Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/153



About a given focus to decribe any trajectory given in pecie, which hall pas thro' given points and touch right lines given by poition.

About the focus (Pl. 7. Fig. 4.) it is required to decribe trajectory ABC, paing thro' two points B, C. Becaue the trajectory is given in pecie, the ratio of the principal axe to the ditance of the foci will be given. In that ratio take KB to BS and LC to CS. About the centres B, C, with the intervals BK. CL decribe two circles, and on the right line KL, that touches the ame in K and L, let fall the perpendicular SG; which cut in A and a, o that GA may be to AS, and Ga to aS, as KB to BS; and with the axe Aa, and vertices A, a, decribe a trajectory. I ay the thing is done. For let H be the other focus of the decribed figure, and eeing GA is to AS as Ga to aS, then by diviion we hall have Ga - GA or Aa to aS - AS or SH in the ame ratio, and therefore in the ratio which the principal axe of the figure to be decribed has to the ditance of its foci; and therefore the decribed figure is of the ame pecies with the figure which was to be decribed. And ince KB to BS, and LC to CS are in the ame ratio, this figure will pas thro' the points B, C, as is manifet from the conic ections

About the focus (Pl. 7. Fig. 5.) it is required to decribe a trajectory, which hall omewhere touch two right lines TR, tr. From the focus On thoe tangents let fall the perpendiculars ST, St,