Mr. J. J. Walker on the Ms seen in Wate7\ 441 



responding line of the primary rainbow lie on a similar cone 

 similarly placed, and whose vertex is at the eye, the vertical 

 distance between corresponding drops for the rainbow and the 

 horizontal iris is double the elevation of the spectator's eye above 

 the surface of the water. Also, since the cones of differently 

 coloured pencils are not similar, those of violet rays in the case 

 of the former bow may, the rain-drops being sufficiently distant, 

 intersect those of red rays forming the horizontal iris in curves 

 of double curvature ; and which will therefore be the loci of 

 drops transmitting one pencil of parallel pencils directly to the 

 eye to form a portion of the primary rainbow, and a differently- 

 tinted pencil, which, after reflexion at the surface of the water, 

 shall also reach the eye and contribute to form the horizontal iris. 



Similarly, on account of the sun's apparent diameter, the 

 height of the observer's eye from the surface of the water not 

 being great, and the rain-drops sufficiently distant, the same 

 drop may transmit pencils of parallel rays of the same tint, one 

 directly, the other by horizontal reflexion ; that forming part of 

 the rainbow corresponding to the more elevated points of the 

 sun's disc, the other forming part of the horizontal bow corre- 

 sponding to the points having a less altitude. 



Suppose the most distant drops of a falling shower to lie in a 

 vertical plane perpendicular to that through the sun's centre and 

 eye of the spectator, situated at a distance 8 from the spectator's 

 eye ; then if, h being given, the sun's altitude be such that 



A cot (/3 — a) > 3, 

 no horizontal bow will be visible, though the primary rainbow 

 be so ; and, the sun's altitude being given, the same absence of 

 the horizontal bow may occur from the increase of the observer's 

 elevation above the siarface of the water. 



It is evident that the similarly tinted parts of the two bows 

 will not coincide at the horizontal surface of the water if h bear 

 a sufficiently great ratio to h ; and that at a considerable eleva- 

 tion of the spectator's eye, the ends of the horizontal bow may 

 appear to lie wholly within the ends at which ''.... bibit ingens 

 Arcus ...."; thus, even to the most unscientific observer, dis- 

 pelling the illusion of its being the " reflexion of the rainbow " 

 in the water. 



The arrangement of the colours is, of course, the same as in 

 the case of the primary rainbow. If the sun be just on the 

 horizon, the angle between the asymptotes will be equal to 2p ; 

 and the bow, therefore, nearly an equilateral hyperbola. 



In the degree of intensity of its colours the primary horizontal 

 bow would, cceteris paribus, coincide with the secondary rainbow. 

 I have found, by such inquiries as I have been able to make 



Phil, Mag, S. 4. Vol, 5. No. 34. June 1853. ^ G 



