Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/181

 line PI, cuts the trajectory in any points X and Y; the point A (by the properties of the conic ections) will come to be o placed, that $$\scriptstyle HA^2$$ will become to $$\scriptstyle AI^2$$ 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 decribed as in the firft cae. 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 decribed.

To decribe a trajectory that hall pas through three given points, and touch two right lines given by poition. Pl. 10. Fig. 4.

Suppoe HI, KL to be the given tangents, and B, C, D, the given points. Through any two of thoe points as B, D, draw the in indefinite right line BD meeting the tangents in the points H, K. Then likewie through any other two of thee 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, o 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 pleaure, either within