450 WAVE THEORY those discovered by Hamilton and Lloyd, generally known as conical refraction. Conical In general there are two refracted rays, corresponding to two refrac- distinct waves. But the refracted waves coalesce when they are tion. perpendicular to either optic axis, and (as we have seen) this wave touches the wave-surface along a circle. Thus corresponding to one wave direction there are an infinite number of rays, lying upon a cone. The division of a single incident ray into a cone of refracted rays is called internal conical refraction. If the second face of the crystal is parallel to the first, each refracted ray resumes on emer gence its original direction, so that the emergent bundle forms a hollow cylinder. External conical refraction depends upon the singular points in the principal plane of sx, where the two sheets of the surface cross one another (fig. 24). At such a point (P) an infinite number of tangent planes may be drawn to the surface, and each of the per pendiculars from represents a wave direction, corresponding to the single ray OP. On emergence these waves will be differently refracted ; and thus corresponding to a single internal ray there are an infinite number of external rays, lying upon a cone. It has already been admitted that the dynamical foundations of Fresnel s theory are unsound; and it must be added that the rigorous theory of crystalline solids investigated by Cauchy and Green does not readily lend itself to the explanation of Fresnel s laws of double refraction. On this subject the reader should con sult Prof. Stokes s Report. Sir W. Thomson has recently shown l that an originally isotropic medium, pressed unequally in different directions, may be so constituted as to vibrate in accordance with Fresnel s laws. It may perhaps be worth while to remark that the equations, analogous to (2) (24), which lead to these laws are d?E dp 0> d 2 rj dp 79 , _. -J^=-r + -V-f, -W = -j- + 1" V 2 7? , &C. . . . (2o , at- dx dtf ay where a, I, c are the principal wave velocities. If we here assume | = A0, 7j = M 9, =1,0, 0/0o =P/Po = e i ^"+ m y+ ra - V( >, and substitute in (25), the condition of transversality leads at once to the desired results. But the equations (25) are not applicable to the vibrations of a crystalline solid. Electro- In the electromagnetic theory double refraction is attributed to magnetic teolotropic inductive capacity, and appears to offer no particular theory, difficulty. If the present position of the theory of double refraction is still somewhat unsatisfactory, it must be remembered that the un certainty does not affect the general principle. Almost any form of wave theory involving transverse vibrations will explain the leading phenomenon, viz., the bifurcation of the ray. It is safe to predict that when ordinary refraction is well understood there will be little further trouble over double refraction. Double The wave-velocity is not the only property of light rendered un- absorp- symmetrical by crystalline structure. In many cases the two tion. polarized rays are subject to a different rate of absorption. Tour malines and other crystals may be prepared in plates of such thickness that one ray is sensibly stopped and the other sensibly transmitted, and will then serve as polarizing (or analysing) ap paratus. Although for practical purposes Nicol s prisms (LIGHT, vol. xiv. p. 612) are usually to be preferred, the phenomenon of double absorption is of great theoretical interest. The explanation is doubtless closely connected with that of double refraction. 22. Colours of Crystalline Plates. When polarized light is transmitted through a moderately thin plate of doubly refracting crystal, and is then analysed, e.g., with a Nicol, brilliant colours are often exhibited, analogous in their character to the tints of Newton s scale. With his usual acuteness, Young at once attributed these colours to interference between the ordinary and extraordinary waves, and showed that the thickness of crystal required to develop a given tint, inversely proportional to the doubly refracting power, was in agreement with this view. But the complete explanation, demanding a fuller knowledge of the laws of interference of polarized light, was reserved for Fresnel and Arago. The subject is one which admits of great development ; 2 but the interest turns principally upon the beauty of the effects, and upon the facility with which many of them may be obtained in experiment. We must limit ourselves to a brief treatment of one or two of the simpler cases. The incident vibration being plane-polarized, we will suppose that its plane makes an angle a with the principal plane of the crystal. On entering the crystal it is accordingly resolved into the two components represented by cos a cos (p, sin a cos <f>, where = Zirt/r. 1 " On Cauchy s and Green s Doctrine of Extraneous Force to explain dynami cally Fresnel s Kinematics of Double .Refraction," Phil. May., Feb. 1888. 2 See Verdet s Lemons, vol. ii. In traversing the crystal both waves are retarded, but we are concerned only with the difference of the retardations. Denoting the difference by p, we may take as the expressions of the waves on emergence cos o cos (f>, sin a cos (0-p). It may be remarked that, in the absence of dispersion, p would be inversely proportional to A ; but in fact there are many cases where it deviates greatly from this law. Now let the plane of analysation be inclined at the angle to that of primitive polarization (fig. 25). Then for the sum of the two resolved components we have cos a cos (a - j8) cos <j) + sin a sin (a - j3) cos (<j) - p} , of which the intensity is {cos a cos (-) + sin a sin (a - 0) cos p } 2 + sin 2 a sin 2 (a - /3) sin 2 = cos 2 y3 - sin 2a sin 2(a-0)sin 2 ^p .... (1). If in (1) we .write /3 + JTT in place of ft, we get sin 2 + sin2asin2(a-j8)sm 2 Ap .... (2); and we notice that the sum of (1) and (2) is unity under all circum stances. The etfect of rotating the analyser through 90 is thus always to transform the tint into its complementary. The two complemen- tary tints may be seen at the same time if we employ a double image prism. In the absence of an analyser we may regard the two images as superposed, and there is no colour. These expressions may be applied at once to the explanation of the colours of thin plates of mica or selenite. In this case the retardation p is propor tional to the thickness, and approxi mately independent of the precise direction of the light, supposed to be nearly perpendicular to the plate, viz., nearly parallel to a principal axis of the crystal. The most important cases are when j8 = 0, j6 = ^7r. In the latter the field would be dark were the plate removed ; and the actual intensity is sin 2 2asin 2 p (3). The composition of the light is thus independent of the azimuth of the plate (a); but the intensity varies greatly, vanishing four times during the complete revolution. The greatest brightness occurs when the principal plane bisects the angle between the planes of polarization and analysis. If /3 = 0, the light is complementary to that represented by (3). If two plates be superposed, the retardations are added if the azimuths correspond ; but they are subtracted if one plate be rotated relatively to the other through 90. It is thus possible to obtain colour by the superposition of two nearly similar plates, although they may be too thick to answer the purpose separately. If dispersion be neglected, the law of the colours in (3) is the same as that of the reflected tints of Newton s scale. The thick nesses of the plates of mica (acting by double refraction) and of air required to give the same colour are as 400 : 1. When a plate is too thick to show colour, its action may be analysed with the aid of a spectroscope. Still thicker plates may be caused to exhibit colour, if the direc- Rings tioii of the light within them makes but a small angle with an from optic axis. Let us suppose that a plate of Iceland spar, or other uniaxal uniaxal crystal (except quartz), cut perpendicularly to the axis, is crystal! interposed between the polarizing and analysing apparatus, and that the latter is so turned that the field is originally dark. The ray which passes perpendicularly is not doubly refracted, so that the centre of the field remains dark. At small angles to the optic axis the relative retardation is evidently proportional to the square of the inclination, so that the colours are disposed in concentric rings. But the intensity is not the same at the various parts of the circum ference. In the plane of polarization and in the perpendicular plane there is no double refraction, or rather one of the refracted rays vanishes. Along the corresponding lines in the field of view there is no revival of light, and the ring system is seen to be tra versed by a black cross. In many crystals the influence of dispersion is sufficient to sensibly modify the proportionality of p to A. In one variety of nniaxal apophyllite Herschel found the rings nearly achromatic, indicating that p was almost independent of A. Under these circumstances a much larger number of rings than usual became visible. In biaxal crystals, cut so that the surfaces are equally inclined to the optic axes, the rings take the form of lemniscates. A medium originally isotropic may acquire the doubly refracting Double property under the influence of strain ; and, if the strain be homo- refracti gcncous, the conditions are optically identical with those found in a due to natural crystal. The principal axes of the wave-surface coincide strain, with those of strain. If the strain be symmetrical, the medium is
optically uniaxal. In general, if P, Q, R be the principal stresses,