15. IF (in fig. 77.) the line B C be perpendicular to A B; then it has been observed above[89], that the centripetal power may be so balanced with the progressive motion of the body, that the body may keep moving round the center A constantly at the same distance; as a body does, when whirled about any point, to which it is tyed by a string. If the centripetal power be too weak to produce this effect, the motion of the body will presently become oblique to the line drawn from itself to the center, after the manner of the first of the two cases, which we have been considering. If the centripetal power be stronger, than what is required to carry the body in a circle, the motion of the body will presently fall in with the second of the cases, we have been considering.
16. IF the centripetal power so change with the change of distance, that the body, after its motion has become oblique to the line drawn from itself to the center, shall again become perpendicular thereto; which we have shewn to be possible in both the cases treated of above; then the body shall in its subsequent motion return again to the distance of A B, and from that distance take a course similar to the former: and thus, if the body move in a space free from all resistance, which has been here all along supposed; it shall continue in a perpetual motion about the center, descending and ascending alternately therefrom. If the body setting out from B (in fig. 78.) in the line B C perpendicular to A B, describe the line B D E, which in D shall be oblique to the line A D, but in E shall again become erect to A E drawn from the body in E to the center A; then from this point E the body shall describe the line E F G altogether like to the line B D E, and at G shall be at the same distance from A, as it was at B. But likewise the line A G shall be erect to the body’s motion. Therefore the body shall proceed to describe from G the line G H I altogether similar to the line G F E, and at I have the same distance from the center, as it had at E; and also have the line A I erect to its motion: so that its following motion must be in the line I K L similar to I H G, and the distance A L equal to A G. Thus the body will go on in a perpetual round without ceasing, alternately inlarging and contracting its distance from the center.