<S4 



A TTRACTIO N. 



*. 



into the same length, will describe an hyperbolic area ; 

 which, taken from the area SLxAB, will leave 

 ANB the area sought. Whence this construction of 

 the problem arises. At the points L, A, B, erect 

 the perpendiculars L/, An, Bi ; of which An may 

 be equal to LB, and BA to LA. ' With the asymp- 

 totes L/, LB let the hyperbola a b be described 

 through the points a, b. And the chord b a will in- 

 close the area aba, equal to the area ANB sought. 



Example 2. If the centripetal force, tending to 

 the several particles df a sphere, is reciprocally as the 

 cube of the distance ; or, which is the same thing, 

 as that cube applied to any given plane ; subtitute 



' J for V, then 2PS x LD for PE 1 ; and DN will 



2AS' 

 become as 



SLxAS 1 AS' ALB x AS' 



PS X LD 2PS 2PS x LD' ' that 

 is, because PS, AS, SI are continually proportional, 



If the three parts of 



-xSI- 



LD * 2LD 1 



this quantity are drawn into the length AB, the first 



LSI 



pp will generate an hyperbolic area; the second 



|SIj the area ABx SI ; the third ^^-^1, the 



ALBxSI ALBxSI 



that is, iABxSI. 



2LA 2LB 



From the first let the sum of the second and third be 

 subducted, and there will remain ANB the area 

 sought. Whence this construction of the problem 

 arises. At the points L, A, S, B, erect the perpen- 

 diculars L/, An, Ss, Bb, of which let Si be equal to 

 SI ; and through the point s with the asymptotes 

 L./, LB, let the hyperbola a s b be described, meet- 

 ing the perpendiculars An, B5 in a and b ; and the 

 rectangle 2ASI, subducted from the hyperbolic area 

 AasbB, will leave the area ANB sought. 



Example 3. If the centripetal force, tending to the 

 several particles of a sphere, decreases in the qua- 

 druplicate ratio of the distance from the particles ; sub- 



PE* 



StitUtC 2AW f r V ' then ^2PS X LD for PE, and 



DN will become as ,____ y 



V2&1 VLD 3 



L_ SI'xALB 1_ 



AD 2v/2sT x VldV 



parts, drawn into the length AB, produce as many 



. 2SI'xSL . 1 I 



areas, namely, r== into =^ . 



7 </2SI -/LA VTF ' 



SI'xALB 



V2ST 

 Whose three 



-/2S1 



into 



VLB VlsA; 



and 



into 



I 



-/LA 3 <v/LB' ' 



3^2ST 



And these, after a due 



2SI* X SL 



.SPandSP-f 



2SI 3 



reduction, become 



LI T 3LI ' 



But these, by taking away the latter terms from the 



4SI 3 

 former, become -yy. Therefore the whole force, 



with which the particle P is attracted to the centre 



SI 3 



Attraction 



r i i . ai J , . __ .utraction 



of the sphere, is as ; that is, reciprocally as PS' of Solids. 



XPL ' ' 



The attraction of a particle placed within a sphere 

 may be determined by the same method ; but more 

 expeditiously by the following proposition. 



If SI, SA, SP are taken continually propor- Prop. VI. 

 tional, in a sphere described about the centre S, 

 with the interval SA ; the attraction of a particle Plait.' 

 within the sphere, in any place I, is to its attraction XLIX. 

 without the sphere in the place P, in a ratio com- l, S- s - 

 pounded of the subduplicate ratio of IS, PS, the dis- 

 tances from the centre, and the subduplicate ratio of 

 the centripetal forces, tending to the centre in those 

 places P and I. 



As, if the centripetal forces of the particles of a 

 sphere arc reciprocally as the distances of a particle 

 attracted by them ; the force, with which a particle 

 placed in I is attracted by the whole sphere, will be 

 to the force, with which it is attracted in P, in a ratio 

 compounded of the subduplicate ratio of the distance 

 SI to the distance SP, and the subduplicate ratio of 

 the centripetal force in the place I, arising from any 

 particle in the centre, to the centripetal force in the 

 place P, arising from the same particle in the centre; 

 that is, in the subduplicate ratio of the distances 

 SI, SP to each other, reciprocally.* These two sub- 

 duplicate ratios compound the ratio of equality ; and 

 therefore the attractions in I and P, produced by the 

 whole sphere, are equal. By a like calculation, if the 

 forces of the particles of a sphere are reciprocally iu 

 the duplicate ratio of the distances, it will be collect- 

 ed, that the attraction in I is to the attraction in P, 

 as the distance SP, to SA the semidiameter of the 

 sphere. If those forces are reciprocally in the tri- 

 plicate ratio of the distances, the attractions in I and 

 P will be to each other as SP 1 to SA 1 : if in a qua- 

 druplicate ratio, as SP 3 to SA 3 . Therefore, since the 

 attraction in P, in this last case, was found to be re- 

 ciprocally as PS' X PI, the attraction in I will be re- 

 ciprocally as SA 3 X PI ; that is, because SA 3 is given, 

 reciprocally as PI. And the progression is the same 

 indefinitely. The theorem is therefore demonstrated. 



Retaining the construction above, and a particle 

 being in any place P, the ordinate DN was found as 



-== . Therefore, if IE is drawn, that ordi- 

 PEx V 



nate for any other place I of the particle will become 



as -_, v (changing PS, and PE, for IS, and 



IE.) Suppose the centripetal forces, flowing from 

 any point E of the sphere, to be to each other at the 

 distances IE, PE, as PE"to IE" (where the number 

 n denotes the index of the powers of PE and IE) ; 



and those ordinates will become as 



DE*xPS 



, and 



PE x PE" 



DE'xIS . , 



Y-p 7t=t- : whose ratio to each other is as PS X 



ILxIE' 



IExIE" to ISxPExPE". Since the triangles 



SPE, SEI are similar, on account of the lines 



SI, SE, SP being continually proportional ; and from 



thence it follows, that IE is to PE, as IS to SE or 



SA ; for the ratio of IE to PE substitute the ratio 



