Page:Newton's Principia (1846).djvu/234

distance PE or PF, conjunctly. But (by the last Lemma) Dd is to Ff as PE to PS, and therefore Ff is equal to $\scriptstyle {\frac {PS\times Dd}{PE}}$; and DE² $\scriptstyle \times$ Ff is equal to Dd $\scriptstyle \times {\frac {DE^{2}\times PS}{PE}}$; and therefore the force of the lamina EFfe is as Dd $\scriptstyle \times {\frac {DE^{2}\times PS}{PE}}$ and the force of a particle exerted at the distance PF conjunctly; that is, by the supposition, as DN $\scriptstyle \times$ Dd, or as the evanescent area DNnd. Therefore the forces of all the laminae 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 same at all distances, and DN be made as $\scriptstyle {\frac {DE^{2}\times PS}{PE}}$ 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 $\scriptstyle {\frac {DE^{2}\times PS}{PE^{2}}}$, the force with which the corpuscle P 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 $\scriptstyle {\frac {DE^{2}\times PS}{PE^{4}}}$, the force with which the corpuscle is attracted by the whole sphere will be as the area ANB.

Cor. 4. And universally if the centripetal force tending to the several particles of the sphere be supposed to be reciprocally as the quantity V; and DN be made as $\scriptstyle {\frac {DE^{2}\times PS}{PE\times V}}$; the force with which a corpuscle is attracted by the whole sphere will be as the area ANB.

PROPOSITION LXXXI. PROBLEM XLI.

The things remaining as above, it is required to measure the area ANB.

From the point P let there be drawn the right line PH touching the sphere in H; and to the axis PAB, letting fall the perpendicular HI, bisect PI in L; and (by Prop. XII, Book II, Elem.) PE² is equal to