Page:Newton's Principia (1846).djvu/247



1. Let LGl be the plane by which the solid is terminated. Let the solid lie on that hand of the plane that is towards I, and let it be resolved into innumerable planes mHM, nIN, oKO, &c., parallel to GL. And first let the attracted body C be placed without the solid. Let there be drawn CGHI perpendicular to those innumerable planes, and let the attractive forces of the points of the solid decrease in the ratio of a power of the distances whose index is the number n not less than 3. Therefore (by Cor. 3, Prop. XC) the force with which any plane mHM attracts the point C is reciprocally as CHn-2. In the plane mHM take the length HM reciprocally proportional to CHn-2, and that force will be as HM. In like manner in the several planes lGL, nIN, oKO, &c., take the lengths GL, IN, KO, &c., reciprocally proportional to CGn-2, CIn-2, CKn-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 CGn-3, and therefore the force of the whole solid is reciprocally as CGn-3. Q.E.D.

2. Let the corpuscle C be now placed on that hand of the plane lGL that is within the solid, and take the distance CK equal to the distance CG. And the part of the solid LGloKO terminated by the parallel planes lGL, oKO, will attract the corpuscle C, 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 CKn-3, that is, (because CG, CK are equal) reciprocally as CGn-3. Q.E.D.

. 1. Hence if the solid LGIN be terminated on each side by two infinite parallel places 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.

. 2. If the more distant part of this solid be rejected, because its attraction compared with the attraction of the nearer part is inconsiderable,