4μ2 sin' i − (2 + √√2) (1 + μ2) sin2 i + 2 + √2 = 0.

This equation gives a real value of sin i for values of μ lying between 14 and 1.6.

=

Fresnel employed a rhomb of St Gobain glass, for which μ=151, which gives i 48° 37' 3" or 54° 37' 20". Now the angles at B and D of the rhomb are each equal to the angle of incidence; if therefore a rhomb of glass, whose index of refraction is 1.51, and whose acute angles are equal to 54° 37' 20" be employed, and light polarized as described above is incident normally on the face AB and is reflected twice, the emergent light ought to be circularly polarized. This result was found to agree with experiment.

If the incident light is polarized in any other plane, the emergent light will be elliptically polarized.

If a be the azimuth of the plane of polarization, and the emergent elliptically polarized light be passed through a second rhomb, the reflected light will be plane polarized, and the plane of polarization will be rotated through an angle 2a.

Theories of Neumann and MacCullagh'.

180. We must now consider the difficulty alluded to at the commencement of § 175.

The surface conditions assumed by Fresnel are, (i) continuity of the rate at which energy flows across the reflecting surface, (ii) continuity of the components of displacement parallel to this surface. Now when the incident light is polarized in the plane of incidence, there is no component displacement perpendicular to this surface; but when the light is polarized perpendicularly to

1 Neumann, Abhand. Berlin Akad. 1835.

MacCullagh, “On Crystalline Reflection and Refraction." Trans. Roy. Irish Acad. vols. XVIII. p. 31, and xxI. p. 17.

the plane of incidence, it is impossible to evade the conclusion, that the components perpendicular to the surface ought also to be continuous. In fact a discontinuity in the normal displacement, would involve something analogous to an area source in Hydrodynamics, and there are no grounds for supposing that anything of the kind occurs.

The condition that the normal displacements should be continuous, is

(A+A) sin i A, sin r.

Multiplying this by (14) we obtain

=

(AA) sin i cos i A, sin r cos r .........(18).

=

From (7) the condition of continuity of energy may be written (A- A') p sin i cos i A, p, sin r cos r,

=

and in order that this may be consistent with (18), we must have P = P1. Accordingly Neumann and MacCullagh assumed this condition in their theories of reflection and refraction; and we shall now trace the consequences of this hypothesis.

When the vibrations are perpendicular to the plane of incidence, the equations are

A+ A' = A1,

It follows from (2), that on this theory, the intensity of light in all transparent media is proportional to the square of the amplitude, and accordingly (19) and (20) give the intensities of the reflected and refracted light. Neumann and MacCullagh further supposed, that the vibrations of polarized light are in instead of perpendicular to the plane of polarization, and on this supposition the formulæ (19) and (20) are in complete agreement with (16) and (11) given by the theory of Fresnel.

181. The two hypotheses of Neumann and MacCullagh are singularly seductive, inasmuch as it will hereafter be shown, that they enable the laws of the propagation of light in crystals, and also the reflection and refraction of light from crystalline surfaces, to be determined in accordance with Green's rigorous theory of elastic media; whereas the contrary assumption, that the density of the ether is different in different media, leads to a variety of difficulties in the application of this theory. There are however grave objections to these hypotheses; for in the first place, there are strong grounds for supposing, that the vibrations of polarized light are perpendicular to the plane of polarization; and in the second place, Lorenz and Lord Rayleigh have shown, as will be explained in Chapter XII., that the hypothesis of equal density, leads to the conclusion that there are two polarizing angles, which is contrary to experiment.

EXAMPLES.

1. A thin layer of fluid of thickness T, floats on the surface of a second fluid of infinitesimally greater refractive power. Light is incident perpendicularly on the layer; show that the intensity of the reflected light is

where μ and μ + Su are the refractive indices of the layer and of the fluid which supports it respectively, and a, λ are the amplitude and wave-length of the incident vibration.

2. If in the separating surface of two media, there be a straight groove of small depth c, inclined at an angle a to the plane of incidence, prove that there will be a groove in the refracted wave of depth c sin (i − r) cosec i, inclined at an angle tan-1 (tan a cosec r) to the plane of refraction, where i and r are the angles of incidence and refraction.

What is the corresponding quantity for the reflected wave?

Explain why the image of a candle from rough glass becomes red, as the angle of incidence is diminished.

3. A ray polarized at right angles to the plane of incidence falls on a refracting surface; if the intensities of the reflected and refracted rays are equal, and the tangent of the polarizing angle lies between 1 and 3, prove that the corresponding angle of incidence is least, when the refracting medium is such that its polarizing angle is π.

4. Circularly polarized light is incident in the usual manner upon a Fresnel's rhomb, so cut that after one reflection in the rhomb, the incident ray emerges from it perpendicularly to the cut face. A uniaxal crystal cut parallel to the optic axis is placed in the path of the emergent ray, with its faces normal to it, and with its principal plane inclined at an angle to the plane of incidence and reflection in the rhomb. Show that the intensities of the refracted rays are as 2−1 : √2+1.

5. If light polarized perpendicularly to the plane of incidence, falls on a thin plate of air between two plates of different kinds of glass, prove that there are two angles at which the colours will disappear, and that between the two angles a change takes place in the order of the colours.

6. A ray of circularly polarized light is incident at the plane surface of separation of two media. If e and e' are the excentricities of the elliptically polarized light reflected and refracted, and i and r the angles of incidence and refraction, show that

(1 − e2) (1 − e'2) = cos2 (i + r).

CHAPTER XI.

GREEN'S THEORY OF ISOTROPIC MEDIA.

182. THE various dynamical theories of the ether, which have been proposed to explain optical phenomena, may be classed under three heads; (i) theories which suppose that the ether possesses the properties of an elastic medium, which is capable of resisting compression and distortion; (ii) theories based upon the mutual reaction of ether and matter; (iii) the electromagnetic theory advanced by the late Prof. Clerk-Maxwell, which assumes that light is the result of an electromagnetic disturbance. We shall now proceed to consider the first class of theories.

183. The dynamical theory proposed by Green', assumes that the ether is an elastic medium, which is capable of resisting compression and distortion. It therefore follows, that if the ether is in equilibrium, and any element is displaced from its position of equilibrium or is set in motion, the ether will be thrown into a state of strain, and will thereby acquire potential energy. Now the potential energy of any element of the ether, must necessarily depend upon the particular kind of displacement to which it is subjected; hence the potential energy per unit of volume must be a function of the displacements, or their differential coefficients, or of both. If therefore we can determine the form of this function, the equations of motion can be at once obtained by known dynamical methods.

According to the views held by Cauchy, the ether is to be regarded as a system of material particles acting upon one another by mutually attractive and repulsive forces, such that the

1 Trans. Camb. Phil. Soc. 1838; and Math. Papers, p. 245.