may be equal to ST; and YH will be equal to the principal axe. Join SP, HP, and SP will be the difference between HP and the principal axe. After this manner if more tangents TR are given, or more points P, we ſhall always determine as many lines YH or PH, drawn from the ſaid points T or P, to the focus H, which either ſhall be equal to the axes, or differ from the axes by given lengths SP; and therefore which ſhall either be equal among themſelves, or ſhall have given differences; from whence (by the preceding lemma) that other focus H is But having the foci and the length of the axe (which is either TH; or, if the trajectory be an ellipſis, PH + SP, or PH - SP if it be an hyperbola) the trajectory is given. Q. E. I.
Scholium.
When the trajectory is an hyperbola, I do not comprehend its conjugate by hyperbola under the name of this trajectory. For a body going on with a continued motion can never paſs out of one hyperbola into its conjugate hyperbola.
The caſe when three points are given is more readily ſolved thus. Let B, C, D (PL 8. Fig. 3.) be the given points. join BC, CD, and produce them to E, F; ſ as EB may be to EC, as SB to SC; and FC to FD, as SC to SD. On EF drawn and produced let fall the perpendiculars SG, BH and in GS produced indefinitely take GA to AS, and Ga to aS, as HB is to BS; then A will be the vertex, and Aa the principal axe of the trajectory: Which, according as GA is greater than, equal to, or leſs than AS, will be either an ellipſis, a parabola
