GRAVITATION. 



its g-ravitntlon in any one dii-ci.'\Ion is exaflly balanced by an 

 equal gravitation in the oppolite dircftion. 



Di-au- tlu-ough p the two ftraight lines dp <, e p !>, making 

 a very fmall angle at p. This may reprefcnt the feftion of a 

 very flender double cone tJpe h p ', having/- for the common 

 vertex, -AwAde'Si for the diameter of the circular bafcs. 

 Tlie o-ravitation of p to the matter in the bafe deh equal to 

 its gravitation to the matter in the bafe o i. For the number 

 of particles in de'vi to the number in i '., as the furface of the 

 bafe </£■ to that of the bafe S s, that is, as d e' to ^ y, that is, 

 iispd' to/>i , that is, as the gravitation to a particle m is 

 to the gravitation to a particle in d e. Therefore the whole 

 gravitation to the matter inde, is the fame with the whole 

 gravitation to the matter in o s : fince it is alio in the oppoiite 

 diredion, the particle p is in equilibrio. The fame thing 

 may be demonilrated of the gravitation to the matter in y r 

 andi s t, and in a fimilar manner, of the gravitation to tlie 

 matter in the fedions of the cones dpe, &p-, by any other 

 concentric furface. Confequently, the gravitation to the 

 whole matter contained in tlie folid d q r e, is equal to the 

 gravitation to the whole matter in the iolid ots ;, and the 

 particle /I is ilill in equilibrio. 



Now iince the lines dp ■, ep c, may be drawn in any 

 direftion, and thus be made to occupy the whole fphere, it 

 is evident that the gravitation of/) is balanced in every 

 direftion, and therefore it has no tendency to move in any 

 direftion, in confequence of this gravitation to the fpherical 

 riiell of matter comprehended between the furfaces A L 1j M 

 and al l> m. 



It is alfo evident, that this holds true with refpeft to all 

 the matter comprehended between ALB M, and the con- 

 centric furface /I n ■!) paffing through/); in fliort/)is in equi- 

 librio in its gravitation, to all the matter more remote than 

 itfelf from the centre of the fphere, and appears as if it did 

 not gravitate at all to any matter more remote from the 



centre. 



We have fuppofed the fpherical fhell to be uniformly 

 denfe. But/ will ftill be in equilibrio, although the fliell be 

 made up of concentric ih-ata of different denlities, provided 

 that each ftratum be uniformly denfe. 



For fliould we fuppofe, that in the fpace comprehended 

 between A L B M and p n v, there occurs a furface al b m 

 of a different denfity from all the reft, the gravitation to 

 the intercepted portions q r and s t are equal, becaufe thofe 

 portions are of equal denfity, and are proportioned to/) q^ and 

 p s' inverfely. The propofuion may therefore be expreffed 

 in the following very general terms, " a particle placed any 

 where within a fpherical (liell of gravitating matter, of equal 

 ■ denfity, at all equal diftanccs from the centre, will be in 

 . equilibrio, and will have no tendency to move in any direc- 

 tion." 



The equality of the gravitation to the furface e d, and to 

 the furface ; i' is affirmed, becaufe the numbers of particles 

 in the two furfaces are inverfely as the gravitations towards 

 one in each. 



For ihe very fame reafon, the gravitations towards the 

 ■furfaces ed and q r and ts are all equal. Hence may be 

 derived an elementary propofition ; which is of great ufe in 

 all enquiries of this kind, namely, 



If a cone, or p'-ra;r.id dpe, of uniform gravitating matter, 

 be divided by parallel ledtions d e, q r. Sec. the gravitation 

 of a particle p in the vertex, to each of thofe fetlions, is the 

 fame, and the gravitations to the folids/i q t; pde, qder, 

 &c. are proportional to their lengths /)y, p d, qd, &c. the 

 firll part of this propofition is already demonilrated. Now, 

 conceive the cone to be thus divided into innumerable flices 

 ei equal thicknefs : it is plain that the gravitation to each 



of thefe is the fame ; and, therefore, the gravitation to the 

 folid qpr, is to the gravitation to the foiid qder, as the 

 number of llices in the firft, to the number of flices in the 

 fecond, that is, as p q, the length of the firft, to q d, the 

 length of the fecond. The cone dpe was fuppofed ex- 

 tremely flender. This was not ncceffary for the demonftra- 

 tion of the particularcafe where all the feCfions were parallel ; 

 but in this elementary propofition, the angle at * is fuppofed 

 fmallcr than any affigned angle, that the cone or pyramid may 

 be confidered as one of the elements into which we may re- 

 folve a bodv of any form. In this refolution, the bafes are 

 fuppofed, if not otherwife exprefsly ftated, to be parallel, 

 and perpendicular to the axes ; indeed, they are fuppofed to 

 be portions .r r, ye, s r, of fpherical furfaces, having their 

 centres in p : the fmall portions .v rq, yed, c e i, &c. are 

 held as infignificant, vanifhing in the ultimate ratios of the 

 whole folids. 



It is eafy, alfo, to fee that the equilibrium of /> is not 

 limited to the cafe of a fpherical (licll, but will hold true of 

 any body compofed of parallel ftrata, or ftrata fo formed, 

 that the lines pd, p^\ are cut in the fame proportion by the 

 feiStions de, qr, ice. In a fphcroidiil fliell, for inftance, 

 whofe inner and outer 'furfaces are fimilar, and fimilarly 

 pofited fpheroids, the particle p will be in equilibrio any 

 where within it ; becaufe, in this cafe, the lines p} and « e 

 are equal ; fo are the lines p i and o d, ' the lines / 5 and r e, 

 the lines j; and q d. Sec. : in mod cafes, however, there is 

 but one fituation of the particle /> thatinfnres this equilibrium. 

 But we may at the fame time infer this very ufeful propofi- 

 tion, 



2. If /here he tivo J'ol'ids, perf<.8l\ fmVar, and of the fame um- 

 forin denfity, the grnintntion to eaeh of thefe fol'uh, by a particle 

 fimilarly placed on or in eaeh, is proportiona! to any homologous 



lines of the folids. 



For, the folids being fimilar, tliey may be refolved into 

 the fame number of fimilar pyramids fimilarly placed in the 

 folids. The gravitations to each of any corrcf'ponding pair 

 of pyramids are proportional to the lengths of thofe pyra- 

 mids. Thefe lengths have the fame proportion in every 

 correfponding pair. Therefore, the abfoliite gravitations to 

 the whole pyramids of one folid have the fame ratio to the 

 abfolute gravitation to the whole pyramids of the other folid. 

 And fince the folids are fimilar, and the particles are at the 

 fimilarly placed vertices of all the fimilar and fimilarly 

 placed pyramids, the gravitation compounded of the ablVilnte 

 gravitations to the pyramids of one Iolid, has the fame ratio 

 to the gravitation fimilarly compounded of the abfolute 

 gravitations to the pyramids of the other. 



3. The gravitation of an external particle to a fpherical fur- 

 face, fhell, or entire fphere, iL-hich is equally denfe at all equal 



diflances from the centre, is the fame as if the •whole matter 'were 

 colic tied in its centre. 



Let A L B M [fg. I24^) reprefent fiich a fphere, and let 

 P be the external particle. Draw PACE througli C, the 

 centre of the fphere, and Crofs it by L C M at right ano-lcs. 

 Draw two right lines P D, P E, containing a very fmall 

 angle at P, and cutting the great circle ALB M, in D, E, 

 D', E'. About P, as a centre with the diftaiice PC, de- 

 fcribe the arc C dm, cutting D P in d, and E P in e. About 

 the fame centre dcfcribe the arc D O. Draw (/F, e G, pa- 

 rallel to A B, and cutting L C inyand^;-. Draw C K per- 

 pendicular to P D, and rfH, D ^', and I F >?, perpendicular 

 to A B. Join C D and C F. 



Now, let the figure be fuppofed to turn round the axis 

 P B. The fcmi-circumference ALB will generate a com- 

 plete fpherical furface ; the arc C d m will generate an- 

 other fpherical furface, having P for the centre ; the fmall 



arcs 



