pC. Theſe are the firſt ratio's of the naſcent lines; and hence , that is, the naſcent lineola mn, and the difference of the forces proportional thereto, are reciprocally as the cube of the altitude pC. Q. E. D.
Cor. 1. Hence the difference of the forces in the places P and p, or K and k, is to the force with which a body may revolve with a circular motion from R to K, in the ſame time that the body P in an immovable orb deſcribes the arc PK, as the naſcent line mn to the verſed ſine of the naſcent arc RK, that is as to , or as mk x ms to the ſquare of rk; that is, if we take given quantities F and G in the ſame ratio to one another as the angle VCP bears to the angle VCp, as GG - FF to FP. And there. fore if from the centre C with any diſtance CP or Cp, there be deſcribed a circular ſector equal to the whole area VPC, which the body revolving in an immovable orbit, has by a radius drawn to the centre deſcribed in any certain time; the difference of the forces, with which the body P revolves in an immovable orbit and the body p in a moveable orbit, will be to the centripetal force, with which another body by a radius drawn to the centre can uniformly deſcribe that ſector in the ſame time as the area VPC is deſcribed, as CG - FF to FP. For that ſector and the area pCk are to one another as the times in which they are deſcribed.
