628 PHILOSOPHICAL TRANSACTIONS. [aNNO 1723. 



as the sine of incidence to the sine of refraction : whence mw being equal to 

 3lw, px shall be to pg as the sine of incidence to 3 times the sine of 

 refraction. 



Also, MW being equal to 3lw, the square of mw will be equal to 9 times 

 the square of lw, and the rectangle under vml, or the rectangle under cma, 

 that is, the excess of the square of km above the square of ka, will be equal 

 to 8 times the square of lw; therefore the square of lw, or the square of 

 half Fx, will be to the square of kl, or of ka, as the excess of the square of 

 KM above the square of ka, to 8 times the square of ka, that is, as the excess 

 of the square of the sine of incidence, above the sinie of refraction, to 8 times 

 the square of the sine of refraction. 



Another Determination. — Draw ay parallel to mn, fig. 7, and az parallel to 

 Mv : then is the angle under yaz, equal to that under lmn, which is equal to 

 that under lca ; whence the arches al, yz are equal; but the arches al, vz 

 are likewise equal, because lv, az are parallel, therefore yv being joined, and 

 lF drawn perpendicular to ac, the chord vy shall be the double of lF ; but vA 

 being likewise let fall perpendicular to ac, because mv is the double of ml, 

 vA shall be the double of lF; and therefore vA and vy shall be equal ; whence 

 the point v shall be in a parabola, whose focus is the point y, its axis perpen- 

 dicular to AC, and the latus rectum, belonging to that axis, equal to twice the 

 perpendicular let fall from y upon ac (Vide de la Hire Sect. Conic, lib. 8, 

 prop. 1, 3.) But if Kv be joined, the angle under lkv is equal to twice the 

 complement to a right angle of the angle under klv, which is equal to the 

 angle of incidence, and exceeds the refracted angle by the angle under akl. 



The determinations of these two propositions, have relation to the first and 

 second rainbow; those of the first proposition respecting the interior, and those 

 of the second the exterior. The first determinations of these two propositions 

 assign the angles, under which each rainbow will appear in any given refracting 

 power of the transparent substance, by which they are produced; the latter 

 determinations of these propositions teach how to find the refracting power of 

 the substance, from the angles under which the rainbows appear ; the angle 

 under cmg, in the determinations of the first proposition, being half the angle 

 which measures the distance of the interior bow from the point opposite to the 

 sun; and in tiie determinations of the second proposition, the angle under cmn 

 is half the complement to a right angle of half the angle that measures the 

 distance of the exterior bow, from the point opposite to the sun. But whereas 

 these latter determinations require solid geometry, it may not be amiss here to 

 show how they may be reduced to calculation, seeing the observation of these 

 angles, as Dr. Halley has already remarked, (Phil. Trans. N° 267,^ aflx)rds no 



