane of the surface, draw tangent 'fiance to the two sheets respectively of the wave surface, and join the point of incidence with the points of contact. The two tangent planes will give the directions of the fronts
of the two refracted waves, while the two -joining-lines will give in direction the courses, and in magnitude the velocities, of propagation, of the two refracted rays ; and if from the point of incidence we let fall perpendiculars on the two tangent planes, they will represent in magnitude and direction the velocities of propagation of the refracted waves, estimated in directions perpendicular to their planes. If the incident waves and the surface of the medium be curved, the same construction will hold good on substituting tangent planes for these curved surfaces, just as in the case of ordinary media, but the front of either refracted wave will of course no longer be plane but curved. A construction based on the same principles gives the courses of inter nally reflected rays ; and it may be remarked that a single incident ray in general gives rise to-two internally reflected rays.
The accuracy of Huygens's construction, as has been already observed, was confirmed by the observations of Wollaston and Malus, and it was for some time supposed that other doubly refracting crys tals resembled Iceland spar, except as to the energy of their double refraction. But Biot discovered that in quartz and several other doubly refracting crystals it is the less refracted instead of the more refracted of the two rays which obeys the ordinary law of refraction, so that in the application of Huygens's construction the oblate must be replaced by a prolate spheroid. This difference does not entail any difference in the state of polarisation : it is still the ordinary ray that is polarised in a principal plane. Crystals in which, as in Iceland spar, the ordinary ray is the more refracted, have been called negative, and those of the other class positive ; terms derived from certain views relating to the corpuscular theory.
If a plate of Iceland spar cut in a direction perpendicular to the axis be interposed between the polariser and the analyser of a polarising apparatus, a most splendid series of coloured rings make their appear ance. If the analyser had been turned till the field was dark, the rings are seen to be interrupted by a black cross, the arms of which are parallel and perpendicular to the plane of primitive polarisation, and the order of the tints, beginning from the centre, agrees with that of the reflected system of Newton's rings. If the analyser be turned through 90% the black cross is replaced by a white one in the same position, and the tints now agree with the transmitted system of Newton's rings, but are much more vivid. In intermediate positions of the analyser, the rings are less vivid, and are interrupted by a double cross containing white light of mean intensity, or, in other words, by eight radii, of which every alternate pair are at right angles. The arms of the crosses are parallel and perpendicular to the planes of polarisation of the polariser and the analyser. Similar rings are seen iu the case of other doubly refracting crystals of the class now known uniaxal; but sometimes, in consequence of a great chromatic variation in the doubly refracting power of the crystal, the succession of tints deviates materially from that of Newton's rings.