It follows from this, that if the ray of light, after passing through the medium from north to south, is reflected by a mirror, so as to return through the medium from south to north, the rotation will be doubled when it results from magnetic action. When the rotation depends on the nature of the medium alone, as in turpentine, &c., the ray, when reflected back through the medium, emerges in the same plane as it entered, the rotation during the first passage through the medium having been exactly reversed during the second.

811.] The physical explanation of the phenomenon presents considerable difficulties, which can hardly be said to have been hitherto overcome, either for the magnetic rotation, or for that which certain media exhibit of themselves. We may, however, prepare the way for such an explanation by an analysis of the observed facts.

It is a well-known theorem in kinematics that two uniform circular vibrations, of the same amplitude, having the same periodic time, and in the same plane, but revolving in opposite directions, are equivalent, when compounded together, to a rectilinear vibration. The periodic time of this vibration is equal to that of the circular vibrations, its amplitude is double, and its direction is in the line joining the points at which two particles, describing the circular vibrations in opposite directions round the same circle, would meet. Hence if one of the circular vibrations has its phase accelerated, the direction of the rectilinear vibration will be turned, in the same direction as that of the circular vibration, through an angle equal to half the acceleration of phase.

It can also be proved by direct optical experiment that two rays of light, circularly-polarized in opposite directions, and of the same intensity, become, when united, a plane-polarized ray, and that if by any means the phase of one of the circularly-polarized rays is accelerated, the plane of polarization of the resultant ray is turned round half the angle of acceleration of the phase.

812.] We may therefore express the phenomenon of the rotation of the plane of polarization in the following manner :-A planepolarized ray falls on the medium. This is equivalent to two circularly-polarized rays, one right-handed, the other left-handed (as regards the observer). After passing through the medium the ray is still plane-polarized, but the plane of polarization is turned, say, to the right (as regards the observer). Hence, of the two circularlypolarized rays, that which is right-handed must have had its phase

814.]

STATEMENT OF THE FACTS.

403

accelerated with respect to the other during its passage through the medium.

In other words, the right-handed ray has performed a greater number of vibrations, and therefore has a smaller wave-length, within the medium, than the left-handed ray which has the same periodic time.

This mode of stating what takes place is quite independent of any theory of light, for though we use such terms as wave-length, circular-polarization, &c., which may be associated in our minds with a particular form of the undulatory theory, the reasoning is independent of this association, and depends only on facts proved by experiment.

813.] Let us next consider the configuration of one of these rays at a given instant. Any undulation, the motion of which at each point is circular, may be represented by a helix or screw. If the screw is made to revolve about its axis without any longitudinal motion, each particle will describe a circle, and at the same time the propagation of the undulation will be represented by the apparent longitudinal motion of the similarly situated parts of the thread of the screw. It is easy to see that if the screw is right-handed, and the observer is placed at that end towards which the undulation travels, the motion of the screw will appear to him left-handed, that is to say, in the opposite direction to that of the hands of a watch. Hence such a ray has been called, originally by French writers, but now by the whole scientific world, a left-handed circularly-polarized ray.

A right-handed circularly-polarized ray is represented in like manner by a left-handed helix. In Fig. 67 the right-handed helix A, on the right-hand of the figure, represents a left-handed ray, and the left-handed helix B, on the lefthand, represents a right-handed ray.

814.] Let us now consider two such rays which have the same wave-length within the medium.

B

N

all respects, except that one is the perversion of the other, like its image in a looking-glass. One of them, however, say 4, has a shorter period of rotation than the other. If the motion is entirely due to the forces called into play by the displacement, this shews that greater forces are called into play by the same displacement when the configuration is like A than when it is like B. Hence in this case the left-handed ray will be accelerated with respect to the right-handed ray, and this will be the case whether the rays are travelling from N to S or from S to N.

This therefore is the explanation of the phenomenon as it is produced by turpentine, &c. In these media the displacement caused by a circularly-polarized ray calls into play greater forces of restitution when the configuration is like A than when it is like B. The forces thus depend on the configuration alone, not on the direction of the motion.

But in a diamagnetic medium acted on by magnetism in the direction SN, of the two screws A and B, that one always rotates with the greatest velocity whose motion, as seen by an eye looking from S to N, appears like that of a watch. Hence for rays from S to N the right-handed ray B will travel quickest, but for rays from N to S the left-handed ray A will travel quickest.

815.] Confining our attention to one ray only, the helix B has exactly the same configuration, whether it represents a ray from S to N or one from N to S. But in the first instance the ray travels faster, and therefore the helix rotates more rapidly. Hence greater forces are called into play when the helix is going round one way than when it is going round the other way. The forces, therefore, do not depend solely on the configuration of the ray, but also on the direction of the motion of its individual parts.

816.] The disturbance which constitutes light, whatever its physical nature may be, is of the nature of a vector, perpendicular to the direction of the ray. This is proved from the fact of the interference of two rays of light, which under certain conditions produces darkness, combined with the fact of the non-interference of two rays polarized in planes perpendicular to each other. For since the interference depends on the angular position of the planes of polarization, the disturbance must be a directed quantity or vector, and since the interference ceases when the planes of polarization are at right angles, the vector representing the disturbance must be perpendicular to the line of intersection of these planes, that is, to the direction of the ray.

817.]

CIRCULARLY-POLARIZED LIGHT.

405

817.] The disturbance, being a vector, can be resolved into components parallel to x and y, the axis of z being parallel to the direction of the ray. Let & and n be these components, then, in the case of a ray of homogeneous circularly-polarized light,

where

η

¿ = r cos 0, n = r sin 0,

0 = nt ― qz+a.

(1)

(2)

In these expressions, denotes the magnitude of the vector, and the angle which it makes with the direction of the axis of a. The periodic time, 7, of the disturbance is such that

The wave-length, A, of the disturbance is such that

(3)

(4)

The phase of the disturbance when t and z are both zero is a. The circularly-polarized light is right-handed or left-handed according as q is negative or positive.

Its vibrations are in the positive or the negative direction of rotation in the plane of (x, y), according as n is positive or negative. The light is propagated in the positive or the negative direction of the axis of %, according as n and q are of the same or of opposite signs.

In all media n varies when q varies, and

dn

dq

is always of the same

n

the value of is

Hence, if for a given numerical value of greater when n is positive than when n is negative, it follows that for a value of q, given both in magnitude and sign, the positive value of n will be greater than the negative value.

Now this is what is observed in a diamagnetic medium, acted on by a magnetic force, y, in the direction of z. Of the two circularlypolarized rays of a given period, that is accelerated of which the direction of rotation in the plane of (x, y) is positive. Hence, of two circularly-polarized rays, both left-handed, whose wave-length within the medium is the same, that has the shortest period whose direction of rotation in the plane of ry is positive, that is, the ray which is propagated in the positive direction of z from south to north. We have therefore to account for the fact, that when in the equations of the system q and are given, two values of n will

satisfy the equations, one positive and the other negative, the positive value being numerically greater than the negative.

818.] We may obtain the equations of motion from a consideration of the potential and kinetic energies of the medium. The potential energy, V, of the system depends on its configuration, that is, on the relative position of its parts. In so far as it depends on the disturbance due to circularly-polarized light, it must be a function of r, the amplitude, and q, the coefficient of torsion, only. It may be different for positive and negative values of q of equal numerical value, and it probably is so in the case of media which of themselves rotate the plane of polarization.

The kinetic energy, T, of the system is a homogeneous function of the second degree of the velocities of the system, the coefficients of the different terms being functions of the coordinates.

819.] Let us consider the dynamical condition that the ray may be of constant intensity, that is, that r may be constant. Lagrange's equation for the force in becomes

d dT dT dv

+ = 0.

dt dr dr dr

(5)

Sincer is constant, the first term vanishes. We have therefore the equation

dT dV

dr

+ = 0,
dr

(6)

in which q is supposed to be given, and we are to determine the value of the angular velocity 0, which we may denote by its actual value, n.

The kinetic energy, T, contains one term involving n2; other terms may contain products of n with other velocities, and the rest of the terms are independent of n. The potential energy, V, is entirely independent of n. The equation is therefore of the form

An2+ Bn + C = 0.

(7)

This being a quadratic equation, gives two values of n. It appears from experiment that both values are real, that one is positive and the other negative, and that the positive value is numerically the greater. Hence, if A is positive, both B and C are negative, for, n2 are the roots of the equation,

if 1 and

The coefficient, B, therefore, is not zero, at least when magnetic force acts on the medium. We have therefore to consider the expression Bn, which is the part of the kinetic energy involving the first power of n, the angular velocity of the disturbance.