GRAVITATION. 



the perpencllculars L/, A a, B^; making An equal to 

 L B, and Bi equal LA. Making L/ and L B afl">Tnp- 

 totes, defcnbe tiirough t!ie points a, b, the hyperbolic 

 curve a b: and the cliord b a being drawn, will enclofe 

 the area aba equal to the area required A N B. 



Example 2. — If the centripetal force tending to the feveral 

 particles of the fphere be repicrocally as the cube of the 

 diftance, or, (wliich is the fame thing,) as that cube 



. PE' 



applied to any given plane ; fubftitute , for V, and 



2 P S X L D for P E- ; and D N will become as 



A L B X A S^ . . ,, . 

 ■ ; that IS, (bccaule 



S L X A S' A S^ 



LSI 



PSxLD 2PS 2PSxLD' 

 P S, A S, S I, are continually proportional,) as • 

 ALB X SI 



SI - 



2LD' 



If thcfe three parts be then drawn 



LSI 

 into the length A B, the firll, y-TT. will generate the area 



of an hyperbola ; the fecond, ^ S I, the area i A B x S I ; 



, ,.,ALBxSI ' 



the tliird, 



2LD' 



the area 



ALB X 

 iLA' 



SI 



, that is, I A B x SI: from the firft fub- 



ALB X SI 

 2LB 



•traft the fum of the fecond and third, and there will remain 

 A N B, the area fought. Whence arifes this conllruftion of 

 the problem. At the points L, A, S, B, [fg. 132.) ereft the 

 perpendiculars L/, A a, S j, V> b, of which fuppofe Ss 

 equal S I ; and through the point j-, to the aifymptotes L /, 

 L B, defcribe the hyperbola asb, meeting the perpendiculars 

 A a, B 3, in a and b, and the reftangle 2 A S I, fubtratled 

 from the hypei'bolic area Aasb'Q, will leave A N B, the 

 area required. 



Example 3 If the centripetal force, tending to the feveral 



particles of the fphere, decreafe in a quadruplicate ratio of 



P E* 



the diftance from the particles, fubftitute for V, 



then V 2 P S + L D for P E, and D N will become 

 S I' X S L I SP ^ _i 



as 



S 1' X A L B 



I 



2 v^ 2 S I 



Thefe three parts, drawn into 

 2SI X SL 



2 V'2 S 1 X 



die length A B, produce as many ai-eas, namely, 

 S I 



VzSI 



into 



_i L ■— into VLB- V -L A, 



v'LA ,/LB' v^2SI 



R P X A L B ^ 

 and -rr^ — into 



TTa^-— L-B'' '"'^'^''■'' 



2SP 



SL 



3 v/ 2 S I 

 after the proper redudlion, become - 



S I' + —~T '• and thefe, by fubtrafting the latter terms 



, S P, and 



3SP 



from the former, become 



4SI' 



therefore the entire force 



3LI' 



with which the particle P is attrafted towards the centre of 



the fphere, is as r— , that is, reciprocally as P S'' x PL. 

 The attraction of a particle, fituated within the fpliercj 



may be determined by the fame method ; but more expe- 

 ditioufly by the following theorem. 



12. If SI, S A, S P, (fy. IT,?,-) I'C taken conthiiiany 

 proportional, in a fphere defcribcd round the centre S, with 

 the railius S A ; then the altraSian of a particle 'within 

 the fphere, in any place I, is to its cUlraSion, ivithout the 

 fphere, in a place P, in a ratio compounded of the fuh-duplicali 

 ralij 0/" I S, PS, the dijlances from the centre, and the fui- 

 duplicaie ratio of the centripetal forces tending to the centre in 

 thofe places, P and I. 



As, if the centripetal forces of the particles of tlie fphei* 

 be reciprocally as tlie diftances ef tlie particle attrafted by 

 them, the force with which the particle fituated at I is 

 attrafted by the entire fphere, will be to the force with 

 which it is attradled at P, in a ratio compounded of the fub- 

 duplicate ratio of the diltance S I to the diftance S P, and 

 the fub-duplicate ratio of the centripetal force in the place 

 I, ariiing from any particle in the centre to the centripetal 

 force in die place P, arifuig from the fame particle in the 

 centre ; that is, in the fub-duplicate ratio of the diftances S I, 

 S P, to each other reciprocally. Thefe two iub-duplicate 

 ratios compofe the ratio of equality ; and, therefore, the 

 attradlions in I and P, produced by the whole fphere, are 

 equal. By a fimilar calculation, if the forces of the par- 

 ticles of the fphere are reciprocally in a duplicate ratio of 

 the diftance, it will be found that the attracSlion in I is to the 

 attraction in P, as the diftance SP to the femi-diameter S A 

 of the fphere. If thofe forces are reciprocally in a triplicate 

 ratio of the diftances, the attradlions in I and P will be to 

 each other as S P' to S A^ ; if in a quadruphcate ratio, as 

 S P^ to S A'. Therefore, fince the attraftion in P, in this 

 laft cafe, was found to be reciprocally as P S^ x PI, the 

 attraftion in I will be reciprocally as S A' into P I ; that is, 

 becaufe S A^ is given reciprocally as P I ; and the progref- 

 fion is the fame indefinitely. The demonftration of thi* 

 theorem is as follows : 



Retaining the fame conftruftion as above, and a particle 

 being in any place P, the ordinate D N was found to be a« 



D E* X P S , . T T T^ ,_ J 



— =r-= -^— ; therefore, 11 1 E be drawn, that ordinate 



PE X V 



for any other place of the particle, as I, will become (fub- 



ftituting PS and PE for I S and I E) as ^! ^' ^ \^ . 



I E X V 



Suppofe the centripetal forces proceeding from any 

 point of the fphere, as E, to be to each other at the 

 diftances I E and P E, as P E" to I E" (where the 

 number n denotes the index of the powers of P E 

 and I E,) and thofe ordinates will become as 



P E^ X P S ^ P E' X I S , . . ^ ^ . 



vtt; s-^^j ^id t^p: TT^j whole ratio to each other is 



P E X P E " I E X I E" 



as P S X I E X I E" to I S X P E X P E". Becaufe 



SI, S E, S P, are continually proportional, the triangles 



S P E, S E I, are alike ; and thence I E is to P E, as 



IS to S E, or S A. Subftitute the ratio of I S to 



S A for the ratio of I E to P E, and the ratio of the 



ordinates becomes that of P S x I E" to S A x P E". 



But the ratio of P S to S A is fub-dupiicate of that 



of the diftances PS, SI; and the ratio of I E" to 



PE", (becaufe I E is to P E as I S to SA) is fub- 



duphcate of that of the forces at the diftances P Sj I S ; 



therefore the ordinates, and confequently the areas which 



the ordinates defcribe, and the attraftions proportional to 



them, are in a ratio compounded of thofe fub-duphcate 



ratios. 



J 3- Tq Jind the force ivitb <wbkb a partkk, placed imhe 



ttntr* 



