The deviation is thus seen to vary with the angle of incidence; and by considering a set of parallel rays passing through the same principal plane of the sphere and incident at all angles, it can be readily shown that more rays will pass in the neighbourhood of the position of minimum deviation than in any other position. (See REFRACTION.) The drop will consequently be more intensely illuminated when viewed along these directions of minimum devia tion, and since it is these rays with which we are primarily con cerned, we shall proceed to the determination of these directions.
Since the angles of incidence and refraction are connected by the relation sin i=,u sin r (Snell's Law),µ being the index of re fraction of the medium, then the problem may be stated as follows : to determine the value of the angle i which makes D = 2 (i-r)-1--n(r - 2r) a maximum or minimum, in which i and r are connected by the relation sin i= ,u sin r, /.1. being a constant. By applying the method of the differential calculus, we obtain cos i= I as the required value; it may be readily shown either geometrically or analytically that this is a minimum. For the angle i to be real, cos i must be a fraction; that is, n2+ or (n+ Since the value of p. for water is about 4, it follows that n must be at least unity for a rainbow to be formed; there is obviously no theoretical limit to the value of n, and hence rainbows of higher orders are possible.
*So far we have only considered rays of homogeneous light, and it remains to investigate how lights of varying refrangibilities will be transmitted. It can be shown, by the methods of the differ ential calculus or geometrically, that the deviation increases with the refractive index, the angle of incidence remaining constant. Taking the refractive index of water for the red rays as l and for the violet rays as we can calculate the following values for the minimum deviations corresponding to certain assigned values of n.
To this point we have only considered rays passing through a principal section of the drop; in nature, however, the rays impinge at every point of the surface facing the sun. It may be readily deduced that the directions of minimum deviation for a pencil of parallel rays lie on the surface of cones, the semi-vertical angles of which are equal to the values given in the above table. Thus,
rays suffering one internal reflection will all lie within a cone of about 42° ; in this direction the illumination will be most intense; within the cone the illumination will be fainter, while, outside of it, no light will be transmitted to the eye.
Fig. 2 represents sections of the drop and the cones containing the minimum deviation rays after 1, 2, 3 and 4 reflections; the order of the colours is shown by the letters R (red) and V (vio let). It is apparent, therefore, that all drops transmitting intense light after one internal reflection to the eye will lie on the sur faces of cones having the eye for their common vertex, the line joining the eye to the sun for their axis, and their semi-vertical angles equal to about 41° for the violet rays and 43° for the red rays. The observer will, therefore, see a coloured band, about 2° in width, and coloured violet inside and red outside. Within the band, the illumination will be faint ; outside the band there will be perceptible darkening until the second bow comes into view. Similarly, drops transmitting rays after two internal reflections will be situated on covertical and coaxial cones, of which the semi-vertical angles are 51° for the red rays and 54° for the violet. Outside the cone of 54° there will be faint illumination ; within it, no secondary rays will be transmitted to the eye. We thus see that the order of colours in the secondary bow is the reverse of that in the primary; the secondary is half as broad again (3°), and is much fainter, owing to the longer path of the ray in the drop, and the increased dispersion.
Similarly, the third, fourth and higher orders of bows may be investigated. The third and fourth bows are situated between the observer and the sun, and hence, to see them, the observer must face the sun when looking at the water drops which produce the bows. But the illumination of the bow is so weakened by the repeated reflections, and the light of the sun is generally so bright, that these bows are rarely, if ever, observed except in artificial rainbows. The same remarks apply to the fifth bow, which differs from the third and fourth in being situated in the same part of the sky as the primary and secondary bows, being just above the secondary.