Q6 PHILOSOPHICAL TRANSACTIONS. [aNNO 1763. 



CDK drawn through d, the square of bc will be to the rectangle under bk, cl, 

 as the rectangle under dp, de, to that under cf, be. Hence, if the ratio of 

 the rectangle under dp, de to the square of a line drawn from d on bc in a 

 given angle, be given; the square of this line being in a given ratio to the rect- 

 angle under cp, be, the ratio of the rectangle under bk, cl to the square of bc, 

 will be given ; whence a conic section passing through d will in all cases be 

 given. 



In the first j)lace let the point d be within the angle bac, fig. 12. Then if 

 bc be bisected by the line am, this will be a diameter to the conic section, which 

 shall touch ba, ac in the points b, c, and bc will be ordinate:y applied to that 

 diameter ; the vertex of this diameter being n, the given ratio of the rectangle 

 under bk, cl to the square of bc, will be compounded of the ratio of the square 

 of MN to the square of na, and of the ratio of the rectangle under bac to the 

 4th part of the square of bc ; and thus the line am will be divided in n in a 

 o-iven ratio, and the point n, one vertex of the diameter, to which bc is ordi- 

 nately applied, will be given. — If an be equal to nm, the point n will be the 

 only vertex of this diameter, and the section will be a parabola. — Otherwise by 

 taking the point o in am extended, so that the ratio of ao to om be the same 

 with that of an to nm, the point o will be the other vertex of the diameter. 

 And here, if the ratio of an to nm be that of a greater to a less, the point o 

 will fall beyond m from a within the angle bac, the conic section being an 

 ellipsis. But if the ratio of an to nm be that of a less to a greater, the point 

 o will fall on the other side of a, and the section will be an hyperbola, fig. 13. 

 And in this case if the opposite section be drawn, that also will be the locus of 

 the point d within the angle vertical to the angle bac. 



In the last place, if d be in either of the collateral angles, am drawn as be- 

 fore will contain a secondary diameter in opposite sections, one of which shall 

 touch ba in b, and the other ca in c, fig. 14. Then if one of these sections 

 pass through d, the sections will be given. For here paq being drawn through 

 A parallel to bc, the given ratio of the rectangle under cl, bk to the square of 

 BC will be the same with that of the given rectangle under bac to the square of 

 AP : therefore ap is given, and thence the sections. For let rs be the secondary 

 diameter, to which bc is ordinately applied, and t the centre of the opposite 

 sections. Then the square of bm will be to the rectangle under amt, as the 

 square of the transverse diameter conjugate to the secondary diameter rs, to the 

 square of this secondary diameter ; and if a line were drawn from m to p, this 

 would touch the hyperbola bp in p, Apoll. conic. 1. 2, prop. 40. and the square 

 of AP will be to the rectangle under mat in the same ratio ; therefore the given 

 ratio of the square of mb to the square of ap, will be that of the rectangle 

 under amt to the rectangle under mat, or the ratio of mt to at ; consequently 



