Section 8. Determination of the orbit under any law of centripetal force. (40) The velocity at a given distance is always the same both iu an orbit and a descent, if it be the same at any one distance in both. 2 Cor. (41) Granting the quadrature of curves, to find the orbit and the time of describing an arc, under any law of force. 3 Cor. (42) The same, the initial velocity and direction being given.
Section 9. On the motion of bodies in moveable orbits, and on the motion of the apsides. (43) How to make a body revolve equiareally in both a moving orbit and in the same fixed. (44) The difference of forces in the two cases is as the inverse cube of the distance. 6 Cor. mostly exhibiting the conclusion in algebraical form. (45) To find the motion of the apsides in orbits nearly circular. 3 Examples ; 2 Cur.
Section 10. On the motion of bodies in given surfaces, and on pen dulous motions. (46) Given a plane, and a centre of force external to it, to find the motion of a point parallel to that plane, the law of force being any whatever. (47) The force in the last being as the distance, the orbit parallel to any plane must be an ellipse, and in all such ellipses the time of revolution is the same, and the same as that of a double ascent and descent. Schol. (48) and (49) Rectification of the epicycloid and hypocycloid. 3 Cor. (50) Way to make a body oscil late in a given hypocycloid. Cor. (51) If the force tending to the centre of the fixed circle in such an oscillation be as the distance, the times of all such oscillations are equal. Cor. (52) Determination of the velocity and time at any point of such an oscillation. 2 Cor.; the second being an application to the common cycloid. (53) On a given curve, to find the law of force which gives isochronous oscillations. 2 Cor. (54) A body moving on a rigid curve, under a given law of centripetal force, to find the time of its oscillations. (55) If a body move on a surface of revolution, the centre of force being in the axis, equal areas are described in equal times on a plane perpendicular to the axis. Cor. (56) To find the curve described in the last case.
Section 11. On the motion of bodies centripetally attracted to each other. (57) Two bodies, mutually attracting, describe similar figures about each other and about their common centre of gravity. (58) And with the same forces, the same curve may be described by either about the other at rest. 3 Cor. (59) Relation of the periodic times about the centre of gravity, and of one body about the other at rest. (60) In the same two cases, relation of the axes* of the ellipses described.
(61) And for any law of force, the bodies move round their centre of gravity as if a third body were placed in that centre, attracting with the same law of force. (62) Determination of the descent towards each other of two mutually attracting bodies. (63) Determination of the orbits of two such bodies, with given initial velocity and direction. (64) The force being as the distance, determination of the relative motions of several bodies. (65) The force being inversely as the square of the distance, and there being several bodies, one may move round another in an ellipse nearly, and describe areas nearly proportional to the times. 3 Cor. (66) The celebrated proposition of the three bodies, showing the diminution of the disturbance by the third body attracting both the others. (In the corollaries following, let the earth and moon, for distinctness sake, be the two bodies, and the sun the disturbing body : but let it be remembered that Newton does not mention the name of any planet nor hint at any application.) Cor. 1, If the earth had more satellites, the same proposition would apply to one as dis turbed by the rest. Cor. 2 and 3, The moon moves quickest, emteris paribus, in conjunction and opposition, and slowest in quadraturea. Cor. 4, The moon's orbit is more curved in quadratures than in syzygies. Cor 5, Hence, excentricity being excluded, the moon is i farther from the earth in quadratures than in syzygies. Cor. 6, Explanation of the effect of the variation of the sun's distance on the moon's period. Cor 7, The moon's apsides progress and regress, but the former more than the latter. Cor. 8, Effect of the position of the apsides with respect to the sun. Cor. 9, Effect on the exeentricity of the moon's orbit. Cor. 10 and 11, Effect on the inclination and place of the nodes. Cor. 12, Disturbance rather greater in conjunction than in opposition. Cor. 13, The same species of effect produced whether the disturbing body is the greater or the less of the three. Cor. 14, 15, 16, 17, On the dependence of the disturbing forces on the distance of the disturbing body. Cor. 18, 19, 20, 21, 22, Explanation of pre cession of equinoxes and tides. (67) The disturbing body describes areas more nearly proportional to the times about the centre of gravity of the other two bodies than about either of them, and an orbit more nearly elliptic. (68) And the more so on account of its attracting the other bodies. Cor. (69) The attracting forces of bodies are, ceteris paribus, as their masses. 3 Cor. and Scholium.