GRAVITATION. 



arcs D E, rJe, !? G, t^'Ht gvnecafe rings, oi' zones, of thofe 

 ftj'icrical fm-faces ; D O will alfo generate a zone of a fur- 

 lace having P for its centre ; fg and F I will generate zones 

 of flat circular furfaces. 



It is evident that the zones generated by D E and D O, 

 fwhich we may call the zones D E, DO,) having the 

 feme radius D i, are to e»ch other as their refpedive breadths 

 D E, DO. In like manner, the zones g'^ncrated by dc, 

 fg, F I, F G, being all at the fame diilance from the axis 

 A B, are alfo as th.eir refpedive breadths d c, fg, F I, F G. 

 But the zone D O is to the zone Je, at P I) to P ,/' : for 

 DO is to de, as PD to P d, and the radius of rotation 

 Do is to the radius JH, alfo a"; P D to P^. The cir- 

 eiimferences, dcfcriljcd by DO and </<•, are, therefore, in the 

 iznri proportion as P D to P ^/ ; thorefore the zones being as 

 thiir breadths, and as their circumferences jointly, are as 

 PDandPJS 



C K and d H, lieing the fines of the fame arc C d, are 

 equal ; therefore KD and/F, the halves of chordu equally 

 dillant from the centre, are alfo equal : thi.refore the trian- 

 gles C D K and C F/"are equal and fimilar. But C D K is 

 hmilar to EDO, for the right angles P D O and C D E 

 are equal. Taking away the common angle C D O, the 

 remainders C D K and EDO are equal. In like manner, 

 C Fy" and G F I are fimilar ; and, therefore, ( fince C D K 

 and C Fy are fimilar,) the elementary triangles EDO and 

 G F I are fimilar, and D O : D E =i F I : G F. The ab- 

 folute gravitation, or tendency of P, to the zone D O, is 

 equal to its abfohite gravitation to the zone de, becaufe the 

 number of particles in the firft, is to the particles in the lalt, 

 as P D' to P <-/', that is, inverfely as the gravitation to a 

 particle in the firlt, to the gi-avitation to a particle in the lall ; 

 therefore let c reprt-fent the circumference of a circle, whofe 

 radius is i. The fnrface of the zone generated by D O, 

 will be D O x <r x D ^, and the gravitation to it will 



. DO X f X DJ ,. , de X c X dB. 



be rr^r^ , to which =j— r > or 



, is equal. This expreffes the abfolute gi-avi- 



F? X c X 



PD' 



dn 



tation to the zone generated by D O, this gravitation being 

 exerted in the diredion P D. 



But it is evident that the tendency of P, arifing from its 

 gravitation to every particle in the zone, muft be in the 

 dircdion P C. The oblique gravitation muft, therefore, be 

 eftiniated in the diredion P C, and muft be reduced in the 

 proportion of P(/ to P H. It is plain that P </ : P H = 

 de ■ fg, becaufe de and fg are perpendicular to P (/ and 

 P H ; therefore the reduced, or central gravitation of P, 

 to the zone generated by D O, will be cxpreffcd by 

 fg X c X d H 



PC' ■ 



But the gravitation to the zone generated by D O, is to 

 the gravitation to the zone generated bv D E, as D O to 

 D E, that is, as FT {or fg) to F G. Tlierefore the cen- 

 tral gravitation to the zone generated by D E, will be ex- 



_,, FGxcxJH ... _^ ITT- 



prefted by — — Now, FG X f X r/H is 



the value of the furface of the zone generated by F G ; 



and if all this matter were colleded in C, the gravitation of 



„, FGxfXr/H ,. ,,, 



P to It would be exaftly —^ , and it would be 



in the diredion PC. Hence it follows, that the central 

 gravitation of P to tiic zone generated by D E, is the lame 

 as its gravitation to all the matter in the- zone geHci-ateJ by 

 F G, if that matter were j)laced in C. 

 Vol.. XVI. 



Wliat has been Jemonltrated refpefting the «re D E, t'l 

 true of ev-er)- portion of the circiiinfcrcnce. Each has a 

 fubftitute F G, which being placed in the centre r, the grs. 

 vitation of P is the fame. If P T touch th- fpliert in T, 

 every portion of the arc T L B will have its fnbftitute in the 

 quadrant L B, and every part of the arc A T has hi fub- 

 ftitute in the quadrant A T L, as is cafily feen. And 

 hence it follows, that the gi-avitation of a particle, P, to a 

 fphcrical furface, A L B M, is the fame as if all tire matter 

 of that furface were colleded in its centre. 



We fee alfo that the gravitation to the furface generated 

 by the rotation of A T round A B is equal to the gravita- 

 tion to the furface generated by T L B, wliich is much 

 larger but more remote. 



What we have now demonftrated with refped to the fur- 

 face generated by the femicircle A L B, is equally true with 

 regard to the furface generated by any concentric femicircle, 

 fuch 7\s a 1 1/. It is true, therefore, with regard to the fliell 

 comprehended between thofe two furfaces ; for this (hell 

 may be refolved into innumerable concentric ftrata, and the 

 propofition may be affirmed with refpcct to each of thorn, 

 and therefore witli refped to the whole. And tliis w ill be 

 ftill true if the whole fphere be thus occupied. 



Lattly, it follows that the propofition is ftill true, al- 

 though thefe llrata fliould differ in deiifity, pronded that 

 each llratuni is uniformly d nfe in every part. 



It may, therefore, be affirmed in the molt genera] terms, 

 that a particle, P, placed without a fpherical furface, (hell, 

 or entire fphere, equally denfe at equal diftances from the 

 centre, tends to the centre with the fame force, as if the 

 whole matter of the furface, fliell, or fphere were colleded 

 there. - 



This will be found to be a vqtj important propofition. 



4. The gravitation of an extrrra! particle to a fpherical furfarff 

 Jlelly or entire fphere, of anif.rm denjity at equcd dijiancet frcm 



the centre, is as the quantity nf matter in that lods dirrSlj ; anet 

 as the fqiiare of the dijlancc from its centre inverjely. 



For, if all the matter were colleded in its centre, th* 

 gravitation would be the fame, and it would then \-ary in the 

 inverfe duplicate ratio of the diftance. 



Cor. I. — Particles placed on the furfaces of fphercs of 

 equal denfity, gravitate to the centres of thofe fpheres with 

 forces proportional to the- radii of the fphercs. 



For the quantities of matter are as die cubes of the radii. 



(/' 

 Tlierefore the graWtation g is as — , that is, as it. 



Cor. 2. — The fame thing holds true if the diftance of the 

 external particles from the centres of the fpheres are as the 

 diameters or radii of the fpheres. 



Cor. 3. — If a particle be placed within the furface of a 

 fjjhere of uniform denfity, its gravitation at different dif- 

 tances from the centre will be as thofe diftances. For, it 

 will not be affcdcd by any matter of the fpiiere that is more 

 remote from the centre, and its gravitation to what is lels 

 remote, is as its diftanc-.^ from ih- centre bv the lalt Cor. 



5. The mtitual gravilaiiyii of t-.:ti fphr.:s rf unifc/rm ilaifly ijt 

 thtir concentric //rata, is in ll.-e iit^'trjc dufJiciiU ratio of toe dif- 

 laiire belti'een lh:ir centres. 



For the gravitation of each particle ia the fphere A, it 

 to the fphiTe B. the fame ;ib if all tiic matter in B were coU 

 leded at its cvntre. Suppofe it fo placed : 



T!ie gravitation of B to A will be the fame as if all the 

 matter in A were colleded at iti centre. Therefore it will 

 be as (/- inverfely. But the grantation of A to B is equal 

 to that of B'to .•\. Tlierefoi-c,-&c.. 



The abiokite jji-a. italion of two fj-.ljcrci wliofe quantit'et 

 4^ '-•♦■ 



