Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/397

(by cor. 3. prop. 90.) the force with which any plane mHM attracts the point C, is reciprocally as $\scriptstyle CH^{n-2}$. In the plane mHM take the length HM reciprocally proportional to $\scriptstyle CH^{n-2}$, and that force will be as HM. In like manner in the ſeveral planes IGL, nIN, oKO, &c. take the lengths GL, IN, KO, &c. reciprocally proportional to $\scriptstyle CG^{n-2}$, $\scriptstyle CI^{n-2}$, $\scriptstyle CK^{n-2}$, &c. and the forces of thoſe planes will be as the lengths ſo taken, and therefore the ſum of the forces as the ſum of the lengths, that is, the force of the whole ſolid as the area GLOK produced infinitely towards OK. But that area (by the known methods of quadratures) is reciprocally as $\scriptstyle Cg^{n-3}$, and therefore the force of the whole ſolid is reciprocally as $\scriptstyle CG^{n-3}$. Q. E. D.

Plate 24, Figure 7

Case 2. Let the corpuſcle (Fig. 7.) be now placed on that hand of the plane IGL that is within the ſolid, and take the diſtance CK equal to the diſtance CG. And the part of the ſolid LGI x KO terminated by the parallel planes IGL, oKO, will attract the corpuſcle, ſituate in the middle, neither one way nor another, the contrary actions of the oppoſite points deſtroying one another by reaſon of their equality. Therefore the corpuſcle C is attracted by the force only of the ſolid ſituate beyond the plane OK. But this force (by caſe 1.) is reciprocally as $\scriptstyle CH^{n-3}$, that is (becauſe CG, CK are equal) reciprocally as $\scriptstyle CG^{n-3}$. Q. E. D.

Cor. 1. Hence if the ſolid LGIN be terminated on each ſide by two infinite parallel planes LG, IN; its attractive force is known, ſubducting from the attractive force of the whole infinite ſolid LGKO, the attractive force of the more diſtant part NIKO infinitely produced towards KO.