200 THE MATHEMATICAL PRINCIPLES [BoOK I. IK will be hindered by causing the system of the bodies T and L on one side, and the body S on the other, with proper velocities, to revolve round the common centre of gravity C. With such a motion the body S, because the sum of the motive forces SD X T and SD X L is proportional to the distance OS, tends to the centre C, will describe an ellipsis round the same centre C; and the point D, because the lines CS and CD are proportional, will describe a like ellipsis over against it. But the bodies T and L, at tracted by the motive forces SD X T and SD X L, the first by the first, and the last by the last, equally and in the direction of the parallel lines TI and LK, as was said before, will (by Cor. 5 and 6, of the Laws of Motion) continue to describe their ellipses round the movable centre D, as before. Q.E.I. Let there be added a fourth body V, and, by the like reasoning, it will be demonstrated that this body and the point C will describe ellipses about the common centre of gravity B ; the motions of the bodies T, L, and S round the centres D and C remaining the same as before ; but accelerated. Arid by the same method one may add yet more bodies at pleasure. Q..E.I. ^This would be the case, though the bodies T and L attract each other mutually with accelerative forces either greater or less than those with which they attract the other bodies in proportion to their distance. Let all the mutual accelerative attractions be to each other as the distances multiplied into the attracting bodies ; and from what has gone before it will easily be concluded that all the bodies will describe different ellipses with equal periodical times about their common centre of gravity B, in an immovable plane. Q.E.I. PROPOSITION LXV. THEOREM XXV. Bodies, whose forces decrease in a duplicate ratio of their distances from their centres, 'may move among" themselves in ellipses ; and by radii drawn to the foci may describe areas proportional to the times very nearly. In the last Proposition we demonstrated that case in which the motions will be performed exactly in ellipses. The more distant the law of the forces is from the law in that case, the more will the bodies disturb each other's motions ; neither is it possible that bodies attracting each other mutually according to the law supposed in this Proposition should move exactly in ellipses, unless by keepirg a certain proportion of distances from each other. However, in the following crises the orbits will not much dif fer from ellipses. CASE I. Imagine several lesser bodies to revolve about some very great one at different distances from it, and suppose absolute forces tending to rvery one of the bodies proportional to each. And because (by Cor. 4, ol the I aws) the common centre of gravity of them all is either at rest, 01