Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/368

282 and erect the perpendicular dn. By the lat theorem the force with which the laminæ EFfe attracts the corpucle P. is as $$\scriptstyle DE^2 \times Ff$$ and the force of one particle exerted at the ditance PE or PF, conjunctly. But (by the lat lemma) Dd is to Ff as PE to PS, and therefore Ff is equal to $$\textstyle \frac {PS \times Dd}{PE}$$ and $$\scriptstyle DE^2 \times Ff$$ is equal to $$\textstyle Dd \times \frac {DE^2 \times PS}{PE}$$ and therefore the force of the laminæ EFfe $$\textstyle Dd \times \frac {DE^2 \times PS}{PE}$$ and the force of particle exerted at the diŧance PF conjunctly; that is uppoition, as DN x Dd, or as the evanecent area DNnd. Therefore the forces of all the laminæ exerted upon the corpucle P are as all the areas DNnd, that is, the whole force of the phere will be as the whole area ANB. Q. E. D.

Hence if the centripetal force tending to the everal particles remain always the æme at all ditances, and DN be made as $$\textstyle \frac {DE^2 \times PS}{PE}$$ the whole force with which the corpucle is attracted by the phere is as the area ANB.

If the centripetal force of the particles be, reciprocally as the ditance of the corpucle attracted by it, and DN be made as $$\textstyle \frac {DE^2 \times PS}{PE^2}$$ the force with which the corpucle B is attracted by the whole phere will be as the area ANB.

If the centripetal force of the particles be reciprocally as the cube of the ditance of the corpucle attracted by it, and DN be made as