GRAVITATION. 



ftnlre of a fphere, is atlraHid toiuanh any figment of that 

 J'pherc. 



Let P {f.g. 134-) be a particle in the centre of a fphcre, 

 and R B S D a fegment thereof containv-d between 

 t!ie plane R D S, and fpherical fuperlicies R B S. Let 

 D B be interfered in F by a fpherical fuperficits E F G, 

 defcribed from the centre P, and let the fegment be divided 

 into the parts B R E F G S, F E D G : let that fnrface 

 be fuppoled not purely mathematical, but phyfical, having 

 foine but a very inconfiderable thicknefs : let that thicknefs 

 be called O, and (by what Archimedes has demonllrated) 

 that fuperficies will be as P F x D F x O : let us fup- 

 pofe befides, the attraclive forces of the particles of the 

 fphere, to be reciprocally as that power of the dill ances of 

 which n is the index ; and the force with which the furface 



E F G attrads the body P will be as 

 2DExO DF'xO 



BE' X O 



that : 



PF» 



as — ppT-l iTp; — ■ ^'^'^ '''^ perpendicular F N, 



drawn into O, be proportional to this quantity, and the 

 curvilinear area D B I, which the ordinate F N, drawn 

 through the length D B, with a continued motion defcribes, 

 will be as the whole force with which the whole fegment 

 R B S D attradls the body P. 



14. To find the force zinih loh'ich a particle, placed without 

 the antre of it fphcre, in the axis of any fegment, is attrailed 

 ly that fegment. 



Let the body P, placed in the axis A D B, of the feg- 

 ment E B K {Jig. 135.) be attraded by that fegment: 

 round the centre P, with the radius P E, let the 

 fpherical furface E F K be defcribed ; and let it divide the 

 fegment into two parts E B K F E and E F K D E. Let 

 the force of the former part be found by Prop. 11. and, 

 the force of the latter part by Prop. 13. and the lum of 

 the forces will be the force of the whole fegment E B K D E. 



Thefe are the principal propofitions by which Newton has 

 inveftigated the nature of theattradion exerciled by Ipheri- 

 cal bodies. In the thirteenth fedion of the Principia, the 

 author lliews the manner in which the law of attradion is in- 

 veftigated for other bodies ; the moft interefting refults are 

 contained in the following propofitions. 



Of the altradive forces of bodies 'which are not of a fpherical 

 fgure. 



15. If a body he nttraBed by another, and its attradion be 

 tonjiderahly Jironger -when it is contiguous to the attratiing body, 

 than luhen they are fparatcd from one another by a very J mall 

 interval; the forces of the particles of th; attracting body de- 

 treafe, as the attraded body recedes, in more than a duplicate 

 ratio of the particles attraded. 



16. If the forces of the particles, of -which an attraHive body 

 is compofed, decreafe as the attraded body recedes, in a triplicate, 

 or more than triplicate ratio of the di/lance from the particles, 

 the attradion will be confiderably Jironger in the place of con- 

 tad, than ivhcn the attrading and attraded bodies are Jeparated 



from each, though by the mofl minute interval. 



17. If fwo bodies, fimilar to each other, and conji fling of 

 matter equally attradive, attrad feparately two particles, pro- 

 portional to thofe bodies, and in a fimilar fituation to them; 

 the accelerative atlradions of the particles towards the entire 

 bodies will be as the accelerative attradions of the particles 

 towards particles of the bodies proportional to the whole, and 



Jimilarly jduated in them. 



18. If th^- attradive forces of the equal particles of any body 

 le as the d fiance of the places from the particles, the farce of 

 the whole body will tend to its centre of gravity j aiul will hi 



the fame with the force of a gTobe, eonjifling of fmiTar and 

 equal matter, and having its centre in the centre of gravity. 



19. If there be fcvcral bodies, confjling of equal particlett 

 whofe fores areas the diftanccs of 'the places Jrom each, the 

 force compounded of all the forces by which any panicle it 

 attraded, will tend to the common centre of gravity of tie 

 attracting bodies; and will be the fame as if thefe attrading 

 bodies, preferring their common centre of gravity, should uniu 

 lucre, and be formed into a globe. 



20. If a folidbe plane on one fide, and infniuh extended on 

 all other fides, and confijl of equal particles equally altradi-oe, 

 whofe forces decreafe, as they recede from the fdid in the ratio 

 of any power greater than the fquare of the dijlances ; and a 

 particle placed towards cither part of the plane is attraded by 

 the force of the whole f lid; then the attradive force of the 

 whole folitt, as it recedes from its plane fuperfcies, will decreafe 

 in the ratio of a power whoj'e fide is the diflance of a particle 

 from the plane, and is indeed lefs by three than the index of 

 the power of the diflances. 



Though the above propofitions are fufficicnt for all ailro. 

 nomical inveftigations, yet there are many queftions in natu- 

 ral philofophy, particularly thofe which relate to the attrac- 

 tion of mountains, which require that tliefc enquiries Ihould 

 be extended to a greater variety of cafes ; ajid the following 

 propofitions form a part of the interefting invcftigations of 

 profeffor Playfair, as he has communicated them to the pub- 

 lic, in the fixth volume of the Edinburgh Tranfadions, a» 

 above-mentioned : they were fuggerted to the learned author 

 by the experiments which have been made of late years» 

 concerning the gravitation of terreflrial bodies, particularly 

 by Dr. Malkelyne on the attradion of mountains, and by- 

 Mr. Cavendilh, on the attradion of leaden balls, as has 

 been defer bed at length under the article Density, to 

 which the reader is referred. 



I. To fnd the folid into which a mafs of homogeneous matter 

 miifl be formed, to attrad a particle given in po/ition with the 

 greateft force poffible in a given diredion. 



Let A Qig. 136.) be tlic particle given in pofition, 

 A B the diredion in which it is to be attraded ; and 

 A C B H a feetion of the folid required, by a plane paffine 

 through A B. r 1- 6 



Since the attradion of the folid is a maximum by hypothe- 

 fis, any fmall variation in the ligure of the folid, provided 

 the quantity of matter remain the fame, will not change 

 the attradion in the diredion A B. If, therefore, a fmall 

 portion of matter be taken from any point C in the fuper- 

 licies of the folid, and placed at D, another point in thj 

 fame fuperficies, tliere v.ill be no variation in the force which 

 the fohd exerts on the particle A in the diredion A B. 



The curve A C B, therefore, is the locus of all the points, 

 in which a body, being placed, will attrad the particle A in 

 the diredion A B, with the fame force. This condition is 

 fufficient to determine the nature of the curve A B C. From 

 C any point in that curve, draw G E perpendicular to A B ; 



then if a mafs of matter, placed at C be called m', — ^ 



^ 'AC 



will be the attradion of that mafs on A, in the diredion 



A C, and ■ 



X AE 



AC 



will be its attradion ia the diredion 



m' 



AB: asthisisconftant, itwillbcequal to -— -J, and, thsre- 



fore, A B' x AE= AC'. 



All the fedions of the required folid, therefore, br 

 planes paCintj ilirough A U; have this jjioperty, tkit A C 



