Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/259

 pC. Thee are the firt ratio's of the nacent lines; and hence $$\textstyle \frac {mk \times ms}{mt}$$, that is, the nacent lineola mn, and the difference of the forces proportional thereto, are reciprocally as the cube of the altitude pC. Q. E. D.

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 ame time that the body P in an immovable orb decribes the arc PK, as the nacent line mn to the vered ine of the nacent arc RK, that is as $$\textstyle \frac {mk \times ms}{mt}$$ to $$\textstyle \frac {rk^2}{2kC}$$, or as mk x ms to the quare of rk; that is, if we take given quantities F and G in the ame 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 ditance CP or Cp, there be decribed a circular ector equal to the whole area VPC, which the body revolving in an immovable orbit, has by a radius drawn to the centre decribed 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 decribe that ector in the ame time as the area VPC is decribed, as CG - FF to FP. For that ector and the area pCk are to one another as the times in which they are decribed.