Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/289



To define the velocities of the pendulums in the everal places, and the times in which both the entire ocillation, and the everal parts of them are performed.

About any centre G (Pl. 20. Fig. 1.) with the interval GH equal to the arc of the cycloid RS, decribe a emi-circle HKM biected by the semi-diameter GK. And if a centripetal force proportional to the ditance of the places from the centre tend to the centre G, and it be in the perimeter HIK equal to the centripetal force in the perimeter of the globe QOS tending towards its centre, and at the ame time that the pendulum T is let fall from the highet place S, a body as L is let fall from H to G; then becaue the forces which act upon the bodies are equal at the beginning, and always proportional to the paces to be decribed TR, LG, and therefore if TR and LG are equal, are alo equal in the places T and L, it is plain that thoe bodies decribe at the beginning equal paces ST, HL, and therefore are till acted upon equally, and continue to decribe equal paces. Therefore by prop. 38; the time in which the body decribes the arc ST is to the time of one ocillation, as the arc HI the time in which the