Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/261

 ame difference in every altitude A will be as $$\textstyle \frac {RGG - RFF}{A^3}$$. Therefore to the force $$\textstyle \frac {FF}{AA}$$, by which the body may revolve in an immovable ellipis VPK, add the exces $$\textstyle \frac {RGG - RFF}{A^3}$$ and the um will be the whole force $$\textstyle \frac {FF}{AA} + \frac {RGG - RFF}{A^3}$$ by which a body may revolve in the ame time in the moveable ellipis upk.

In the ame manner it will be found that if the immovable orbit VPK be an ellipis having its centre in the centre of the forces C; and there be uppoed a moveable ellipis upkk imilar, equal, and concentrical to it; and 2 R be the principal latus reclum of that ellipis, and 2 T the latus tranverum or greater axis; and the angle VCp be continually to the angle VCP as G to F; the forces with which bodies may revolve in the immovable and moveable ellipis in equal times, will be as $$\textstyle \frac {FF}{T^3}$$ and $$\textstyle \frac {FFA}{T^3} + \frac {RGG - RFF}{A^3}$$ reperctively.

And univerally, if the greatet altitude CV of the body be called T, and the radius of the curvature which the orbit VPK has in V, 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 K be called $$\textstyle \frac {VFF}{TT}$$, and in other places P be indefinitely stiled X; and the altitude CP be called