in which a and a' denote (as before) the angles which the planes of polarization form with the plane of incidence, before and after refraction. This law was discovered experimentally by Sir David Brewster: it is a necessary consequence of the theory already given, and is deduced by a process exactly similar to that of the preceding article.

When the refracted ray meets a second surface parallel to the former, and emerges into air, the azimuth of the plane of polarization of the emergent ray will be given by the formula

wherefore, since cos (ri) = cos (ir), we have, for the change of the plane of polarization, produced by transmission through a plate bounded by parallel surfaces,

CHAPTER XI.

ELLIPTIC POLARIZATION.

(192) WHEN an ethereal molecule is displaced from its position of equilibrium, the forces of the neighbouring molecules are no longer balanced, and their resultant tends to drive the particle back to its position of rest.* The displacement being supposed to be very small, in comparison with the intervals between the molecules, the force thus excited will be proportional to the displacement; and from this it follows, according to known mechanical principles, that the trajectory described by the molecule will be an ellipse, whose centre coincides with the position of equilibrium. Hence the vibration of the ethereal molecules is, in general, elliptic, and the nature of the light depends on the direction and relative magnitude of the axes. By the principle of transversal vibrations, these elliptic vibrations are all in the plane of the wave; their axes, however, may either preserve constantly the same direction in that plane, or they may be continually shifting. In the former case, the light is said to be polarized; in the latter, it is unpolarized, or common light.

The relative magnitude of the axes of the ellipse determines the nature of the polarization. When the axes are equal, the ellipse becomes a circle, and the light is said to be circularly polarized. On the other hand, when the lesser axis vanishes, the ellipse becomes a right line, and the light is

media.

This is not strictly true, except in homogeneous or uncrystallized

plane-polarized, the vibrations being in this case confined
to a single plane passing through the direction of the ray.
In intermediate cases, the polarization is called elliptical; and
its character may vary indefinitely between the two extremes
of plane polarization and circular polarization.

(193) An elliptic vibration may be regarded as the resultant of two rectilinear vibrations, at right angles to one another, which differ in phase.

For, let x and y denote the distances of the molecule of the ether from its position of rest, in the two rectangular " directions; a and b the amplitudes of the component vibrations; and t the time. Then

whence

x = a sin (vt − a), y = b sin (vt - ẞ);

=

a-Baro (sin-2)-are (sin-2).

Taking the cosines of both sides, and clearing the result of
radicals, we obtain

This is the equation of an ellipse referred to its centre.

When the component rays are in the same phase, or a = ß, the equation is reduced to

which is the equation of a right line passing through the
origin. In this case, therefore, the vibration becomes recti-
linear, and the light is plane-polarized.

When the component vibrations are equal in amplitude,
and differ 90° in phase,

and the preceding equation becomes

y2 + x2 = a2.

The path described by the molecule is then a circle. The same thing is true, when a - ẞ is any odd multiple of 90°.

(194) The nature of the elliptic polarization is completely defined, when we know the direction of the axes of the ellipse, and the ratio of their lengths.

These may be determined experimentally. In fact, when the elliptically-polarized ray is transmitted through a doublerefracting prism, whose principal section is parallel to one of the axes of the ellipse, it is resolved into two plane-polarized rays, one of which has the greatest possible intensity, and the other the least. Accordingly, the direction of the principal section, for which the two pencils are most unequal, is the direction of one of the axes; and the square roots of the intensities are in the ratio of their lengths.

The direction of the axes of the ellipse may be more conveniently determined by turning the prism until the two pencils are of equal intensity: the principal section is then inclined at an angle of 45° to each of the axes.

(195) When a plane-polarized ray undergoes reflexion, the reflected light is, generally, elliptically polarized. For a plane-polarized ray may be resolved into two, polarized respectively in the plane of incidence, and in the perpendicular plane; and we shall presently see that the effect of reflexion is, in general, to alter the phases of these two portions, and by a different amount. Hence the reflected light is compounded of two plane-polarized rays, whose vibrations are at right angles, and whose phases are no longer coincident; it is therefore elliptically polarized (193).

The first case in which this effect was observed was that of total reflexion.

When the angle of incidence exceeds the angle of total reflexion (the light passing from the denser into the rarer medium), the expressions for the amplitudes of the reflected vibrations, given in (184, 5), become imaginary. But it is obvious that, in this case, the intensity of the reflected light is simply equal to that of the incident, there being no refracted pencil. How, then, are the imaginary expressions to be interpreted? They signify, according to Fresnel, that the periods of vibration of the incident and reflected waves, which had been assumed to coincide at the reflecting surface, no longer coincide there when the reflexion is total; or, in other words, that the ray undergoes a change of phase at the moment of reflexion. The amount of this change has been deduced by Fresnel, by an ingenious train of reasoning, based upon the interpretation of imaginary formula. It varies with the incidence; and is different for light polarized in the plane of incidence, and in the perpendicular plane.

In the case of light polarized in any azimuth, we have only to conceive the incident vibration resolved into two, one in the plane of incidence, and the other in the perpendicular plane. The phases of these vibrations being differently altered by reflexion, the reflected vibration will be the resultant of two vibrations at right angles to one another, and differing in phase,--the amount of the difference depending upon the angle of incidence: this vibration, consequently, will be elliptic, and the reflected light elliptically-polarized. When the azimuth of the plane of polarization of the incident ray is 45°, the amplitudes of the resolved vibrations will be equal; and if, moreover, their difference of phase is a quarter of an undulation, the ellipse will become a circle, and the light will be circularly-polarized.

(196) Reducing his formula to numbers, in the case of St. Gobain's glass, Fresnel found that the difference of phase