Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/196

 132

Under the preceding propoitions are comprehended thoe problems wherein either the centres or aymptotes of the trajectories are given. For when points and tangents and the centre are given, as many other points and as many other tangents are given at an equal ditance on the other ide of the centre. And an aymptote is to be conidered as a tangent, and its infinitely remote extremity (if we may ay o) is a point of contact. Conceive the point of contact of any tangent removed in infinitum, and the tangent will degenerate into an aymptote, and the contructions of the preceding problem will be changed into the contructions of thoe problems wherein the aymptote is given.

After the trajectory is decribed, we may find its axes and foci in this manner. In the contruction and figure of lem. 21. (Pl. 12. Fig. 1.) let thoe legs BP, CP, of the moveable angles PBN, PCN, by the concoure of which the trajectory was decribed, be made parallel one to the other; and retaining that poition, let them revolve about their poles in that figure. In the mean while let the other legs CN, BN of thoe angles, by their concoure K or k, decribe the circle BKGC. Let O be the centre of this circle; and from this centre upon the ruler MN, wherein thoe legs CN, BN did concur while the trajectory was decribed, let fall the perpendicular OH meeting the circle in K and L. And when thoe other legs CK, BK meet in the point K that is nearet to the ruler, the firt legs CP, BP will be parallel to the greater axis and