Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/367

Proposition LXXX. Theorem XL.

Plate 22, Figure 6

If to the ſeveral equal parts of a ſphere ABE, (Pl. 22. Fig. 6.) deſcribed about the centre S, there tend equal centripetal forces; and from the ſeveral points D in the axis of the ſphere AB in which a corpuſcle, as, is placed, there be erected the perpendiculars DE meeting the ſphere in E, and if in thoſe perpendiculars the lengths DN be taken as the quantity $\textstyle {\frac {DE^{2}\times PS}{PE}}$ and as the force which a particle of the ſphere ſituate in the axis exerts at the diſtance PE upon the corpuſcle P, conjunctly; I ſay that the whole force with which the corpuſcle P is attracted towards the ſphere is as the area ANB, comprehended under the axis of the ſphere AB, and the curve line ANB, the locus of the point N.

For ſuppoſing the conſtruction in the laſt lemma and theorem to ſtand. conceive the axis of the ſphere AB to be divided into innumerable equal particles Dd, and the whole ſphere to be divided into ſo many ſphærical concavo-convex laminæ 'EFfe;