VOL. XXXII.] PHILOSOPHICAL TRANSACTIONS. 627 



the point g in the line gh, that ep and hg being produced, till they meet in i, 

 the angle under eih shall be given. 



Let K be the centre of the circle, and kp, kg be joined; let the semi- 

 diameter LK be parallel to the refracted ray pg; and, mk being taken to the 

 semidiameter of the circle in the ratio of the sine of incidence to the sine of 

 refraction, let lm be joined ; and lastly, make the angle under kmn equal to 

 half the given angle under eih. Then, if pg be produced to o, po shall be 

 to Ko, as the sine of the angle of incidence to the sine of the refracted angle, 

 that is as mk to kl; so that kl being parallel to po, and the angle under mkl 

 equal to that under pok, the angle under mlk shall be equal to that under pko, 

 and the angle under kml equal to that under kpo equal to that under fgk, or 

 half that under pgh ; whence the angle under kmn being equal to half the 

 angle under pih, the residuary angle under nml will be equal to half the angle 

 under ipg or to half that under mkl. Therefore lc being drawn, the angle 

 under lmn will be equal to that under mcl ; and in the last place, if mc be 

 divided into two equal parts in p, and pqr be drawn parallel to cl, the angle 

 under qmr will be equal to that under rpm, and the triangles qmr, mpr similar, 

 so that the rectangle under prgi shall be equal to the square of mr. Whence 

 RL being equal to mr, the point l shall be in an equilateral hyperbola, touching 

 the line mn in the point m, and having the point p for its centre. (Apoll. 

 Conic, lib. 1, prop. 37, compared with lib. 7> prop. 23.) But this hyperbola 

 is given in position, and consequently the point l, the angle under mlk, and 

 the equal angle under ckp will be given, and therefore the ray ep is given in 

 position. 



Determination. — Let the hyperbola touch the circle in the point l, fig. 6, 

 and let their common tangent be ls; draw lt parallel to mn, so as to be ordi- 

 nately applied in the hyperbola to the diameter cm. Whence ls touching the 

 hyperbola in l, pt will be to tl as tl to ts, (Apoll. Conic, lib. 1 , prop. 37, 

 compared with lib. 7, prop. 23,) and the angle under tsl equal to that under 

 TLP, but as the angle under scl is equal to that under nml, the same is equal 

 to the angle under tlm; therefore the angle under slc is equal to the angle 

 under mlp. Further, ml being produced to v, and vc joined, the angle under 

 Lvc is equal to that under slc, because ls touches the circle in l; hence the 

 angles under lvc and under mlp are equal, lp, vc are parallel, and mp being 

 equal to pc, ml is equal to lv ; and kw being let fall perpendicular to lv, mw 

 is equal to 3lw. But now if the incident ray ep be produced to x, the angle 

 under mlk being equal to that under ckp, or to that under epk, px shall be 

 equal to lv, equal to 2lw ; and the angle under kml being equal to that under 

 kpg ; since kw is perpendicular to mw, pg shall be to 2mw as mk to kp, or 



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