GRAVITATION. 



cf matter are A and B, and (/ the diflancc ot iheir centres, 



is — . For the tendency of one particle of A to B, 



d' 

 being the aggregate of its tendencies to every particle of B, 



rj 



is -. Therefore, the tendency of the whole of A to B 

 d' 



muft be . And the tendency of B to A is equal to 



</" 



that of A to B. 



6. But if the centripetnl forces ■which tend to the different 

 points ef fpheres are proportional to the ftmple diflnnces from 

 the atlracled bodies, then the compounded force •with -which 

 two fpheres attraH each other mutually, is as the dijlance be- 

 tween the centres of the fpheres. 



- Cafe I. — Let A EB F {fg. 11$.) be a fphere ; S its 

 centre; Pa particle attracted; PASB the axis of the 

 fphere paffing through the centre of the particle ; E F, ef, 

 two planes cutting the fphere and perpendicular to the axis, 

 and equidillant one on one lide, the other on the other, 

 from the centre of the fphere ; G and g the interfettions of 

 the planes and the axis ; and H any point in the plane 

 E F. The centripetal force of the point H on the particle 

 P, exerted in the direftion of the line P H, is as the dif- 

 tance P H ; and the fame exerted in the direftion of the 

 line P G, or towards the centre S, is as the length P G. 

 Therefore, the force of all the points in the plane E F, 

 {that is, of the whole plane) by which the particle P is at- 

 tracted towards the centre S, is as the diftance P G multi- 

 plied by the number of thofe points, that is, as the folid con- 

 tained under that plane E F, and the diftance P G. And, 

 in like manner, the force of the plane ef, by which the par- 

 ticle P is attracted towards the centre S, is as that plane 

 multiplied into its diftance P ^ ; or as the equal plane E F 

 multiplied into that diftance P^; and the fum of the forces 

 of both planes is as the plane E F, multiplied into the fum 

 of the diftances P G + P^, that is, as that plane multiplied 

 into twice the diilance P 6 of the centre and the particle ; 

 that is, as twice the plane E F, multiplied into the diftance 

 P S, or as the fum of the equal planes E F -)- ef multiplied 

 into the fame diftance. And by a fimilar train of reafon- 

 ing, the forces of all the planes in the whole fphere, 

 eqnidiftant on each fide from the centre of the fphere, are as 

 the fum of thofe planes, multiplied into the diftance P S, 

 that is, as the whole fphere, and the diftance P S jointly. 



Cafe 2. — Let the particle P now attraft the fphere 

 A E B F, and, by the fame reafoning, it will appear that 

 the force with which the fphere is attracted is as the dif- 

 tance P S. 



Cafe 3. — If another fphere be now compofed of innume- 

 rable particles P, and becaufe the force with which every 

 particle is attrafted is as the diftance of the particle from 

 the centre of the firft fphere, and as the fame iphere con- 

 jointly, and is therefore the fame as if the whole proceeded 

 from a fingle particle fituated in the centre of the fphere ; 

 the entire force with which all the particles in the fecond 

 fphere are attrafted, that is, with which the whole fphere 

 is attrafted, will be the fame as if that fphere were at- 

 trafted by a force proceeding from a fingle particle in the 

 centre of the firft fphere, and is therefore proportional to 

 the diftance between the centres of the fpheres. 



Cafe 4. — Let the fpheres attraft each other mutually, 

 and the force will be doubled, but the proportion will re- 

 main the fame. 



Cafe 5. — Let the particle P be placed within the fphere 

 AEBF, [Jig. 126.) and becaufe the force of the plane ef, 

 upon the particle, is as the folid contained under that plane 



and the diftance p g ; and the contrary force of the plan? 

 E F, is as the folid contained under that plane and the dif- 

 trince p G ; the force compounded of both will be as the 

 difference of the folids, that is, as the fum of the equal 

 planes multiplied into half the difference of the diftances ; 

 that is, as that fum multiplied into^ S, the diftance of the 

 particle from the centre of the fphere. And, by a fimilar 

 train of reafoning, the attradiion of all the planes E F, e f, 

 throughout the fphere, that is, the attraction of the whole 

 fphere is conjointly as the fum of all the planes or as the 

 whole fphere, and as j!> S the diftance of the particle from 

 the centre of the fphere. 



Cafe 6. — And if a new fphere be now compofed of innu- 

 merable particles, fuch as p, fituated within the firft fphere 

 AEBF, it may be proved, as before, that the attraftion, 

 whether tingle of one fphere towards the other, or mutual 

 of both towards each other, will be as the diilance p S 

 of the centres. 



7. If the flruSure of the fpheres he dijfimilar and unequal, 

 proceeding direSly from the centre loiuards the circumference, but 

 fimilar ami equal throughout every circumference, at equal dijlances 

 from the centre, and if the attracli-v: force be as the dylance of 

 the attracted body, then the entire force •with nuhich t<wo fpheres 

 of this hind attrall each other mutually is proportional to tlit 

 dijlance betiueen the centres of the fpheres. — T'his is denionftrated 

 from the preceding propofition. 



The above inveftigations relate to the principal cafes of 

 attraftion, namely, when the centripetaf forces decreafe in 

 a duplicate ratio, or increaie in the fimple ratio of the dif- 

 tance. And it is remarkable that both thefe fuppolitions 

 caufe bodies to revolve in conic fections, and compufe iphe- 

 rical bodies, whole centripetal forces obferve the fame law 

 of increafe or decreafe, in the recefs from the centre, as 

 the forces of the particles therafelves do. 



%. If a circle AEB [fig. 127.) be defcribed round 

 the centre S, and two circles E F, ef, be alfo defcribed 

 round the centre P, inlerfeSing the former in E and e, and the 

 line P S ;n F anil f ; and if E. D, e d, be drawn perpendicular 

 /o P S ; then if the dijlance of the arcs E F, ef, be fuppefd 1 9 

 be injnitcly dimini/lxd, the limiting ratio of the eiianefcenl line D d 

 to the evanefcetit line Y f is the fame as that of the line P E /o 

 the line P S. 



For if the line P e interfeft the arc E F in q, and the 

 right line E e, which coincides with the evanefcent arc 

 E e, be produced and meet the right line PS in T ; and 

 the perpendicular S G be drawn from S to P E, becaufe 

 the triangles DTE, dT e, D E S, are fimilar, T) d will 

 be to E « as D T to T E, or D S to E S ; and becaufe 

 the triangles Y. e q, E S G, are fimilar, E e will be to <• ^ 

 or Y f as E S to S G ; and ex aquo D d is to Y f as D E 

 to S G ; tliat is, (becaufe the triangles P D E, P G S, 

 are fimilar,) as PE to PS. 



<). If a fupeijcies, as Y.Y fe (Jg. 1 28.) be fuppofed 

 to have its breadth injnitely diminijhed, and that by its 

 revolution round the axis P S /'/ djcribes a fpherical concavo- 

 con-vex folid to the fe-ueral equal particles of which equal centri- 

 petal forces tend : then the force -zvith -which that folid at- 

 tracts a particle placed at P is in a ratio compounded of the ratio 

 of the folid 1) Y.' X Y f, and the ratio of the force with which 

 the given particle in the place Y f -would attroQ the fame 

 pai'ticle. 



For if the force be firft confidered of the fpherical fu- 

 perficies F E, which is generated by the revolution of the 

 arc F E, and is interfered any where, as in r, by the line 

 de, the annular part of the fiiperficies generated by the 



revolutiofi 



II 



