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

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282
Mathematical Principles
Book I.

and erect the perpendicular dn. By the laſt theorem the force with which the laminæ EFfe attracts the corpuſcle P. is as and the force of one particle exerted at the diſtance PE or PF, conjunctly. But (by the laſt lemma) Dd is to Ff as PE to PS, and therefore Ff is equal to and is equal to and therefore the force of the laminæ EFfe and the force of particle exerted at the diſŧance PF conjunctly; that is ſuppoſition, as DN x Dd, or as the evaneſcent area DNnd. Therefore the forces of all the laminæ exerted upon the corpuſcle P are as all the areas DNnd, that is, the whole force of the ſphere will be as the whole area ANB. Q. E. D.

Cor. 1. Hence if the centripetal force tending to the ſeveral particles remain always the ſæme at all diſtances, and DN be made as the whole force with which the corpuſcle is attracted by the ſphere is as the area ANB.

Cor. 2. If the centripetal force of the particles be, reciprocally as the diſtance of the corpuſcle attracted by it, and DN be made as the force with which the corpuſcle B is attracted by the whole ſphere will be as the area ANB.

Cor. 3. If the centripetal force of the particles be reciprocally as the cube of the diſtance of the corpuſcle attracted by it, and DN be made as