passage backwards. The principle of reversibility holds in this case also, provided we reverse the direction of the magnetic field as well as the direction of the ray. 161. Analytical representation of the rotation of the plane of polarization. Consider plane waves travelling in the direction of x, with a uniformly rotating direction of vibration. As each wavefront reaches a given position, the direction of vibration is a definite one, and the angle which that direction forms with one fixed in space is therefore a function of a only. If it be a linear function of x, the plane of polarization rotates through an angle which is proportional to the distance traversed. Let n and be the projections of the displacement, and put η The equations satisfy the condition laid down for the direction of vibration, for if & be the angle between it and the axis of z from which it follows that & is a linear function of x, and that r measures the angle of rotation per unit length of path. We call the quantity r the "gyric coefficient." Equations (1) also satisfy the conditions of ordinary wave propagation, as the displacements may be expressed as a sum of terms, each of which has the form f(x- vt). To show this we need only transform the products of the circular functions in a well-known manner. The disturbance is now expressed in terms of four parts, each of which is of the homogeneous type, but while the periodic time for each of these four waves is the same, the wave-lengths are in groups of two: 2/4, and 2/2 respectively. The displacements 7 and form together a right-handed circularly polarized ray, propagated with velocity v = w/(1+r), while the displacements 72 and 2 combine to form a left-handed circularly polarized ray propagated with velocity vr=w/(l-r). The gyric coefficient may be deduced from v, and v by means of The important conclusion that a wave travelling with a uniform rotation of its planes of polarization is equivalent analytically to the superposition of two circularly polarized rays of opposite directions and propagated with different velocities is due to Fresnel. A simple geometrical illustration may be given. If two points P and Q are imagined to revolve in opposite directions with uniform and identical velocities round the circumference of a circle (Fig. 171), they will cross B B A, at two opposite ends A and B of a diameter, and their combined motion is equivalent to a simple periodic motion along AB as diameter. The two points may be considered to represent the displacements of two waves polarized circularly in opposite directions, having for their resultant a plane polarized wave. If the two circularly polarized waves are transmitted with different velocities, there is, as the waves proceed, a gradual retardation of one circular motion relative to the other, so that the crossing points gradually shift to one side or the other. The combined motion. always remains a simple periodic motion along a diameter, but that diameter rotates uniformly as we proceed along the wave normal. A1, B1 are the crossing points in a wave-front which is at unit distance from that originally considered, AOA, represents the angle through which the plane of polarization is turned in unit length of path. 1 162. Isotropic Substances. There is no satisfactory representation of the mechanism by means of which an asymmetrical molecular structure turns the plane of polarization, but we may easily extend our former equation so as to include rotatory effects. Our equation (41) Chapter XI., for the displacement of an electron, assumed that the electron suffers no constraint in its motion, but if other forces act which depend on the displacements of other electrons, the resultant force may involve not only the three components P, Q, R of electric force, but also their nine differential coefficients with respect to the three independent space variables. Considering small motions only, we need only take linear terms into account. The complicated general equation which would result from the substitution on the right-hand side of (4) of twelve linear terms is much simplified by the restriction that our investigation shall only apply to isotropic substances. In such substances a luminous wave is affected equally in whatever direction it passes, and the resultant differential equation must therefore be independent of the direction of the coordinate axes. If for instance we turn the system of axes through 180° round the axis of z, the simultaneous reversal of the signs of P, Q, x and y must leave the equations unaltered. This consideration shows that there are no dP dQ dR dR terms involving Q, R, ̃dz dz' dx and because all these terms if dy existing would reverse their sign by the supposed change in the coordinate axes. Similarly if we rotate the axes through 180° round the axis of a, the left-hand side of (4) changes sign, hence the general term to be substituted on the right must also reverse its sign. This dP dQ dR The only remaining excludes the terms depending on dady' dz dP dQ differential coefficients are and and these must occur in the combination dP dQ dy dx dy dx' as may be seen by turning the system through 90° round the axis of z and introducing the condition that the equation remains unaltered. We may therefore write the resulting differential Confining ourselves to insulators, equation (35) of Chapter XI. is d2 R K = √2R – 4π NeŸ dt2 ·(7). If the displacements are proportional to e-it, so that in (5) we may write/w2 for , we obtain by substitution in (7) If we consider plane waves parallel to yz so that the electric forces are independent of y and z, equations (8) may be written where M1 = w2 (K+m), _M2 = w2ms. From equations (9) we derive, if i = √−1 : .(9), Reversing the sign of i in (10) and assuming a solution are The positive roots of (12) and (14) which alone need be considered 211 = M1⁄2 + √ M22 + 4M1 2l2 = − M1⁄2 + √ M2 + 4M1 .(15). Separating and retaining only the real parts in the solutions (11) and (13), it is seen by comparison with (2) that (11) represents a right-handed circular polarization, while (12) represents a left-handed circular polarization. The two waves are propagated with velocities w/l and w/l respectively. The superposition of both solutions represents a plane polarized wave, the plane of polarization rotating per unit length of optical path through an angle r which is obtained from (3): r = √(h− ↳2) = {M, 2π Ne's w2 n2 ..(16). The above investigation shows that the only terms depending on the first differential coefficients of the electric forces which can be added to the general equations of light and are consistent with isotropy indicate a turn of the plane of polarization. This does not of course furnish an explanation of the rotatory effect, which would require a knowledge of the physical cause for the existence of the terms. We may however take one step forward towards an explanation by considering that the terms in equation (5) which have been added represent a torsional electric force having the axis of as axis. The equations mean therefore that a displacement of the electron in the z direction may be produced not only by a force acting in that direction, but also by a couple acting round it. A rifle bullet lying in its rifle barrel would be displaced in a similar manner along the barrel both by a pulling and twisting force. But if we take the dimensions of a single electron to be very small, we exclude the possibility of a constraint which would enable a couple to cause a motion in one direction. We must in that case draw the conclusion that the vibrations of the electron which give rise to the rotatory effect are motions of systems of electrons united together by certain forces which are such that a couple of electric forces produces a displacement of the positive electrons in one direction or of the negative electrons in the opposite direction along the axis of the couple. In view of the fact that a single electron cannot be acted on by a torsional force, it would have been more appropriate to base our investigation on equation (44) Chapter XI. The generalized force, would in the present problem dQ depend not only on R but on dy dx (-4), and if the investigation in the second part of Art. 149 is modified by the addition of appropriate terms, the result arrived at would, for a single variable, remain the same as that represented by (16). 163. Allogyric Double Refraction. Equations (2) show that the analytical representation of plane polarized waves travelling through an optically active medium necessarily involves two different wave velocities. In any question concerning the refraction and reflexion of light, we may take all four displacements represented by (2) separately and apply the formula obtained for homogeneous disturbances. It is clear that the wave on emergence must be split into two separate waves which are circularly polarized in opposite directions. This double refraction, due to the rotational effect, is verified by experiment and has some practical importance. Quartz, as has already been mentioned, turns the plane of polarization of waves travelling parallel to the optic axis, and in consequence, a ray travelling along the optic axis is doubly refracted at emergence. Quartz is very useful in optical investigations on account of its transparence to ultra-violet rays, and it is a serious drawback that it is impossible to avoid double refraction in a prism made of that substance. The difficulty is overcome by combining two prisms made of two specimens, one of which has a right-handed and the other a lefthanded rotatory power. These two prisms ABC and A,BC (Fig. 172) are right-angled at C and have their optic axes parallel to AA,. They |