SEC. X.] OF NATURAL PHILOSOPHY. 193 the superficies, and therefore with which it is again repelled by the super ficies towards M ; PTF a right line parallel to the axis and passing through the body, and OF, IH right lines let fall perpendicularly from the points G and I upon that parallel PHTF. I say, now. that the area AGP, de scribed by the radius OP from the beginning of the motion, is proportional to the time. For the force TG (by Cor. 2, of the Laws of Motion) is re solved into the forces TF, FG ; and the force TI into the forces TH, HI ; but the forces TF, TH, acting in the direction of the line PF perpendicular to the plane AOP, introduce no change in the motion of the body but in a di rection perpendicular to that plane. Therefore its motion, so far as it has the same direction with the position of the plane, that is, the motion of the point P, by which the projection AP of the trajectory is described in that plane, is the same as if the forces TF, TH were taken away, and the body wei e acted on by the forces FG, HI alone ; that is, the same as ,f the body were to describe in the plane AOP the curve AP by means of a centripetal force tending to the centre O, and equal to the sum of the forces FG and HI. But with such a force as that (by Prop. 1) the area AOP will be de scribed proportional to the time. Q.E.D. COR. By the same reasoning, if a body, acted on by forces tending to two or more centres in any the same right line CO, should describe in a free space any curve line ST, the area AOP would be always proportional to the time. PROPOSITION LVI. PROBLEM XXXVII. Granting the quadratures of curvilinear figures, and supposing that there are given both the law of centripetal force tending to a given cen tre, and the curve superficies whose axis passes through that centre ; it is required to find the trajectory which a body will describe in that superficies, when going off from a given place with a given velocity, and in a given direction in that superficies. The last construction remaining, let the body T go from the given place S, in the di rection of a line given by position, and turn into the trajectory sought STR, whose ortho graphic projection in the plane BDO is AP. And from the given velocity of the body in the altitude SC, its velocity in any other al titude TC will be also given. With that velocity, in a given moment of time, let the body describe the particle Tt of its trajectory, and let P/? be the projection of that particle described in the plane AOP. Join Op, and a little circle being described upon the curve superficies about the centre T 13