Proposition XLII. Problem XXIX.
The law of centripetal force being given, it os required to find the motion of a body ſetting out from a given place, with a given velocity, in the direction of a given right line.
Suppoſe the ſame things as in the three preceding propoſitions; and let the body go off from the place L (Pl. 17. Fig. 6.) in the direction of the little line IK, with the ſame velocity as another body, by falling with an uniform centripetal force from the place P, may acquire in D; and let this uniform force be to the force with which the body is at firſt urged in I, as DR to DF. Let the body go on towards k; and about the centre C with the interval Ck, deſcribe the circle ke, meeting the right line PD in e, and let there be erected the lines eg, ev, ew, ordinately applied to the curves BFg, abv, acw. From the given rectangle PDRQ and the given law of centripetal force, by which the firſt body is acted on, the curve line BFg is alſo given, by the conſtruction of prop. 27. and its cor. 1. Then from the given angle CIK is given the proportion of the naſcent lines IK, KN; and thence by the conſtruction of prob. 28. there is given the quantity Q, with the curve lines abv, acw; and therefore, at the end of any time Dbve, there is given both the altitude of the body Ce or Ck, and the area Dcwe, with the ſector
