Prop. XV. 



Plate 

 XUX. 

 Fig. 11. 



PE 



P towards A, will be as PE X F/ and 



jointly ; that is, as the quantity contained under 

 r/'X I'K X AP ; or as the area FK/ multiplied 

 into AP. And therefore the sum of the forces, 

 with which all the annuli in the circle, which is de- 

 scribed about the centre A with the interval AD, 

 attract the body P towards A, is as the whole area 

 AHIKL multiplied into AP. 



Cor. 1. Hence it appears, that, if the forces of 

 the points decrease in the duplicate ratio of the dis- 

 tances, that is, if FK is as ^^ 2 , and therefore the 



ATTRACTION. 



APxFK 



87 



area AHIKL 



as RT 



PF" 



1 

 'PH 



the attraction of the 

 PA 



PH' 



particle P towards the circle will be as 1 



AH 

 PH 



Cor. 2. And universally, if the forces of 

 points at the distances D, are reciprocally as 



that 



is, as =rv 



the 

 any 

 1 



power D" of the distances ; that is, if FK is as =^- t 



and therefore the area AHIKL as p _, 



1 



"PH"-' ' 

 the attraction of the particle P towards the circle will 



1 PA 



beas PA-' - PTF^' 



Cor. 3. And, if the diameter of the circle is in- 

 creased indefinitely, and the number n is greater than 

 unity, the attraction of the particle P towards the 

 whole plane indefinitely increased will be reciprocally 



PA 



as PA 3- * ; because the other term ,,_, willbe less 



than any assignable quantity. 



To find the attraction of a particle, placed in the 

 axis of a round solid, to the several points of which 

 there tend equal centripetal forces, decreasing in any 

 ratio of the distances. 



Let the particle P, placed in the axis AB, be at- 

 tracted towards the solid DECG. Let this solid be 

 cut by any circle RFS perpendicular to this axis ; 

 and in its semidiameter FS, in any plane PALKB 

 passing through the axis, let the length FK be taken 

 (by Prop. XIV.) proportional to the force, with 

 which the particle P is attracted towards that circle. 

 And let the point K touch the curve line LKI, meet- 

 ing the planes of the exterior circles A L and BI in L 

 and I ; and the attraction of the particle P towards 

 the solid will be as the area LABI. 



Car. 1. Hence it appears, that if the solid is a 

 cylinder, described by the parallelogram ADEB re- 

 volved about the axis AB, and the centripetal forces 

 tending to its several points are reciprocally as the 

 squares of the distances from the points ; the attrac- 

 tion of the particle P towards this cylinder will be as 

 AB PE + PD. For the ordinate FK (by Cor. 1. 



Prop. XIV. ) will be as 1 * The part I, of this 



quantity, drawn into the length AB, describes the 



area 1 xAB. 



PF 



And the ether part p^-, drawn into 



the length PB, describes the area 1 into PE AD, A ""io 



which may be easily shewn from the quadrature of the ' ' SoliJs - _ 

 curve LKl : and, in like manner, the same part, ^~" v~ ' 

 drawn into the length PA, describes the area 1 into 

 PD AD ; and drawn into AB, the difference of 

 PB, PA, describes 1 into PE PD, the difference of 

 the areas. From the first content 1 into AB let the last 

 content 1 into PE PD be taken away, and the area 

 LABI will remain equal to 1 into AB PE + PD! 

 Therefore the force, proportional to this area, is as 

 AB PE + PD. 



Cor. 2. Hence also the force ; is known by which Pi.at& 

 a spheroid AGBC attracts any body P, placed ex- XLIX. 

 ternallyin its axis AB. Let NKRM be a conic i S- J 3. 

 section, whose ordinate ER, perpendicular to PE, 

 may be always equal to the length of the line PD, 

 which is drawn to that point D, in which that ordi- 

 nate cuts the spheroid. From the vertices A, B of 

 the spheroid let AK, BM, be erected, perpendicu- 

 lar to its axis AB, respectively equal to AP, BP ; 

 and therefore meeting the conic section in K and M : 

 and let KM be joined, cutting off from it the segment 

 KMRK. Let S be the centre of the spheroid, 

 and SC its greatest semidiameter ; and the force, 

 with which the spheroid attracts the body P, will 

 be to the force, with which a sphere, described, 

 with the diameter AB, attracts the same body, as< 

 ASxCS 1 PSxKMRK AS 3 . , 



PS J +CS I ^A"S 3 t0 W' And ' b >' the 



same principles of calculation the forces of the seg- 

 ments of the spheroid might be found. 



Cor. 3. But, if the particle is placed within the 

 spheroid in its axis, the attraction will be as its dis- 

 tance from the centre. Which may be more easily 

 collected by the following reasoning, whether the 

 particle is in the axis, or in any other given diameter. 

 Let AGOF be the attracting spheroid, S its centre, Fl > J4i 

 and P the body attracted. Let the semidiameter 

 SPA, and also two right lines DE, FG, meeting the 

 spheroid in D and E, F and G, be drawn through 

 that body : and let PCM, HLN, be the surfaces of 

 two interior spheroids, similar and concentric to the 

 exterior ; of which let the former pass through the 

 body P, and cut the right lines DE and FG in B 

 and C ; and let the latter cut the same right lines 

 in H, I, and K, L. Let all the spheroids have one 

 common axis, and the parts of the right lines inter- 

 cepted on each side DP and BE, FP and'CG, DH 

 and IE, FK and LG will be mutually equal ; be- 

 cause the right lines DE, PB, and HI, are bisected 

 in the same point ; as also the right lines FG, PC, 

 and KL. Conceive now DPF, EPG to represent 

 opposite cones, described with the indefinitely small 

 verticle angles DPF, EPG, and the lines DH, EI, 

 to be also indefinitely small : and the particles of the 

 cones DHKF, GLIE, cut off by the surfaces of 

 the spheroids, by reason of the equality of the lines 

 DH, EI, will be to each other as the squares of 

 their distances from the particle P, and therefore 

 will attract that particle equally. And, by a like 

 reasoning, if the spaces DPF, EGCB are divided in- 

 to particles by the surfaces of innumerable similar 

 spheroids, concentric to, and having a common axis 

 with the former, all these will equally attract the 



