Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/199

 and perpendicular on the leer; and the contrary will happen if thoe legs meet in the remotet point L. Whence if the centre of the trajectory is given, the axes will be given; and thoe being given, the foci will be readily found.

But the quares of the axes are one to the other as KH to LH, and thence it is eay to decribe a trajectory given in kind through four given points. For if two of the given points are made the poles C, B, the third will give the moveable angles PCK, PBK; but thoe being given, the circle BGKC may be decribed. Then, becaue the trajectory is given in kind, the ratio of OH to OK, and therefore OH it elf will be given. About the centre O, with the interval OH, decribe another circle. and the right line that touches this circle and paes through the concoure of the legs CK, BK, when the firt legs CP, BP, meet in the fourth given point, will be the ruler MN, by means of which the trajectory may be decribed. Whence alo on the other hand a trapezium given in kind (excepting a few caes that are impoible) may be incribed in a given conic ection.

There are alo other lemma's by the help of which trajectories given in kind may be decribed through given points, and touching given lines. Of uch a fort is this, that if a right line is drawn through any point given by poition. that may cut a given conic ection in two points, and the ditance of the interections is biected, the point of biection will touch another conic ection of the ame kind with the former, and having its axes parallel to the axes of the former. But I haten to things of greater ue.