If a body moves in the semi-circumference PQA; it is proposed to find the law of the centripetal force tending to a point S, so remote, that all the lines PS, RS drawn thereto, may be taken for parallels. From C, the centre of the semi-circle, let the semi-diameter CA be drawn, cutting the parallels at right angles in M and N, and join CP. Because of the similar triangles CPM, PZT, and RZQ, we shall have to as to ; and, from the nature of the circle, is equal to the rectangle , or, the points P, Q coinciding, to the rectangle . Therefore is to as to ; and , and . And therefore (by Corol. 1 and 5, Prop. VI.), the centripetal force is reciprocally as ; that is , reciprocally as . Q.E.I. And the same thing is likewise easily inferred from the preceding Proposition. SCHOLIUM. And by a like reasoning, a body will be moved in an ellipsis, or even in an hyperbola, or parabola, by a centripetal force which is reciprocally as the cube of the ordinate directed to an infinitely remote centre of force. PROPOSITION IX. PROBLEM IV. If a body revolves in a spiral PQS, cutting all the radii SP, SQ, &c., in a given angle; it is proposed to find the law of the centripetal force tending to the centre of that spiral. Suppose the indefinitely small angle PSQ to be given; because, then, all the angles are given, the figure SPRQT will be given in specie. Therefore the ratio is also given, and is as QT, that is (because the figure is given in specie), as SP. But if the angle PSQ is any way changed, the right line QR, subtending the angle of contact QPR[Pg 114] (by Lemma XI) will be changed in the duplicate ratio of PR or QT. Therefore the ratio remains the same as before, that is, as SP. And is as , and therefore (by Corol. 1 and 5, Prop. VI) the centripetal force is reciprocally as the cube of the distance SP. Q.E.I. The same otherwise. The perpendicular SY let fall upon the tangent, and the chord PV of the circle concentrically cutting the spiral, are in given ratios to the height SP; and therefore is as , that is (by Corol. 3 and 5, Prop. VI) reciprocally as the centripetal force. LEMMA XII. All parallelograms circumscribed about any conjugate diameters of a given ellipsis or hyperbola are equal among themselves. This is demonstrated by the writers on the conic sections. PROPOSITION X. PROBLEM V. If a body revolves in an ellipsis; it is proposed to find the law of the centripetal force tending to the centre of the ellipsis.