Finding the orbits from the focus given

PROPOSITION 51 THEOREM 18

If a centripetal force tending on all sides to the centre C of a globe, be in all places as the distance of the place from the centre, and by this force

oscillate (in the manner above de alone acting upon it, the body the in the / say, that all the oscil perimeter of scribed] cycloid how in soever will be performed in equal lations, tfiemselves, unequal QRS ; times. W T infinitely produced let fall the perpendicular CT. and Because the centripetal force with which the body T CX, join towards C is as the distance CT, let this is impelled (by Cor. 2, of the For upon the tangent I ,aws) be resolved into body directly from P the parts CX, TX, of stretches the thread PT, which CX impelling the and by the resistance the rhread makes to it is totally employed, producing no other effect but the 3ther part TX, impelling the body transversely or towards X, directly Then it is plain that the accelera accelerates the motion in the cycloid. ; tion of the body, proportional to this accelerating force, will bo every

moment WV, as and the length TX, TX, [BOOK that is (because 1 CV
TW proportional to them are given), TW, as the length that is (by Cor. 1, Prop. XLIX) as the length of the arc of the cycloid TR. If there fore two pendulums APT, Apt, be drawn unequally aside from the perpendicular AR, and let fall together, their accelerations will be always as the arcs to be de tR. But the parts described at the are as the accelerations, thai of the motion beginning are to be described at the be that the wholes as is, scribed TR, ginning, and therefore the parts which remain to be described, and the subsequent accelerations proportional to those parts, are also as the wholes, and so on. Therefore the accelerations, and consequently the velocities generated, and the parts described with those velocities, and the parts to be described, are always as the wholes and therefore the parts to be described preserving a given ratio to each other will vanish together, that is, the two bodies oscillating will arrive together at the perpendicular AR. ; And since on the other hand the ascent of the pendulums from the lowest place the same cycloidal arcs with a retrograde motion, is retarded in through the several places they pass through by the same forces by which their de R scent was accelerated : it is plain that the velocities of their ascent and de and consequently performed in equal and two the since and, therefore, parts of the cycloid lying on either side of the perpendicular are similar and equal, the two pendu lums will perform as well the wholes as the halves of their oscillations in scent through the times same arcs are equal, RQ RS ; the same times.

Corollary

The force with which the body T is accelerated or retarded in any place T of the cycloid, is to the whole weight of the same body in the highest place S or Q, as the arc of the cycloid TR is to the arc SR or QR