line PI, cuts the trajectory in any points X and Y; the point A (by the properties of the conic ſections) will come to be ſo placed, that will become to in a ratio that is compounded out of the ratio of the rectangle XHY to the rectangle BHD, or of the rectangle CGP to the rectangle DGB; and the ratio of the rectangle BHD to the rectangle PIC. But after the point of contact A is found, the trajectory will be deſcribed as in the firft caſe. Q. E. F. But the point A may be taken either between or without the points H and I; upon which account a twofold trajectory may be deſcribed.
Proposition XXIV. Problem XVI.
To deſcribe a trajectory that ſhall paſs through three given points, and touch two right lines given by poſition. Pl. 10. Fig. 4.
Suppoſe HI, KL to be the given tangents, and B, C, D, the given points. Through any two of thoſe points as B, D, draw the in indefinite right line BD meeting the tangents in the points H, K. Then likewiſe through any other two of theſe points as C, D, draw the indefinite right line CD, meeting the tangents in the points I, L. Cut the lines drawn in R and S, ſo that HR may be to KR, as the mean proportional between BH and HD is to the mean proportional between BK and KD; and IS to LS, as the mean proportional between CI and ID is to the mean proportional between CL and LD, But you may cut, at pleaſure, either within
