580 ON THE LIGHT EMITTED FBOM A [436
It is hoped that the explanations and calculations here given may help to remove the difificulties which have been felt in connexion with this subject. The main point would seem to be the interpretation of the n- term as representing the surface reflexion when a cloud is supposed to be abruptly terminated. For myself, I have always regarded the light internally dispersed as proportional to n, even when n is very great, though it may have been rather by instinct than on sufficiently reasoned grounds. Any other view would appear to be inconsistent with the results of my son's recent laboratory experiments on dust-free air.
The reader interested in optics may be reminded of the application of similar ideas to a grating on which fall plane waves of homogeneous light. If the spacing be quite uniform, the light behind is limited to special directions. Seen from other directions the interior of the grating appears dark. But if the ruling be irregular, light is emitted in all directions and the interior of the grating, previously dark, becomes luminous.
In the problems considered above the space occupied by a source, whether primary or secondary, has been supposed infinitely small. Probably it would be premature to try to include sources of finite extension, but merely as an illustration of what is to be expected we may take the question of n phases distributed at random over a complete period (27r), but under the limitation that the distance between neighbours is never to be less than a fixed quantity S. All other situations along the range are to be regarded as equally probable.
As we have seen, the expectation of intensity may be equated to
09...d0n + ... d01...d0n,...
J J J
and the question turns upon the limits of the integrals.
The case where there are only two phases (n = 2) is simple. Taking #2 as coordinates of a representative point, Fig. 3, the sides of the square OAGB are 2?r. Along the diagonal 0(7, dl and Q» are equal. If DJ3, B FG be drawn parallel to 0(7, so that OD, OF are equal to 8, the prohibited region is that part of the square lying between these lines. Our inte- p grations are to be extended over the remainder, viz. the triangles FBG, DAE, and every point, "° D or rather every infinitely small region of given •Plg' 3*
area, is to be regarded as equally probable. Evidently it suffices to consider one triangle, say the upper one, where 0Z > 6l.
For the denominator in (40) we have
* = area of triangle FBG = K2?r ~ S)2-