Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/285



If a centripetal force tending on all ides to the centre C of a globe (Pl. 19. Fig. 4.) be in all places as the ditance of the place from the centre, and by this force alone acting upon it, the body T ocillate (in the manner above decrid) in the perimeter of the cycloid QRS; I ay that all the ocillations bow unequal oever in themeves will be performed in equal times.

For upon the tangent TW infinitely produced let fall the perpendicular CX and join CT. Becaue the centripetal force with which the body T is impelled towards C is as the ditance CT; let this (by cor. 2. of the laws) be reolved into the parts CX, TX of which CX impelling the body directly from P tretches the thread PT and by the reitance the thread makes to it is totally employed, producing no other effect; but the other part TX, impelling the body tranverely or towards X directly accelerates the motion in the cycloid. Then it is plain that the acceleration of the body, proportional to this accelerating force, will be every moment as the length TX, that is, (becaue CV, WV, and TX, TW proportional to them are given) a the length TW that is (by cor. 1. prop. 49.) as the length of the arc of the cycloid TR. If