Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/151

 RS, TV are equal, therefore the right lines SR, VR, as well as the angles TRS, TRV, will be alo equal. Whence the point R will be in the conic ection, and the perpendicular TR will touch the ame: and the contrary. Q. E. D.

From a focus and the principal axes given, to decribe elliptic and hyperbolic trajectories, which hall pas through given points, and touch right lines given by poition. Pl. 7. Fig. 2.



Let S be the common focus of the figures; AB the length of the principal axis of any trajectory; P a point through which the trajectory hould pas; and TR a right line which it hould touch. About the centre P, with the interval AB - SP, if the orbit is an ellipis, or AB + SP if the orbit is an hyperbola, decribe the circle HG. On the tangent TR let fall the perpendicular ST and produce the ame to V, o that TV may be equal to ST; and about V as a centre with the interval AB decribe the circle FH. In this manner whether two points P, p, are given, or two tangents TR, tr, or a point P and a tangent TR, we are to decribe two circles. Let H be their common intersection, and from the foci S, H with the given axis decribe the trajectory. I ay the thing is done. For (becaue PH + SP in the ellipis, and PH - SP in the hyperbola is equal to the axis) the decribed trajectory will pas through the point P, and (by the preceding lemma) will touch the right line TR. And by the ame argument it