Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/350

and pi to pq; that is (becauſe of the like triangles PIQ and PSF, piq and psf) as PS to PF and ps to pf. Thence ex equo, the attraction of the corpuſcle P towards S is to the attraction of the corpuſcle p towards s, as $\scriptstyle {\frac {PF\times pf\times ps}{PS}}$ is to $\scriptstyle {\frac {pf\times PF\times PS}{ps}}$, that is, as $\scriptstyle {ps^{2}}$ to $\scriptstyle {PS^{2}}$. And by a like reaſoning the forces with which the ſuperficies deſcribed by the revolution of the arcs KL, klattract thoſe corpuſcles, will be as $\scriptstyle {ps^{2}}$ to $\scriptstyle {PS^{2}}$. And in the ſame ratio will be the forces of all the circular ſuperficies into which each of the ſphærical ſuperficies may be divided by taking sd always equal to SD, and se equal to SE. And therefore by compoſition, the forces of the entire ſphærical ſuperficies exerted upon thoſe corpuſcles will be it: the ſame ratio. Q. E. D.

Proposition LXXII. Theorem XXXII.

If to the ſveral points: of a ſphere there tend equal centripetal forces decreaſing in a duplicate ratio of the diſtances from thoſe points; and there be given both the denſity of the ſphere and the ratio of the diameter of the ſphere to the diſtance of the corpuſcle from its centre; I ſay that the force with which the corpuſcle is attra-

cted