468 Professor Tyndall [Jan. 18, 



will act upon it as it acted upon its neighbour, the incident and 

 emergent rays inclosing in this instance a larger angle than before. 

 As we retreat further from the central ray the enlargement of this 

 angle continues up to a certain point, where it reaches a maximum, 

 after which further retreat from the central ray diminishes the angle. 

 Now, a maximum resembles the ridge of a hill, or a watershed, 

 from which the land falls in a slope at each side. In the case 

 before us the divergence of the rays when they quit the raindrop 

 would be represented by the steepness of the slope. On the top of 

 the watershed — that is to say, in the neighbourhood of our maximum 

 • — is a kind of summit level, where the slope for some distance almost 

 disappears. But the disappearance of the slope indicates, in the 

 case of our raindrop, the absence of divergence. Hence we find that 

 at our maximum, and close to it, there issues from the drop a sheaf 

 of rays which are nearly, if not quite, parallel to each other. These 

 are the so-called " effective rays " of the rainbow.* 



Let me here point to a series of measurements which will illus- 

 trate the gradual augmentation of the deflection just referred to 

 until it reaches its maximum, and its gradual diminution at tho 

 other side of the maximum. The measures correspond to a series 

 of angles of incidence which augment by steps of ten degrees. 



i d 



10° 10° 



20° 19° 36' 



30° 28° 20' 



40° 35° 36' 



50° 40° 40' 



i d 



60° 42° 28' 



70° 39° 48' 



80° 31° 4' 



90° 15° 



The figures in the column i express these angles, while under d we 

 have in each case the accompanying deviation, or the angle enclosed 

 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 colours rise 

 to their maximum, and fall on passing beyond it ; the maximum 



* There is, in fact, a bundle of rays near the maximum, which, when they 

 enter the drop, are converged by refraction almost exactly to the same point at 

 its back. If the convergence were quite exact, then the symmetry of the liquid 

 sphere would cause the rays to quit the drop as they entered it — that is to say, 

 perfectly parallel. But inasmuch 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 colours. (See Engel and 

 Shellbach's diagrams of the rainbow.) 



