WAVE THEORY 451 the difference of velocities for waves propagated parallel to R is evidently proportional to (P - Q), and so on. More often it happens that the strain is not homogeneous. Even then the small parts may be compared to crystals, but the optical constants vary from point to point. The comparatively feeble doubly refracting power thus developed in glass may best be made evident by the production of the colours of polarized light. Thus, in an experiment due to Brewster, a somewhat stout slab of glass, polished on the edges, is interposed between crossed Nicols. When the slab is bent in a plane perpendicular to that of vision, a revival of light takes place along the edges, where the elongation and con traction is greatest. If the width (in the direction of vision) be sufficient, the effect may be increased until the various colours of Newton s scale are seen. These colours vary from point to point of the thickness in the plane of bending, the " neutral axis " remaining dark. The optic axis, being everywhere coincident with the direction of elongation (or contraction), is parallel to the length of the slab. To this direction the plane of polarization should be inclined at about 45. Effects The condition of internal strain is not necessarily due to forces )f heat, applied from without. Thus, if glass originally free from strain be unequally heated, the accompanying expansions give rise to internal strains which manifest themselves in polarized light. If the heating be moderate, so as not to approach the softening point, the state of ease is recovered upon cooling, and the double refrac tion disappears. But if the local temperature be raised further, the hot parts may relieve themselves of the temporary strain, and then upon cooling they and other parts may be left in a condition of permanent strain. Sudden cooling of glass heated to the soften ing point leads to a similar result. The outer parts harden while the interior is still at a higher temperature, so that, when the whole is cooled down, the outside, being as it were too large for the inside, is in a condition of radial tension and circumferential compression. An examination in polarized light shows that the strains thus occasioned are often very severe. If any small part be relieved by fracture from the constraint exercised upon it by the remainder, the doubly refracting property almost or wholly disappears. In this respect unannealed glass differs essentially from a crystal, all parts of which are similar and independent. It may be remarked that it is difficult to find large pieces of glass so free from internal strain as to show no revival of light when examined between crossed Nicols. 23. llotatory Polarization. Rotation In general a polarized ray travelling along the axis of a uniaxal ay crystal undergoes no change ; but it was observed by Arago that, juartz. if quartz be used in this experiment, the plane of polarization is found to be rotated through an angle proportional to the thickness of crystal traversed. The subject was further studied by Biot, who ascertained that the rotation due to a given thickness is in versely as the square of the wave-length of the light, thus varying very rapidly with the colour. In some specimens of quartz (called in consequence right-handed) the rotation is to the right, while in others it is to the left. Equal thicknesses of right- and left- handed quartz may thus compensate one another. Fresnel has shown that the rotation of the plane may be inter preted as indicating a different velocity of propagation of the two I ircularly-polarized components into which plane-polarized light may always be resolved. In ordinary media the right- and left- handed circularly-polarized rays travel at the same speed, and at any stage of their progress recompound a ray rectilinearly-polar- ized in a fixed direction. But it is otherwise if the velocities of propagation of the circular components be even slightly different. Th he first circularly-polarized wave may be expressed by 7j 2 = - and the second (of equal amplitude) by !;> = cos (nt-kg), The resultant of (1) and (2) is = 1 + la = 2r cos i(&, -kjs cos {nt - %(L + ) : } , 77 = 7?! + r; 2 = 2r sin (& 2 - so that (1); (2). which shows that for any fixed value of z the light is plane-polar ized. The direction of this plane, however, varies with z. Thus, if VI = tan 9, so that 6 gives the angular position of the plane in reference to {, we have 0i(* a -*i)= ....... (4), indicating a rotation proportional to z. The quantities k lf k are inversely as the wave-lengths of the two circular components for the same periodic time. When the relative retardation amounts to an entire period, ( 2 -,)3 = 2ir, and then, by (4), = ir. The revolution of the plane through two right angles restores the original state of polarization. In quartz the rotation is very rapid, amounting in the case of yellow light to about 24 for each milli metre traversed. It is interesting to observe with what a high degree of accuracy Delicacy the comparison of the velocities of the two waves can be effected, of test. If the plane of polarization be determined to one minute of angle, a relative retardation of A/10800 is made manifest. If I be the thickness traversed, v and v+Sv the two velocities, the relative retardation is I8v/v. To take an example, suppose that Z = 20 inches, A. = T5&T!7 ^ nc ^ so t na ^ if *> v / v exceed 10 , the fact might be detected. In quartz the rotation of the plane depends upon the crystalline structure, but there are many liquids, e.g., oil of turpentine and common syrup, which exhibit a like effect. In such cases the rota tion is of course independent of the direction of the light ; it must be due to some peculiarity in the constitution of the molecules. A remarkable connexion has been observed between the rotatory Right- property and the crystalline form. Thus Herschel found that in and left- many specimens the right-handed and left-handed varieties of handed quartz could be distinguished by the disposition of certain sub- varieties, ordinate faces. The crystals of opposite kinds are symmetrical in a certain sense, but are yet not sivpcrposdble. The difference is like that between otherwise similar right- and left-handed screws. The researches of Pasteur upon the rotatory properties of tartaric acid have opened up a new and most interesting field of chemistry. At that time two isomeric varieties were known, ordinary tartaric acid, which rotates to the right, and racemic acid, which is optically in active, properties of the acids shared also by the salts. Pasteur found that the crystals of tartaric acid and of the tartrates possessed a right-handed structure, and endeavoured to discover correspond ing bodies with a left-handed structure. After many trials crystal lizations of the double racemate of soda and ammonia were obtained, including crystals of opposite kinds. A selection of the right- handed specimens yielded ordinary dextro-tartaric acid, while a similar selection of the left-handed crystals gave a new variety Irevo-tartaric acid, rotating the plane of polarization to the left in the same degree as ordinary tartaric acid rotates it to the right. A mixture in equal proportions of the two kinds of tartaric acid, which differ scarcely at all in their chemical properties, 1 reconsti tutes racemic acid. The possibility of inducing the rotatory property in bodies other- Magnetic wise free from it was one of the finest of Faraday s discoveries, rotation. He found that, if heavy glass, bisulphide of carbon, &c., are placed in a magnetic field, a ray of polarized light, propagated along the lines of magnetic force, suffers rotation. The laws of the pheno menon were carefully studied by Verdet, whose conclusions may be summed up by saying that in a given medium the rotation of the plane for a ray proceeding in any direction is proportional to the difference of magnetic potential at the initial and final points. In bisulphide of carbon, at 18 and for a difference of potential equal to unity C. G. S., the rotation of the plane of polarization of a ray of soda light is 0402 minute of angle. 2 A very important distinction should be noted between the magnetic rotation and that natural to quartz, syrup, &c. In the latter the rotation is always right-handed or always left-handed with respect to the direction of the ray. Hence when the ray is reversed the absolute direction of rotation is reversed also. A ray which traverses a plate of quartz in one direction, and then after reflexion traverses the same thickness again in the opposite direc tion, recovers its original plane of polarization. It is quite other wise with the rotation under magnetic force. In this case the rotation is in the same absolute direction even though the ray be reversed. Hence, if a ray be reflected backwards and forwards any number of times along a line of magnetic force, the rotations due to the several passages are all accumulated. The non-reversibility of light in a magnetized medium proves the case to be of a very exceptional character, and (as was argued by Thomson) indicates that the magnetized medium is itself in rotatory motion independ ently of the propagation of light through it. 3 The importance of polarimetric determinations has led to the con- Polari- trivance of various forms of apparatus adapted to the special require- metry. ments of the case. If the light be bright enough, fairly accurate measurements may be made by merely rotating a Nicol until the field appears dark. Probably the best form of analyser, when white light is used and the plane is the same for all the coloured com ponents, is the Jellet, 4 formed by the combination of two portions of Iceland spar. By this instrument the field of view is duplicated, and the setting is effected by turning it until the tvo portions of the field, much reduced in brightness, appear equally dark. A similar result is attained in the Laurent, which, however, is only Laurent applicable to homogeneous light. In this apparatus, advantage is analyser, taken of the action of a half-wave plate. In passing such a plate the plane of polarization is as it were reflected by the principal section, that is, rotated until it makes the same angle with the principal section as at first, but upon the further side. The plate covers only half of the field of view, and the eye is focused upon the 1 It would seem that the two varieties could be chemically distinguished only by their relations with bodies themselves right-handed or left-handed. 2 Phil. Tranf., 1885, p. 343. 3 Maxwell s Electricity awl Afaynetism, vol. ii. chap. xxi.
4 A description is given in Gluzebrook s Physical Optics, London, 1883.