ATTRACTION. 



83 



Attraction PD X D</. Let the line DF be now supposed to be 

 of Solids, divided into innumerable equal particles, each of 

 * which may be called Dd; and the surface FE will be 



divided into as many equal annuli, whose forces will 

 be as the sum of all the rectangles PD X Dd ; that is, 

 as *PF : ^PD 2 , and therefore as DE 2 . Let the 

 surface FE be now multiplied into the altitude F/"; 

 and the force of the solid EF/e, exerted upon the 

 particle P, will be as DE ! X Xf; supposing that the 

 force is given, which any given particle F/*exerts up- 

 on the particle P at the distance PF. But, if that 

 force is not given, the force of the solid EF/e will be 

 as the solid DE' X ~Ff, and that force not given, 

 jointly. 

 Prop. IV. If equal centripetal forces tend to the several equal 

 parts of any sphere ABE, described about the centre 

 S ; and, from the several points D, perpendiculars 

 DE are erected to the axis of the sphere AB, in which 

 any particle P is placed, meeting the sphere in E ; 

 and in those perpendiculars the lengths DN are ta- 



ken, which are as the quantity 



DE 2 xPS, 

 PE 



and the 



force, which a particle of the sphere, placed in the 

 axis at the distance PE, exerts upon the particle P, 

 jointly ; I say, that the whole force, with which the 

 particle P is attracted towards the sphere, is as the 

 area ANB, contained between AB the axis of the 

 sphere, and the curve line ANB, which the point N 

 continually touches. 



For, supposing the construction in the last lemma 

 and theorem to remain, conceive the axis of the sphere 

 AB to be divided into innumerable equal parts Dd, 

 and the whole sphere to be divided into as many sphe- 

 rical concavo-convex laminae EF/e ; and let the per- 

 pendicular dn be erected. By the last theorem, the 

 force, with which the lamina EF/e attracts the par- 

 ticle P, is as DE 1 X F/> and the force of one particle 

 exerted at the distance PE or PF, jointly. But, by 

 the last lemma, Dd is to F/as PE to PS ; and there- 

 fore F/is equal to =7, -; and DE 1 X F/" ' s equal 



PE 



to D^x 



DE'xPS 

 PE 



and therefore the force of the 



lamina EF/e is as Dd X 



DE'xPS 

 PE ' 



and the force of 



a particle exerted at the distance PF, jointly ; that is, 

 from the supposition, as DN X Dd, or as the inde- 

 finitely small area DNnrf. Therefore the forces of 

 all the laminae, exerted upon the particle P, are as all 

 the areas DNnrf ; that is, the whole force of the 

 sphere is as the whole area ANB. 



Corol. 1. Hence, if the centripetal force tending 

 to the several particles remains always the same at all 



distances, and Dw be made as 



D E'xP S 

 PE 



the whole 



force, with which the particle P is attracted by the 

 *pherc, is as the area ANB. 



Corol. 2. If the centripetal force of the particles 

 is reciprocally as the distance of the particle attracted 



DE 1 x PS 



by it, and DN is made as ~ ~ ; the force, with 



PE* 

 which the particle P is attracted by the whole sphere, 

 will be as the area ANB. 



Cor. 3. If the centripetal force of the particles Attraction 

 is reciprocally as the cube of the distance of the par- of Solid*. 



DE 2 xPS *~y~ / 

 tide attracted by it, and DN is made as ==3 ; 



the force, with which the particle P is attracted by 

 the whole sphere, will be as the area ANB. 



Cor. 4. And universally, if the centripetal force, 

 tending to the several particles of a sphere, is sup- 

 posed to be reciprocally as the quantity V, and DN 



DE 2 x PS 

 is made as -=r= Tf , the force, with which a par- 

 PL XV r 



tide is attracted by the whole sphere, will be as the 



area ANB. 



Supposing what has been already established, it is Prop. V. 

 required to measure the area ANB. 



From the point P let the right line PH be drawn, Platb 

 touching the sphere in H ; and having let fall HI XLIX, 

 perpendicular to the axis PAB, let PI be bisected in Fi S- 5 * 

 L; and PE 1 will be equal to PS 2 + SE 2 + 2PSD. 

 But, because the triangles SPH, SHI are similar, 

 SE 2 or SH 1 is equal to the rectangle PSI. There- 

 fore PE 2 is equal to the rectangle contained under 

 PS and PS + SI + 2SD; that is, under PS and 

 2LS + 2SD ; that is, under PS and 2LD. More- 

 over, DE 1 is equal to SE 2 SD 2 , or SE 2 LS 2 -f 

 2SLD LD 2 ; that is, 2SLD LD 2 ALB. For 

 LS 2 SE 1 , or LS 2 SA Z , is equal to the rectangle 

 ALB. Let therefore 2SLD LD 1 ALB be 



DE* x PS 



substituted for DE' ; and the quantity -== ^= , 



n ' PExV 



which, according to the fourth corollary of the pre- 

 ceding proposition, is as the length of the ordinate 



DN, will resolve itself into three parts, 



LD'xPS ALB x PS 



PExV 



. where, if instead of V 



the inverse ratio of the centripetal force is substi- 

 tuted ; and, instead of PE, the mean proportional 

 between PS and 2LD ; those three parts will become 

 ordinates of as many curve lines, whose areas are found 

 by the common methods. 



Example 1. If the centripetal force, tending to 

 the several particles of a sphere, is reciprocally as 

 the distance, for V substitute the distance PE ; 

 then 2PS X LD for PE 2 ; and DN will become as 



ALB 

 SL fjLD 2TD"' Suppose DN equal to the 



AT B 

 double of this, 2SL LD j^-; and 2SL, the 



given part of the ordinate, drawn into the length 

 AB, will describe the rectangular area 2SL x AB : 

 and the indefinite part LD, drawn perpendicularly 

 into the same length, by a continual motion, made 

 according to such a law, that, in its motion, it may 

 either by increasing or decreasing be always equal to 



Ljj LA* 



the length LD, will describe the area ; 



that is, the area SL X AB ; which, taken from the 

 former area 2SL X AB, leaves the area SL X AB. 



ALB 



But the third part -*-*r > drawn after the sameman- 



ner, by a continual motion, perpendicularly into thr 



