Newton has determined by computation that when the angle A E 0 (fig. 3) =40°17', the violet rays alone, after two refractions and one reflection, will enter the eye of the spectator at E, the other rays falling below; and when L B E 0 =42° 2', the red rays alone will enter tho eye, the violet rays passing above. Again, when L 050= 50' 59', the red rays only will enter the eye, after two refractions and two reflections, the violet rays falling below ; and when L DE 0=54' 9', the violet rays alone will enter, the red passing above. If the interval between the drops A and n, and also between the drops c and a, were occupied by other drops, it may readily be imagined that the pencils of parallel rays which come from them to the eye would be of all the prismatic colours between the red and violet, and that thus there would appear in the heavens two narrow spectra : the length of that between A and B would bo 1° 45', and of that between o and D would be 3°10'. Therefore, if all the lines drawn to E from the drops in the two spectra were to revolve conically about o as an axis, the drops on these lines would be in situations to send to the eye rays of their own proper colours, and thus there would exist the appearance in the heavens of two concentric bands of variously coloured light.
But it has been hero supposed that the pencils s A, B n, lac., come from the centre only of the sun's disc, whereas each point of the diso produces two bows similar to those which have been described : therefore the lower extremity of the interior bow will be a violet band whose breadth is equal to half the diameter of the sun (suppose 15'), and which is situated immediately below the violet line formed by the centre of the disc; and in like manner the upper extremity of the interior bow will be a red band whose breadth is also = 15', and which is situated immediately above the red line formed by the centre of the disc : consequently the whole breadth of the interior bow is about = 2° 15'. Similarly 30' (the measure of the sun's diameter) must be added to the breadth of the outer bow, as before determined, which thus becomes about = 3° 40'. In both bows, the colours between the violet and red are lees distinct than those two colours, because of the mixture of the coloured light from all parts of the disc.
On account of the two reflections which take place in the interior of the drops which give rise to the outer bow, while there is but one reflection in those which produce the inner bow, there must be a greater quantity of light lost by transmission through the drops in the former case than in the latter; and hence the outer bow is always fainter than the other.
The effect of the light of any given colour which comes out of the drop in a divergent state has been neglected in the preceding inves tigation, but it is by no means insensible, especially in the neighbour hood of the bow, where the divergence is not great. If a ray which emerges after one or two internal reflections be first supposed to be ncident along a line passing through the centre of the drop, and the ins of incidence be then conceived to move parallel to itself until laving passed through a I in fig. 1 or 2, it just grazes the surface, it
will be found by calculation that the deviation, at first 180° or 360°, lecreases until the position is reached in which consecutive rays come nit parallel, after which it increases again. In fig. 1, the deviation akes place in the direction of the motion of the hands of a watch, and n fig. 2 in the:coutrary direction. Hence the rays which emerge with ;neater deviation than E, are situated (as to their directions) above C 2 in fig. 1, and below K E in fig. 2 ; and therefore in fig. 3 the drops to vhich they refer would be situated below A or s in the former case, end above C or D in the latter. Hence, considering light of one colour mly, and again regarding the sun as a mere point, we ought instead :f two bright circles in the sky, in the positions determined above, xf have two bows, in each of which the brightness terminates abruptly in the side towards the other, and fades off on the other side. Hence, :onsidering the joint effect of all the colours, we see that they must As more mixed together, especially about the blue and violet, than would otherwise have been the case. In a vivid solar rainbow the darkness of the space between the bows compared with the space mmediately within the primary bow is readily seen.
There is one part of the phenomenon which cannot be explained merely by geometrical optics, namely, the existence of what have been failed supernumerary bows. In the upper part of the primary solar rainbow two or even three maxima of red may frequently be seen on :he inside of the principal maximum, forming as many additional or supernumerary bows, decreasing in vividness and in breadth on receding from the principal bow inwards. Even the secondary bow, notwithstanding its comparative faintness, shows symptoms of the fame phhnomenon. The existence of these bows was first explained by Dr. Young, who pointed out that the light corresponding to any par ticular direction within the bow was double, and that the two portions were in a condition to interfere. The explanation, however, resulting from the application of the principle of interference to two systems of rays whose courses are determined by geometrical optics is imperfect, Ind would make the illumination increase to infinity on passing from the illuminated side to the place of the geometrical bow, and then ceases abruptly. Whenever such results follow from the calculations of geometrical optics, it is found that they are modified in practice by the phenomena of diffraction. The complete explanation of the super numerary bows, according to the principles of the undulatory theory, has been given by Mr. Airy, in a paper On the Intensity of Light in the neighbourhood of a Caustic' (` Camb. Phil. Trans.,' vol. vii., p. 379), and his results have been verified by the observations of Professor ?filler.