332 THE MATHEMATICAL PRINCIPLES [BOOK I. place P. the ordinate DN was found to be as T)F2 \" PS — 00 ^~\r- Therefore if I Cj X V IE be drawn, that ordinate for any other place of the corpuscle, as I, will DE2 X IS become (mutatis mutandis] as ~T~p~rry~- Suppose the centripetalsorces flowing from any point of the sphere, as E, to be to each other at the dis tances IE and PE as PE'1 to IE11 (where the number u denotes the index DE2 X PS of the powers of PE and IE), and those ordinates will become as ^p - -57^7, 2 \x IS and — ~" --- TT7, whose ratio to each other is as PS X IE X IEn to IS X IE X IE" PE X PEn. Because SI, SE, SP are in continued proportion, the tri angles SPE, SEI are alike ; and thence IE is to PE as IS to SE or SA. For the ratio of IE to PE write the ratio of IS to SA ; and the ratio of the ordinates becomes that of PS X IE" to SA X PEn. But the ratio of PS to SA is snbduplicate of that of the distances PS, SI ; and the ratio of IE" to PE1 (because IE is to PE as IS to SA) is subduplicate of that of the forces at the distances PS, IS. Therefore the ordinates, and conse quently the areas whioifi the ordinates describe, and the attractions propor tional to them, are in a ratio compounded of those subduplicate ratios. Q.E.D. PROPOSITION LXXXIII. PROBLEM XLII. To find the force with which a corpuscle placed in the centre of a sphere is attracted towards any segment of that sphere whatsoever. Let P be a body in the centre of that sphere and RBSD a segment thereof contained under the plane RDS, and thesphrcrical superficies RBS. Let DB be cut in F by a sphaerical superficies EFG described from the centre P, and let the segment be divided into the parts _B BREFGS, FEDG. Let us suppose that segment to be not a purely mathematical but a physical superficies, having some, but a perfectly inconsiderable thickness. * Let that thickness be called O, and (by what Archi medes has demonstrated) that superficies will be as PF X DF X O. Let us suppose besides the attrac tive forces of the particles of the sphere to be reciprocally as that power of r.he distances, of which n is index ; and the force with which the superficies DE2 X O EFG attracts the body P will be (by Prop. LXXIX) as -- — that, 2DF X O is, as ---?— -,DF2 X O ~"~ppn — * ppn the perpendicular FN drawn into