attracted by the whole spheroid AGOD as the distance PS to the distance AS. Q.E.D. PROPOSITION XCII. PROBLEM XLVI. An attracting body being given, it is required to find the ratio of the decrease of the centripetal forces tending to its several points. The body given must be formed into a sphere, a cylinder, or some regular figure, whose law of attraction answering to any ratio of decrease may be found by Prop. LXXX, LXXXI, and XCI. Then, by experiments, the force of the attractions must be found at several distances, and the law of attraction towards the whole, made known by that means, will give the ratio of the decrease of the forces of the several parts; which was to be found. PROPOSITION XCIII. THEOREM XLVII. If a solid be plane on one side, and infinitely extended on all other sides, and consist of equal particles equally attractive, whose forces decrease, in the recess from the solid, in the ratio of any power greater than the square of the distances; and a corpuscle placed towards either part of the plane is attracted by the force of the whole solid; I say that the attractive force of the whole solid, in the recess from its plane superficies,[Pg 241] will decrease in the ratio of a power whose side is the distance of the corpuscle from the plane, and its index less by 3 than the index of the power of the distances. CASE 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 . In the plane mHM take the length HM reciprocally proportional to , 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 , , , &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 , and therefore the force of the whole solid is reciprocally as . Q.E.D.