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

 266 and pi to pq; that is (becaue 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 corpucle P towards S is to the attraction of the corpucle 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 reaoning the forces with which the uperficies decribed by the revolution of the arcs KL, klattract thoe corpucles, will be as $$\scriptstyle{ps^2}$$ to $$\scriptstyle{PS^2}$$. And in the ame ratio will be the forces of all the circular uperficies into which each of the phærical uperficies may be divided by taking sd always equal to SD, and se equal to SE. And therefore by compoition, the forces of the entire phærical uperficies exerted upon thoe corpucles will be it: the ame ratio. Q. E. D.

If to the veral points: of a phere there tend equal centripetal forces decreaing in a duplicate ratio of the ditances from thoe points; and there be given both the denity of the phere and the ratio of the diameter of the phere to the ditance of the corpucle from its centre; I ay that the force with which the corpucle is  cted