720 PHILOSOPHICAL TRANSACTIONS. [aNNO ]7()3 



between the two series, thus ; a, b, c, z, c, hy a; and that the rays, in their 

 passage through z, are parallel to one another. 



But to give the rays this parallelism in their passage through z, and to explain 

 the several phenomena of refracted light, we shall need the following 



Lemma, a Problem. Given, in fig. 2, dcb the difterence of two angles acd, 

 acb, and the ratio of di the sine of the greater to bh the sine of the lesser 

 being also given, to find the angles. — For dp, the sine of the given difl^erence, 

 write s, and for its cosine cf write c; for the lesser sine bh, the letter z, and let the 

 given ratio of di to bh, be that of m to w, the radius cb being unity. Then, 

 having drawn fg perpendicular to di ; from the similar triangles in this figure, 

 we shall have cb : ch : : dp : dg, or 1 : i/ 1 — z^ : : * : dg = aV 1 — z^ ; and 

 cb : BH : : CF : Gi, or 1 : z : t c ; gi = cz. But (by hypoth.) di : bh : : wi : n ; 

 that is DG + gi, or s^\ — z^ + cz : z : : ot : rz ; which gives i/ \ — z^ : z, or 



ch : BH, or 1 : tang, acb : -. m — nc '. ns \ that is, tane;. acb == - — . In words 



*=" ^ m — uc 



— multiply the sine of the given difference by the least term of the given ratio 

 for a dividend : from the greater term subtract the product of the cosine of the 

 difference and the lesser term for a divisor ; and the quotient shall be the tangent 

 of the less angle acb. — Or, if you prefer a geometrical construction ; in the 

 semidiameter cb produced take cm to cb as di to bh ; and in the tangent to the 

 circle at b, make BLto bc, as dp to pm, and bcl shall be the less angle sought. 

 Or you need only join dm and draw the semidiameter ca parallel to it. 



But before we apply this solution, it may be proper to give a table of the re- 

 fractive powers of glass, water and spirit of wine, whether contiguous to the air, 

 or perhaps the fluids contiguous to glass : these being the substances in which 

 experiments may be most conveniently made : and it is also necessary to know 

 the limitations that arise from those several powers. 



1 . When light passes from air into glass, and the angle of incidence is near 

 go", whose sine is unity; 

 The sine of the refraction of the red rays = ff is .6493508 = sin. 40° 29' 33.6' 



And of the violet w = .6410256 = sin 39 52 6 



And their difference O 37 27.6, 



is the greatest angle at which the violet and red rays can diverge in the refraction 

 from air into glass, wanting very little of 374'- And when an unrefracted pencil 

 passes from glass into air, as soon as the angle of incidence exceeds 39° 52' &, 

 the violet rays will begin to be reflected; and when the incidence exceeds 40** 

 29' 33. "6 the rays will be totally reflected. 

 2. From Air into Water. Thesineofrefractionoftheredis.7517905 = j.48° 44' 44* 



Of the violet .7454080 = ^, 48 J J 39 



And the greatest divergence 33 5, 



the angle of beginning reflection from water into air being 48° 11- 39". 



