Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/298

 any the ame right line CO, hould decribe in a free pace any curve line ST; the area AOP would always proportional to the time.

Granting the quadratures of curvilinear figures and uppoing that there are given both the law of centripetal force tending to a given centre, and the curve upeficies who axis paes through that centre; it if required to find the trajectory which a body will decribe in that uperificies, when going of from a given place with a given velocity, and in a given direction in that uperficies.

The lat contruction remaining, let the body T go from the given place S (Pl. 20. Fig. 6.) in the direction of a line given by poition, and turn into the trajectory ought STR whoe orthographic projection in the plane BLO is AP. And from the given velocity of the body in the altitude SC, its velocity in any other altitude TC will be alo given. With that velocity in a given moment of time let the body decribe the particle Tt of its trajectory, and let Pp be the projection of that particle decribed in the plane AOP. Join Op, and a little circle being decribed upon the curve uperficies about the centre T with the interval Tt, let the projection of that little circle in the plane