GRAVITATION. 



revolution of tlic arc r E will be as the fmall line D f/, 

 the radius of the fphcic rpmaining the fame ; as Archi- 

 medes has demon [Iratcd in Iiis book on tlie fphcre and cy- 

 linder. And the torce of this fuperficies exerted in the 

 direction of the lines PE or Pr fituated all round in tlic 

 conical fiiperficies, will be as this annular fuperficies itfelf ; 

 that is, as the line D d, or, w-liich is the fame, as the 

 rectangle under the given radius P £ of the fphcre, and 

 the line D^/; but that force, exerted in the dircftion of 

 the line P S, tending to the centre S, will be lefs in the 

 ratio of P D to P E, and therefore will be as P D x D (/. 

 If the line D F be now confidercd as divided into in- 

 numerable equal particles, each of which may be called 

 D (/, then the fuperficies F E will be divided into fo 

 many equal annuli, whofe forces will be as the fum of 

 all the rectangles P D X D J; that is, as i P F' — 4 P D' 

 and tlurefore as D E'. Let now the fuperficies F E be 

 multiplied into the altitude F/", and the force of the folid 

 E Fy"-? exerted upon the particle P will be as D E' x Fy'.- 

 that is, it the force be given which any given particle, as 

 Vf, exerts upon the particle P at the diilance P F. But 

 if that force be not given, the force of the fohd E ¥ fe 

 will be as the folid D E' x Ff, and that force not given 

 conjointly. 



I o. If to the feveral equal parts of a fphere ABE 

 Jig. 129.) defcribed about the centre S, there tend equal 

 centripetal forces ; and f from the fevcral points D, perpendi- 

 culars D, E, be drawn to the axis of the fphere A B, in ivhich 

 a particle P is placed, meeting the fphere in E ; and if in tkefe 

 perpendiiulars the lengths D, N, be taken as the quantity 



— — , and as the force -which a particle of the fphere 



fituated in the axis exerts at the diflance P E upon the particle 

 P conjoinllv ; then the •whole force -wilh luhich the particle P 

 is attraHed toivards the fphere is as the area A N B, conipre- 

 h:ndcd under the axis of the fphere A B, and the curve line 

 A N B, the locus of the point N. 



For fuppofing the conftruttion in the lafl lemma and theo- 

 rem to remain, let the axis of the fphere A B be fuppofed to 

 be divided into innumerable equal particles D, d, and the whole 

 fphere to be divided into fo many fpherical concavo-convex 

 laminae E ¥ J e, and let the perpendicular </ n be drawn. 

 By the laft theorem the force with which the laminas E ¥ fe, 

 attracts the particle P, is as D E^ x ¥ f, and the force of 

 one particle exerted at the diitance P E or P F conjointly. 

 But (by the laft lemma) D 4/ is to ¥f as P E to P S, and 



therefore ¥ f is equal to —- ; and D E^ x ¥ f is 



D N be made as 



D E' X P 5 

 PE' ' 



the force with which the 



PE 



equal to D</ x 



DE'- 



PS 



PE 



and therefore the force of the 



laminae E ¥ fe, is as D </ X 



DE^x_PS 

 PE 



, and the force of 



a particle exerted at the diflance P F conjointly ; that is, 

 by the fuppofition, as D N x Y) d, or as the evanefcent area 

 1) N H d. Therefore the forces of all the lamina exerted 

 upon the paiticle P are as the areas D N n ^, that is, the 

 fphere will be as the whole area A N B. 



Cor. I . — Hence if the centripetal force tending to the feveral 

 particles remain always the fame at all diftances, and D N be 



made as '^'^ ' '^^ whole force with which the ])ar- 



ticle is attracted by the fphere is as the area A N B. 



Cor. 2. — If the centripetal force of the particles vary reci- 

 procally as the diilance of the particle attrailcd by it, and 



particle P is attradcd by the whole fphcre wll be as the 

 area A N B. 



Cor. 3 — [fthe centripetal force of the particles var)- rrci- 

 procally as the cube of the diflance of the particle attraded 



D E' X PS 

 by it, and D N be made a» -— , the force with 



which the particle is attracted by the wliole fphere will be 

 as the area A N B. 



Cor. 4.— And univerfally of the centripetal force tendin;cf 



to the feveral particles of the fphere be fuppofed to be 



reciprocally as the quantity V ; and D N be made as 



DE-x PS ^ ^ 



P E V~' ''^ ^' which a particle is attraded by 



the whole fphere will be as the area A N B. 



II. Suppofmg every thing fo remain as above, it is required 

 to meafure the area A N B fig. I ^o 



From the point P let the riglit line P H be drawn touch, 

 ing the fphere in H ; and having drawn H I perpendicular 

 to the axis P A B, bifcct P I in L, and P E^ will be equal 

 to PS- X SE' X 2 PSD. But bccaufe the triangles 

 S P H, SHI, are fimilar, S E" or S IP is equal to the 

 redtangle P S I. Therefore PE is equal to the rectangle 

 contained under P S and P S x S I x 2 S D ; that is, 

 under P S and 2 L S x 2 S D ; that is, under P S and 

 2 L D. Moreover D E' is equal to S E' — S D', or 

 S E- - L S + 2 S L D - L D% that is, 2 S L D - L D* 

 -ALB. For L S' - S E , or L S' - S A>, is equal 

 to the rectangle ALB. Therefore if inftcad of D E' 

 we write 2 S T. D — T. IT- _ A T 'R, tt„ 

 DE' X PS 



PE X V 

 will now refolve 

 L D- X PS 



SLD — LD'- — ALB, tJie quantity 

 which is as the length of the ordinate D N, 



2SLD_>^PS 



7v~ 



itfclf into three 



parts, 



PE 



PS 



. — ; where, if inllcad of V 



A^LB_ 

 P E X y P E~x V 



we write the inverfe ratio of the centripetal force, and in- 

 ftead of P E, the mean proportional between P S and 2 L D, 

 thofe parts w ill become the ordlnates of fo many curve hncs, 

 whofe areas may be found by the common methods. 



Ex. I. — If the centripetal force tending to the feveral 

 particles of the fphcre be reciprocally as the ditlance, in- 

 ftead of V write P E the diftance, then 2 P S x L D for 



P E'-; and D N will become as S L — i L D - 



Suppofe D N equal to the double 2 S L — L D — 

 iven part of the ordinate drawn 



ALD 

 2 L D. 

 ALB 



LD 



into the 



and 2 S L the 



length A B will dcfcribe the rectangular area 2 S L x A B j 

 and the indefinite part L D drawn pcrpendicul.'rlv into the 

 fame length with a continued motion, according to fuch a law 

 thatitsniotion in either dircftion may, byincreiiling or dicieaf. 

 ing, remain always equal to the length L D, will defcribe 



L B' - L A' , . , 



the area , that is the area t> L x AB; 



which taken from the former area 2 S L 

 area S L x A B. But the iliird Dart 



drawn in 



X A B, leaves tlie 

 ALB 



Td 



a fimilar manner with a continued motion perpendicularly 

 into the fame length, will uetcribe the area of an hyper- 

 bola, which fubtractcd from the area S L x A B wiU 

 leave A N B the area fought. Whence this conilruftion of 

 the problem arifcs. At the points L, A, B, (f^. 151.) erect 

 g 2 lie 



