by the incident and emergent rays. It will be seen that as the angle i increases, the deviation also increases up to 42° 28', after which, although the angle of incidence goes on augmenting, the deviation becomes less. The maximum 42° 28' corresponds to an incidence of 60°, but in reality at this point we have already passed, by a small quantity, the exact maximum, which occurs between 58° and 59°. Its amount is 42° 30'. This deviation corresponds to the red band of the rainbow. In a precisely similar manner the other colors rise to their maximum, and fall on passing beyond it; the maximum for the violet band being 40° 30'. The entire width of the primary rainbow is therefore 2°, part of this width being due to the angular magnitude of the sun.
We have thus revealed to us the geometric construction of the rainbow. But though the step here taken by Descartes and Newton was a great one, it left the theory of the bow incomplete. Within the rainbow proper, in certain conditions of the atmosphere, are seen a series of richly-colored zones, which were not explained by either Descartes or Newton. They are said to have been first described by Mariotte,[1] and they long challenged explanation. At this point our difficulties thicken, but, as before, they are to be overcome by attention. It belongs to the very essence of a maximum, approached continuously on both sides, that on the two sides of it pairs of equal value may be found. The maximum density of water, for example, is 39° Fahr. Its density when 5° colder, and when 5° warmer, than this maximum is the same. So, also, with regard to the slopes of our water-shed. A series of pairs of points of the same elevation can be found upon the two sides of the ridge; and, in the case of the rainbow, on the two sides of the maximum deviation we have a succession of pairs of rays having the same deflection. Such rays travel along the same line, and add their forces together after they quit the drop. But light, thus re-enforced by the coalescence of non-divergent rays, ought to reach the eye. It does so; and were light what it was once supposed to be—a flight of minute particles sent by luminous bodies through space—then these pairs of equally deflected rays would diffuse brightness over a large portion of the area within the primary bow. But inasmuch as light consists of waves and not of particles, the principle of interference comes into play, in virtue of which waves can alternately re-enforce and destroy each other. Were the distance passed over, by the two corresponding rays within the drop, the same, they would emerge exactly as they entered. But in no case are the
- ↑ Prior of St. Martin-sous-Beaune, near Dijon, member of the French Academy of Sciences; died in Paris, May, 1684.
as the convergence is not quite exact, the parallelism after emergence is only approximate. The emergent rays cut each other at extremely sharp angles, thus forming a "caustic" which has for its asymptote the ray of maximum deviation In the secondary bow we have to deal with a minimum, instead of a maximum, the crossing of the incident and emergent rays producing the observed reversal of the colors. (See Engel and Shellbach's diagrams of the rainbow.)
