626 PHILOSOPHICAL TRANSACTIONS. [aNNO J 723. 



angle of incidence to the sine of the refracted angle, and the ratio of ih to 

 IG is given, as also the ratio of ik to ig. Therefore ik being perpendicular 

 to AC, the point i is in a conic section given in position, whose axis is perpen- 

 dicular to AC, and one of its foci is the point g. (See Papp, 1. 7. prop. 238. 

 Milnes Conic, part. 4. prop. 9.) Consequently the points i and p are given, 

 and lastly the ray ep given in position. 



Determination. It is evident, that this conic section may either cut the 

 circle in two points, touch it in one point, or fall wholly without it. Therefore 

 let the section touch the circle in the point i, fig. 3, and let il touch both 

 the section and the circle in the same point i. Then gl being joined, the 

 angle under igl, on account of the conic section, is a right one ; so that pgl 

 is one continued right line, and ip is to il as fg to gi; also, m being the 

 centre of the circle, mi to il, or ph to hi, as fg to 2gi, because mi is to if 

 as GI to 2gi. Hence by permutation ph is to fg as hi to 2gi ; that is, as the 

 sine of the angle of incidence to twice the sine of the refracted angle. 



Further, fh being to hi as fg to 2gi, the square of ph will be to the 

 square of hi, as the square of fg to 4 times the square of gi. Therefore, by 

 composition, as the square of ph to the square of pi or of ac, so is the square 

 of fg to the square of pi together with 3 times the square of gi, and so like- 

 wise is the excess of the square of pg above the square of ph, which equals 

 the excess of the square of ih above the square of ig, to 3 times the square of 

 gi ; for as one antecedent to one consequent, so is the difference of the ante- 

 cedents to the difference of the consequents. Hence in the last place, the 

 square of half fh will be to' the square of am, as the excess of the square of 

 ih above the square of ig to 3 times the square of ig, or as the excess of the 

 square of the sine of incidence above the square of the sine of refraction, to 

 3 times the square of the sine of refraction. 



Another Determination. — Draw the diameter go, fig. 4, and the tangent op, 

 meeting gp produced in a: then the angle under ifg is equal to the angle under 

 OGP, the angle under pil equal to that under gog, both being right, and pi is 

 equal to go; whence the triangles goq, pil are similar and equal; so that gq 

 is equal to pl, and the point p in an hyperbola passing through g, whose 

 asymptotes are ac and op. (Apoll. Conic. 1. 2, prop. 8.) 



Pbop. 2. A refracting Circle and its refracting Power being given, the 

 Ray is given in Position, which passing Parallel to a given Diameter of the 

 Circle, after its Refraction, is so reflected from the farther Surface of the 

 Circle, as to be inclined to its incident Course in a given Angle. — Let abcd, 

 fig. 5, be the given circle; let ac be the given diameter, ep the incident ray 

 parallel to it, which, being refracted into the line fg, shall be so reflected from 



