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245

draw PT perpendicular to OP in the plane POz. Then the preceding equations show, that the direction of vibration is along PT, and its magnitude is equal to

Restoring the time factor ebt and realizing, this becomes

This is the expression for the disturbance produced by a simple source of light at a point r, whose distance from the source is large compared with the wave-length.

The motion is, as might be expected, symmetrical with respect to the axis of z, and vanishes on that axis where =0 or π; and it is a maximum on the plane ay where π.

=

234. In order to obtain the most general expression for a singular point of the second order, we must put n = 1; whence =(B-C)x+(C − A ) y2 + (A − B) z2 + 2A'yz + 2B′zx + 2Cxy, X1 = ai +By+yz;

d p

.(49).

The expression for a singular point of the second order accordingly contains eight constants, and is therefore a function of considerable generality. Let us now suppose, as a particular case, that

then, if we confine our attention to points at a considerable distance from the origin, we may put

It therefore follows, that the magnitude of the displacement represented by (50) is

and that its direction is along PT. Restoring the time factor, adding (48) and (51), and writing cdS/2x for F, we obtain

We therefore see that Stokes' expression for the disturbance produced by an element of a plane wave of light is equivalent to the combination of a simple and a double source.

At the same time, if we were to carry out the investigation on the same lines as I have done in the case of sound in the paper referred to, there can, I think, be little doubt, that we should find that there is an infinite number of combinations of multiple sources which would produce the required effect, and consequently Stokes' law although the simplest, is only one out of an infinite number. The question is not, however, of very much importance in the case of light, inasmuch as, in problems relating to diffraction, we may with sufficient accuracy take sin : = cos 0=1, in which case the disturbance due to the element will be

235. The physical explanation of the intensely blue colour of the sky, which cannot fail to have attracted the attention of those who have resided in warm countries, has formed the subject of various speculations. It has also been found by experiment, that a beam of light which is emitted by a bright cloud, exhibits decided traces of polarization, and that the direction of maximum larization is perpendicular to that of the beam. The experi

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ments of Tyndall1 on precipitated clouds, point to the conclusion, that both these phenomena are due to the existence of small particles of solid matter suspended in the atmosphere, which modify the waves of light in their course; and we shall now proceed to give an account of a theory due to Lord Rayleigh, by means of which these phenomena may be explained.

236. The theory of Lord Rayleigh in its original form, was an elastic solid theory; but it is equally applicable to the electromagnetic theory of light, since we shall hereafter see, that the equations, which are satisfied by the electric displacement, are of the same form as those which are satisfied by those portions of the displacements of an elastic solid, upon which distortion unaccompanied by dilatation depends.

If we suppose that the particles are spherical, it follows that when a plane wave of light impinges upon a particle, the latter will be thrown into a state of vibration; and the only possible motion which the particle can have, will consist of a motion of translation in the plane containing the directions of propagation and vibration of the impinging wave, and a motion of rotation about an axis perpendicular to this plane. If the ether be regarded as a medium, which possesses the properties of an elastic solid, the motion of the particles will give rise to two scattered waves, one of which will be a longitudinal wave, and therefore produces no optical effects, whilst the other will be a distortional wave, which will give rise to the sensation of light. If on the other hand, the ether be regarded as an electromagnetic medium, only one wave, viz. an optical wave, will be propagated. In order to obtain a complete mathematical solution, it would be necessary to introduce the boundary conditions, and to proceed on

1 Phil. Mag. May 1869, p. 384.

2 Ibid. Feb., April and June 1871; Aug. 1881.

3 See Phil. Mag. Aug. 1881.

If the ether be regarded as a medium which possesses the properties of an elastic solid, three suppositions may be made respecting the boundary conditions. (i) We may suppose that no slipping takes place, which requires that the velocity of the ether in contact with the sphere should be equal to that of the sphere itself; but, inasmuch as there are reasons for thinking that the amplitudes of the vibrations of the matter are very much smaller than those of the ether in cont with it, except in the extreme case in which one of the free periods of the m equal to the period of the ethereal wave, this hypothesis is improbable.

(ii) We may suppose that partial slipping takes place. This hy

the same principles as in the corresponding acoustical problem'. It will not however be necessary to enter into any considerations of this kind, if we assume that the principal effect of the incident wave is to cause the particle to perform vibrations parallel to the direction of vibration of this wave.

237. To fix our ideas, let us suppose that the direction of propagation of the primary wave is vertical, and that the plane of vibration is the meridian. The particle will accordingly vibrate north and south, and its effect will be the same as that of a simple source of light, whose axis is in this direction. Accordingly if be the angle which any scattered ray makes with the line running north and south, it follows from (48), that the displacement will be of the form

and is therefore a maximum for rays, which lie in the vertical plane running east and west, for which = 7; whilst there is no scattered ray along the north and south line for which p=0. If the primary wave is unpolarized, the light scattered north and south is entirely due to that component which vibrates east and west. Similarly any other ray scattered horizontally is perfectly polarized, and the vibration is performed in a horizontal plane. In other directions, the polarization becomes less and less complete as we approach the vertical, and in the vertical direction altogether disappears.

238. The preceding argument also shows, that the vibrations of polarized light must be perpendicular to the plane of polarization. For if the light scattered in a direction perpendicular to that of a primary wave be viewed through a Nicol's prism, it will be found that no light is transmitted, when the principal section is open to the objection that the law of slipping is unknown, and would therefore involve an additional assumption; and also that it would introduce frictional resistance.

(iii) We may suppose that perfect slipping takes place. In this case the boundary con litions are continuity of normal motion, and zero tangential stress. This hypothesis has much to commend it on the ground of simplicity, since the action of the ether on the matter consists of a hydrostatic pressure, and in the case of a sphere is consequently reducible to a force; whereas, if no slipping or partial slipping took place, the action would (except in special cases) consist of a couple as well as a force.

Lord Rayleigh, Theory of Sound, vol. II. § 334.

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parallel to the direction of the primary wave. Hence the vibrations of the extraordinary wave in a uniaxal crystal, lie in the principal plane.

239. We must now consider the colour of the scattered light. The experiments of Tyndall showed, that when the particles of foreign matter were sufficiently fine, the colour of the scattered light is blue. The simplest way of obtaining a theoretical explanation of this phenomenon, is by means of the method of dimensions. The ratio I of the amplitudes of the scattered and the primary light, is a simple number, and is therefore of no dimensions. This ratio must however be a function of T the volume of the disturbing particle, p' its density, r the distance of the point under consideration from it, b the velocity of propagation of light, and p the density of the ether. Since I is of no dimensions in mass, it follows that p and p' can only occur under the form p/p', which is a number and may be omitted; we have therefore to find out how I varies with T, r, λ and b.

Of these quantities b is the only one depending on the time; and therefore since I is of no dimensions in time, b cannot occur. We are therefore left with T, r and X.

Now it is quite clear from dynamical considerations, that I varies directly as T and inversely as r, and must therefore be proportional to T/Xr, T being of three dimensions in space. In passing from one part of the spectrum to another, λ is the only quantity which varies, and we thus obtain the important law :—

When light is scattered by particles, whose dimensions are small compared with the wave-length of light, the ratio of the amplitudes of the vibrations of the scattered and incident light, varies inversely as the square of the wave-length, and the ratio of the intensities, as the inverse fourth power.

From this law we see, that the intensity of the blue light is the greatest. Hence the blue colour of the sky may be accounted for on the supposition, that it is due to the action of minute particles of vapour, and also probably to the molecules of air, which scatter the waves proceeding from the sun.