458 W A V E THEORY Fig. 27 exceed 3 + /8. So large a value of jj? not being available, the conversion of plane-polarized into circularly-polarized light by one reflexion is impracticable. The desired object may, however, be attained by two successive reflexions. The angle of incidence may be so accommodated to the index that the alteration of phase amounts to period, in which case a second reflexion under the same conditions will give rise to light circularly polarized. Putting (2e- 2e ) = Jir, we get 2/rsin 4 e = (l + V!){(l+M 2 )sin 2 + l} . . . (25), an equation by which 9 is determined when p. is given. It appears that, when /j. = l 51, 6 = 48 37 or 54 37 . These results were verified by Fresnel by means of the rhomb shown in fig. 27. The problem of reflexion upon the elastic solid theory, when the vibrations are executed in the plane of incidence, is more complicated, on account of the tendency to form waves of dilatation. In order to get rid of these, to which no optical phenomena correspond, it is necessary to follow Green in supposing that the velocity of such waves is infinite, or that the media are incompressible. 1 Even then we have to introduce in the neighbour hood of the interface waves variously called longitudinal, pressural, or surface waves ; other wise it is impossible to satisfy the conditions of continuity of strain and stress. These waves, analogous in this respect to those occurring in the second medium when total reflexion is in progress (19), extend to a depth of a few wave-lengths only, and they are so constituted that there is neither dilatation nor rotation. On account of them the final formulae are less simple than those of Fresnel. If we suppose the densities to be the same in the two media, there is no correspondence whatever between theory and observation. In this case, as we have seen, vibrations perpendicular to the plane of incidence are reflected according to Fresnel s tangent-formula ; and thus vibrations in the plane of incidence should follow the sine-formula. The actual result of theory is, however, quite different. In the case where the relative index does not differ greatly from unity, polarizing angles of 22^ and 67^ are indicated, a result totally at variance with observation. As in the case of diffraction by small particles, an elastic solid theory, in which the densities in various media are supposed to be equal, is inadmissible. If, on the other hand, following Green, we regard the rigidities as equal, we get results in better agreement with observation. To a first approximation indeed (when the refraction is small) Green s formula coincides with Fresnel s tangent-formula ; so that light vibrating in the plane of incidence is reflected according to this law, and light vibrating in the perpendicular plane according to the sine-formula. The vibrations are accordingly perpendicular to the plane of polarization. The deviations from the tangent-formula, indicated by theory when the refraction is not very small, are of the same general character as those observed by Jamin, but of much larger amount. The minimum reflexion at the surface of glass (/t = f) would be -^, 2 nearly the half of that which takes place at perpendicular incidence, and very much in excess of the truth. This theory cannot there fore be considered satisfactory as it stands, and various suggestions have been made for its improvement. The only variations from Green s suppositions admissible in strict harmony with an elastic solid theory is to suppose that the transition from one medium to the other is gradual instead of abrupt, that is, that the transitional layer is of thickness comparable with the wave-length. This modification would be of more service to a theory which gave Fresnel s tangent-formula as the result of a sudden transition than to one in which the deviations from that formula are already too great. It seems doubtful whether there is much to be gained by further discussion upon this subject, in view of the failure of the elastic solid theory to deal with double refraction. The deviations from Fresnel s formulae for reflexion are comparatively small ; and the whole problem of reflexion is so much concerned with the condition of things at the interface of two media, about which we know little, that valuable guidance can hardly be expected from this quarter. It is desirable to bear constantly in mind that reflexion depends entirely upon an approach to discontinuity in the properties of the medium. If the thickness of the transitional layer amounted to a few wave-lengths, there would be no sensible reflexion at all. Another point may here be mentioned. Our theories of reflexion take no account of the fact that one at least of the media is disper sive. The example of a stretched string, executing transverse vibrations, and composed of two parts, one of which in virtue of 1 The supposition that the velocity is zero, favoured by some writers, is in admissible. Even dilatational waves involve a shearing of the medium, and must therefore be propagated at a finite rate, unless the resistance to compression were negative. But in that case the equilibrium would be unstable. 2 Green s Papers, by Ferrers, p. 333. stiffness possesses in some degree the dispersive property, shows that the boundary conditions upon which reflexion depends are thereby modified. We may thus expect a finite reflexion at the interface of two media, if the dispersive powers are different, even though the indices be absolutely the same for the waves under consideration, in which case there is no refraction. But a know ledge of the dispersive properties of the media is not sufficient to determine the reflexion without recourse to hypothesis. 3 28. The Velocity of Light. According to the principles of the wave theory, the dispersion of refraction can only be explained as due to a variation of velocity with wave-length or period. In aerial vibrations, and in those propagated through an elastic solid, there is no such variation ; and so the existence of dispersion was at one time considered to be a serious objection to the wave theory. Dispersion in vacua would indeed present some difficulty, or at least force upon us views which at present seem unlikely as to the constitution of free aether. The weight of the evidence is, however, against the existence of dis persion in vacua. Were there a difference of one hour in the times of the blue and red rays reaching us from Algol, this star would show a well-marked coloration in its phases of increase and decrease. No trace of coloration having been noticed, the differ ence of times cannot exceed a fraction of an hour. It is not at all probable that the parallax of this star amounts to one-tenth of a second, so that its distance, probably, exceeds two million radii of the earth s orbit, and the time which is required for its light to reach us probably exceeds thirty years, or a quarter of a million hours. It is therefore difficult to see how there can be a difference as great as four parts in a million between the velocities of light coming from near the two ends of the bright part of the spectrum." 4 For the velocity of light in vacua, as determined in kilometres per second by terrestrial methods (LIGHT, vol xiv. p. 585), New- comb gives the following tabular statement : Michelson, at Naval Academy, in 1870 299,910 Michelson, at Cleveland, 1882 299,853 Newcomb at Washington, 1882, using only results sup posed to be nearly free from constant errors 299,860 Newcomb, including all determinations 299,810 To these may be added, for reference Foucault, at Paris, in 1862 298,000 Cornu, at Paris, in 1874 298,500 Cornu, at Paris, in 1878 300,400 This last result, as discussed by Listing 299,990 Young and Forbes, 1880-1881 301,382 Newcomb concludes, as the most probable result- Velocity of light in t ewo = 299,86030 kilometres. It should be mentioned that Young and Forbes inferred from their observations a difference of velocities of blue and red light amount ing to about 2 per cent. , but that neither Michelson nor Newcomb, using Foucault s method, could detect any trace of such a difference. When we come to consider the propagation of light through ponderable media, there seems to be little reason for expecting to find the velocity independent of wave-length. The interaction of matter and aether may well give rise to such a degree of compli cation that the differential equation expressing the vibrations shall contain more than one constant. The law of constant velocity is a special property of certain very simple media. Even in the case of a stretched string, vibrating transversely, the velocity becomes a function of wave-length as soon as we admit the existence of finite stiffness. As regards the law of dispersion, a formula, derived by Cauchy from theoretical considerations, was at one time generally accepted. According to this, M = A + BA- 2 + C;v- 4 + (1); and there is no doubt that even the first two terms give a good representation of the truth in media not very dispersive, and over the more luminous portion of the spectrum. A formula of this kind treats dispersion as due to the smallness of wave-lengths, giving a definite limit to refraction (A) when the wave-length is very large. Recent investigations by Langley on the law of disper sion for rock-salt in the ultra red region of the spectrum are not very favourable to this idea. The phenomena of abnormal disper- No dis persion in Table of velocities. Formula; for dis persion. 3 The reader who desires to pursue this subject may consult Green, "On the Laws of Reflexion and Refraction of Light at the Common Surface of Two Non- Crystallized Media," Camb. Trans., 1838 (Green s Worts, London, 1871, pp. 242, 283) ; Lorenz, " Ueber die Reflexion des Lichts an der Granzfliiche zweier iso- tropen, durchsichtigen Mittel," Pogg. Ann., cxi. p. 460 (I860), and " Bestimmung der Schwingungsrichtung des Lichttethers durch die Reflexion und Brechung des Lichtes," ibid., cxiv. p. 238 (1861); Strutt (Rayleigh), "On the Reflection of Light from Transparent Matter," Phil. Mag., [4] xiii. (1871); Von der Miihll, " Uebcr die Reflexion und Brechung des Lichtes an tier Grenze unkrystallinischen Medien," Math. Ann., v. 470 (1872), and "Ueber Greens Thcorie der Reflexion und Brechung des Lichtes." Math. Ann., xxvii. 500(1886); Thomson, JiaUimore Lectures; Glazebrook, "Report on Optical Theories," Brit. Ass. Rep., 1886; Rayleigh, " On Reflection of Vibrations at the Confines of Two Media between which the Transition is gradual," Proc. Math. Soc., xi.; and Walker, " An Account of Cauchy s Theory of Reflection and Refraction of Light," Pfiif. Mag., 151 (1887). References to recent German writers, Ketteler, Lommel, Voigt, &c., will be found in Glazebrook s Report,
4 Newcomb, Astron. Papers, vol. ii. parts iii. and iv.. Washington, 1885.