176 THE MATHEMATICAL PRINCIPLES [BOOK I. idd the excess — -:-= , and the sum will be the whole force -r-r -\A AA RGG — RFF, .-5 — by which a body may revolve in the same time in the motA. •able ellipsis upk. COR. 3. In the same manner it will be found, that, if the immovable or bit VPK be an ellipsis having its centre in the centre of the forces C} and there be supposed a movable ellipsis -upk, similar, equal, and concentrical to it ; and 2R be the principal latus rectum of that ellipsis, and 2T the latus transversum, or greater axis ; and the angle VCjo be continually to the angle TCP as G to F ; the forces with which bodies may revolve in the imFFA FFA movable and movable ellipsis, in equal times, will be as ^ and -p™~ RGG — RFF + — .-3 respectively. A COR. 4. And universally, if the greatest altitude CV of the body be called T, and the radius of the curvature which the orbit VPK has in Y, that is, the radius of a circle equally curve, be called R, and the centripetal force with which a body may revolve in any immovable trajectory VPK at the place VFF V be called -f-=Trri , and in other places P be indefinitely styled X ; and the altitude CP be called A, and G be taken to F in the given ratio of the angle VCjD to the angle VCP ; the centripetal force with which the same body will perform the same motions in the same time, in the same trajectory •upk revolving with a circular motion, will be as the sum of the forces X -fVRGG — VRFF ~ A* COR. 5. Therefore the motion of a body in an immovable orbit being given, its angular motion round the centre of the forces may be increased or diminished in a given ratio; and thence new immovable orbits may be found in which bodies may revolve with new centripetal forces. COR. 6. Therefore if there be erected the line VP of an indeterminate -p length, perpendicular to the line CV given by po sition, and CP be drawn, and Cp equal to it, mak ing the angle VC/? having a given ratio to the an gle VCP, the force with which a body may revolve in the curve line Vjo/r, which the point p is con tinually describing, will be reciprocally as the cube C of the altitude Cp. For the body P, by its vis in ertia alone, no other force impelling it, will proceed uniformly in the right line VP. Add, then, a force tending to the centre C reciprocally as the cube of the altitude CP or Cp, and (by what was just demonstrated) the