VOL. XXII. 3 FHILOSOPHICAL TRANSACTIONS. 329 



Then let ca be divided in e, so that ce may be to ae, as unity is to the num- 

 ber of reflections which a ray of the sun suffers, proper to produce the proposed 

 rainbow. Then with the diameter ae let the semicircle abe be described, and 

 with centre c and radius cd draw the arch bd, meeting the semicircle abe in 

 the point b. Lastly, drawing the right lines cb, and ab, upon ah produced let 

 fall the perpendicular of, and eb parallel to it. I say the angle cbf will be the 

 angle of incidence, and cab the refracted angle, as were required ; and these 

 will produce the proposed rainbow. 



Demonstration. — Since the triangles acf and aeb are similar, it will be af to 

 BF, so is AC to EC, that is, as the number of reflexions encreased by unity, is to 

 unity, by the construction. Therefore the moment of the angle cbf will be 

 to the moment of the angle cap, in the same proportion, by the foregoing 

 lemma. But the sine of the angle cbf is to the sine of the angle caf, in the 

 ratio of the sides ca and cb, that is, the ratio of the given refraction, also by 

 construction. Therefore the angle of incidence cbf has its corresponding re- 

 fracted angle caf, and their moments are in the ratio proposed ; therefore they 

 are the angles required, a. e. d. 



And now multiplying the refracted angle by the number of reflexions encreased 

 by unity, and from the product subtracting the angle of incidence, we shall 

 have half the distance of the rainbow from the sun, if the number of the re- 

 flexions is even, or from the point opposite to the sun, if odd, as said before. 



Hence by a construction that is neat enough and not inelegant, we may 

 exhibit, by way of synopsis, the incidences of all rainbows in order, in any 

 liquid, the refraction of which is known. For if the assumed line AC be divided 

 in two equal parts at e, in three at e, in four at £, in five at n, and so on ; and 

 with the diameters ae, Ae, ki, An, &c. be described the semicircles abe, Abe, 

 A(3£, AuM, all which are met by the circular arch DBb|3u ; described with centre c 

 and radius CD, (which radius is to ac in the given ratio of refraction,) in the 

 points B, b, (3, u ; I say that drawing the lines ab, Ab, a|3, a-j, they will consti- 

 tute with the line ac the angles cab, CAb, CA(3, cAu, equal to the refracted 

 angles ; and with the rays cb, cb, c(3, Cu, respectively, angles equal to the 

 angles of incidence required. That is, abc, or rather its complement to a 

 semicircle, for the primary rainbow, Abe for the secondary, a,Qb for the third, 

 and Auc for the fourth ; and so on. 



Now if any one is desirous to investigate these angles by an exact calculation, 

 from the same source an analyst will easily discover, that making radius =1, 



and the ratio of refraction as r to s, the sine of incidence will be »/ z — :;— ; but 



VOL. IV. 3 Y 



