We will suppose that N is the total number of atoms in a small solid angle, the illumination from which is to be found. Then, from the formula for line-waves, we shall have an intensity proportional to and here is the intensity of the direct sunlight. Now in fact p, the scattering constant of the atom, does not depend very much on the wave-length, as long as we only consider visible light and we can therefore say that the light scattered is inversely proportional to the fourth power of the 'wave-length. This explains why the sky is blue even though sunlight is rather weaker in the blue part of the spectrum than the red, for the wave-length of red light is about 1.8 times that of blue, and so factor is about Io times as large for blue light as for red.
Another property of the sky-light is its polarization. Consider a point of the sky at right-angles to the sun. The unpolarized light from the sun may be broken into two polarized components, one of which has electric force pointing at the observer. The line waves induced by this component will have the observer at their poles, and so will give no light towards him. He will therefore only receive the light of the other component, and this will be polarized in the direction perpendicular to the line joining the sun to the point observed. At other angles both polarized components are present, one in constant intensity and the other proportional to the squared cosine Jf the distance from the sun. If the sky is actually observed at right-angles to the sun with a nicol, it will be found that the polarization is not complete. This is partly to be attributed to rays that have been scattered several times on their way to the eye, and also to the fact that, though we have spoken only of atoms, the air is mostly composed of diatomic mole cules, and for these the line-wave need not have its pole exactly coincident with the direction of the incident force. There is also usually a complication due to dust, which acts by direct reflection and makes the sky much brighter near the sun than at broad angles.
A most interesting application of the theory of sky-light was made by Lord Rayleigh (3rd baron). The barometer shows the mass of the atmosphere, and so, by a direct comparison between the brightness of the sky and that of the sun, it is possible to deduce how much light is scattered by one cubic centimetre of air at ground level. In fact, if N is the number of atoms in I cu.cfn., we can evaluate Now, as we shall see, we can also find Alp by a study of refraction, and hence we can estimate N. The process led to one of the earliest good determinations of the funda mental constant of Avogadro, the number of molecules in a gram molecule. Similar processes have since been applied in the laboratory, with the advantage that the incident light can be itself polarized, and similar results are obtained.
Take a thin sheet of atoms spread over a plane on which mono chromatic light falls perpendicularly. The diagram of fig. 7 will describe the process, provided that we now regard the plane as composed of matter. Each atom will emit a line-wave, and the effect at P will consist of the superposition of these waves on the original beam, which is supposed to arrive at P undisturbed. The process is very like Fresnel's discussion of diffraction, though there we imagined that the original wave was suppressed at the plane AB. Suppose that there are N atoms per unit volume, in a thin sheet of thickness 1 spread over the z plane, and let the inci dent wave be The effects that all the atoms produce at P can be summed just as in Fresnel's construction, and the result is an amplitude The important point to notice is that the phase differs by a quarter wave-length from that of the original wave ; this is due to the fact that the scattered waves are in phase with the inci dent, and is in contrast with Fresnel's construction, in which, in order to get the right result, the phase had to be advanced by a quarter wave-length. We now add the two waves together, and. taking advantage of the smallness of the scattered wave, we find If we adopt the ordinary process of refraction and attribute the change of phase to the changed wave-velocity during the passage through the thickness 1 of the sheet of matter, we should say the emergent wave was and so we may identify n— I with This is the physical origin of refraction. We see also how the reflected wave arises, for the line-wave from each atom will be exactly the same at the point which is image of P in the plane AB as it is at P, and so the total amplitudes of the scattered waves will be the same at the two points; but for the reflected wave there is no interference with the incident light. It is easy to verify that the actual reflected intensity is that which should arise from a thin sheet of refractive index n and thickness 1.