ON RAINBOWS. 663 



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 maxi- 

 mum, 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 Des- 

 cartes or Newton. They are said to have been first described by 

 Mariotte,* and they long challenged explanation. At this point our 

 difficulties thicken, but, as before, they are to be overcome by atten- 

 tion. It belongs to the very essence of a maximum, approached con- 

 tinuously 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 rain- 

 bow, 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 dif- 

 fuse 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 



as the convergence is not quite exact, the parallelism after emergence is only approxi- 

 mate. 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 Shell- 

 bach's diagrams of the rainbow.) 



* Prior of St. Martin-sous-Beaune, near Dijon, member of the French Academy of 

 Sciences ; died in Paris, May, 1684. 



