446 able of transverse vibrations. But there seems to be no reason a priori for preferring one kind of vibration to another ; and the phenomena of polarization prove conclusively that, if luminous vibrations are analogous to those of a material medium, it is to solids, and not to fluids, that we must look. An isotropic solid is capable of propagating two distinct kinds of waves. the first dependent upon rigidity, or the force by which shear is resisted, and the second analogous to waves of sound and dependent upon compressibility. In the former the vibrations are transverse to the direction of propagation, that is, they may take place in any direction parallel to the wave front, and they are thus suitable representatives of the vibrations of light. In this theory the lumiuiferous ether is distinctly assimilated to an elastic solid, and the velocity of light depends upon the rigidity and density assigned to the medium. Medium The possibility of longitudinal waves, in which the displacement incom- is perpendicular to the wave-front, is an objection to the elastic pressible. solid theory of light, for there is nothing known in optics corre sponding thereto. If, however, we suppose with Green that the medium is incompressible, the velocity of longitudinal waves be comes infinite, and the objection is in great degree obviated. Such a supposition is hardly a departure from the original idea, inas much as, so far as wo know, there is nothing to prevent a solid material possessing these properties, and an approximation is actu ally presented by such bodies as jelly, for which the velocity of longitudinal vibrations is a large multiple of that of transverse vibrations. 20. Interference of Polarized Light. The conditions of interference of polarized light are most easily deduced from the phenomena of the colours of crystalline plates, if we once admit Young s view that the origin of the colours is to be sought in the interference of the differently refracted rays. Independently of any hypothesis of this kind, the subject was directly investigated by Fresnel and Arago, 1 who summarized their conclusions thus : Laws of (1) Under the same conditions in which two rays of ordinary interfer- light appear to destroy one another, two rays polarized in contrary ence. (viz., perpendicular) directions are without mutual influence. (2) Two rays of light polarized in the same direction act upon one another like ordinary rays ; so that, with these two kinds of light, the phenomena of interference are identical. (3) Two rays originally polarized in opposite directions may after wards be brought to the same plane of polarization, without tlicreby acquiring the power to influence one another. (4) Two rays polarized in opposite, directions, and afterwards brought to similar polarizations, react in the same manner as natural rays, if they arc derived from a beam originally polarized in one direction. The fact that oppositely polarized rays cannot be made to inter fere may of itself be regarded as a proof that the vibrations are transverse ; and the principle, once admitted, gives an intelligible account of all the varied phenomena in this field of optics. The only points on which any difficulty arises are as to the nature of ordinary unpolarized light, and the rules according to which in tensity is to be calculated. It will be proper to consider these questions somewhat fully. Plane In ordinary (plane) polarized light the vibrations are supposed to polarizi- be in one direction only. If jc and y be rectangular coordinates tion. in the plane of the wave, we may take, as representing a regular vibration of plane-polarized light, x=acos((j)- a) ....... (1), where = 27r</T, and a, a denote constants. It must be remembered, however, that in optics a regular vibration of this kind never pre sents itself. In the simplest case of approximately monochromatic light, the amplitude and phase must be regarded ( 4) as liable to incessant variation, and all that we are able to appreciate is the mean intensity, represented by M(a 2 ). If a number of these irre gular streams of light are combined, the intensity of the mixture cannot be calculated from a mere knowledge of the separate inten sities, unless we have assurance that the streams are independent, that is, without mutual phase-relations of a durable character. For instance, two thoroughly similar streams combine into one of four fold intensity, if the phases are the same ; while, if the phases are opposed, the intensity falls to zero. It is only when the streams are independent, so that the phase-relation is arbitrary and variable from moment to moment, that the apparent resultant intensity is necessarily the double of the separate intensities. If any number of independent vibrations of type (1) bo super posed, the resultant is [2a x cos aj] cos (p + [Sa x sin oj sin </> , and the momentary intensity is [2 a 1 cos fll ] 2 + [2 a l sin aj 2 , or a + a -i + + Sffljfl^ cos (a l - a.,) + . . . 1 Fresnel s Works, vol. i. p. 521. The phase-relations being unknown, this quantity is quite indeter minate. But, since each cosine varies from moment to moment, and on the whole is as much positive as negative, the mean intensity is that is to say, is to be found by simple addition of the separate intensities. Let us now dispense with the restriction to one direction of Elliptic vibration, and consider in the first place the character of a regular polariza- vibration, of given frequency. The general expression will be tion. x = acos((f>- a), y=--bcos(<f>- 0) . . . (2), where a, a, b, /3 are constants. If /3 = a, the vibrations are executed entirely in the plane x/y = a/b, or the light is plane-polarized. Or if ft = TT a, the light is again plane-polarized, the plane of vibration being x/y^-a/b. In other cases the vibrations are not confined to one plane, so that the light is not plane-polarized, but, in con formity with the path denoted by (2), it is said to be elliptically- polarized. If one of the constituents of elliptically-polarized light be suitably accelerated or retarded relatively to the other, it may be converted into plane-polarized light, and so identified by the usual tests. Or, conversely, plane-polarized light may be converted into elliptically-polarized by a similar operation. The relative acceleration in question is readily effected by a plate of doubly refracting crystal cut parallel to the axis. If j8 = a4 7r > whether in the first instance or after the action of a crystalline plate, = cos(0- a), y= &sin(<- a) .. . (3). The maxima and minima values of the one coordinate here occur synchronously with the evanescence of the other, and the co ordinate axes are the principal axes of the elliptic path. An important particular case arises when further b = a. The Circular path is then a circle, and the light is said to be circularly -polarized, polariza- According to the sign adopted, in the second equation (3), the circle tion. is described in the one direction or in the other. Circularly -polarized light can be resolved into plane-polarized components in any two rectangular directions, which are such that the intensities are equal and the phases different by a quarter period. If a crystalline plate be of such thickness that it retards one component by a quarter of a wave-length (or indeed by any odd multiple thereof) relatively to the other, it will convert plane- polarized light into circularly-polarized, and conversely, in the latter case without regard to the azimuth in which it is held. The property of circularly-polarized light whereby it is capable of resolution into oppositely plane-polarized components of equal intensities is possessed also by natural unpolarized light; but the discrimination may be effected experimentally with the aid of the quarter-wave plate. By this agency the circularly-polarized ray is converted into plane-polarized, while the natural light remains apparently unaltered. The difficulty which remains is rather to explain the physical character of natural light. To this we shall presently return ; but in the meantime it is obvious that the con stitution of natural light is essentially irregular, for we have seen that absolutely regular, i.e., absolutely homogeneous, light is necessarily (elliplically) polarized. In discussing the vibration represented by (2), we have considered the amplitudes and phases to be constant ; but in nature this is no more attainable than in the case of plane-polarized light. In order that the elliptic polarization may be of a definite character, it is only necessary that the ratio of amplitudes and the difference of phases should be absolute constants, and this of course is consistent with the same degree of irregularity as was admitted for plane vibrations. The intensity of elliptically-polarized light is the sum of the intensities of its rectangular components. This we may consider to be an experimental fact, as well as a consequence of the theory of transverse vibrations. In whatever form such a theory may be adopted, the energy propagated will certainly conform to this law. When the constants in (2) are regarded as subject to variation, the apparent intensity is represented by ....... (4). We are now in a position to examine the constitution which must Unpolar- be ascribed to natural light. The conditions to be satisfied are ized light, that when resolved in any plane the mean intensity of the vibra tions shall be independent of the orientation of the plane, and, further, that this property shall be unaffected by any previous rela tive retardation of the rectangular components into which it may have been resolved. The original vibration being represented by or, as we may write it, since we are concerned only with phase differences, x = acos<f), y = bcos((f>-8) . . . (5), let us suppose that the second component is subjected to a retarda tion e. Thus x=acos(j), y=bcos((j)- 5 - e) . . . (6),
in which a, b, 5 will be regarded as subject to rapid variation,