83 



A TTRACTIO N. 



.Attraction 



ids. 



Prop.XVU 



body P on both sides toward-; contrary part*. Thcrc- 

 forc the forces of the cone DPI', and of the conical 

 ~"" v "* segment EGCB, are ccinal, and by their contrary 

 actions destroy each other mutually. And the case 

 is the same of the forces of all the matter without 

 the interior spheroid PCBM. Therefore the body P 

 is attracted by the interior spheroid PCBM alone ; 

 and therefore (by Chap. IV. Physical Astronomy,) 

 its attraction is to the force, with which the body A 

 is attracted by the whole spheroid AGOD, as the 

 distance PS to the distance AS. 

 Prop. XVI. An attracting body being given, it is required to 

 find the ratio of the decrease of the centripetal forces, 

 tending to its several points. 



Of the body given, a sphere, or a cylinder, or 

 some other regular figure is to be formed, whose law 

 of attraction, agreeing to any ratio of decrease, may 

 be found by Prop. IV. V. and XV. Then the force 

 of attraction must be found by experiment at diffe- 

 rent distances ; and the law of attraction towards the 

 whole, thence discovered, will give the ratio of the 

 decrease of the forces of the several parts. 



If a solid, plane on one side, and indefinitely ex- 

 tended on all other sides, consists of equal particles 

 equally attractive, whose forces, in receding from the 

 solid, decrease in the ratio of any power of the dis- 

 tances greater than the square ; and a particle, pla- 

 ced towards either part of the plane, is attracted 

 by the force of the whole solid ; the attractive 

 force of the solid, in receding from its plane surface, 

 will decrease in the ratio of a power, whose side is 

 the distance of the particle from the plane, and whose 

 index is less by three than the index of the power of 

 the distances. 



Case 1. Let LG/ be the plane by which the solid 

 is terminated. And let the solid lie on the side of 

 the plane towards I ; and let it be resolved into innu- 

 merable planes HM, IN, oKO, &c. parallel to 

 GL. And first let the attracted body C be placed 

 without the solid. Let CGHI be drawn perpendi- 

 cular to those innumerable planes ; and let the attrac- 

 tive forces of the points of the solid decrease in the 

 ratio of a power of the distances, whose index is 

 the number n not less by three. Therefore (by Cor. 3. 

 Prop. XIV.) the force with which any plane ;HM 

 attracts the point C, is reciprocally as CH"~ *. In 

 the plane wHM let the length HM be taken reci- 

 procally proportional to CH" ', and that force will 

 be as HM. In like manner, in the several planes 

 /GL, hIN, oKO, &c. let the lengths GL, IN, KO, 

 &c be taken reciprocally proportional to CG" % 

 CI" -1 , CK" -1 , &c. and the forces of those planes 

 will be as the lengths so taken ; and therefore the 

 sum of the forces as the sum of the lengths ; that 

 is, the force of the whole solid as the area GLOK, 

 produced indefinitely towards OK. But that area, 

 by the known methods of quadratures, is recipro- 

 cally as CG H 3 , and therefore the force of the whole 

 solid is reciprocally as CG' : 3 . 



2. Let the particle C be now placed on the 

 side of the plane /GL. within the solid ; and let the 

 distance CK be taken equal to the distance CG. 

 And the part of the solid LG/oKo, terminated by 

 the parallel planes /CL, oKO, will attract the par- 



Plate 

 XLIX. 

 Fig. 15. 



I Iff. IG. 



tide C, placed in the middle, to neither side; the Attl 

 contrary actions of the opposite points mutually de- ' s " ll<1: '- 

 bt roving each other by their equality. Therefore 

 the particle C is attracted by the force only of the 

 solid placed beyond the plane OK. But this force, 

 by the first case, is reciprocally as CK'-~ 3 ; that is, 

 because CG, CK, are equal, reciprocally as CG" - 3 . 

 i. Hence, if the solid LGIN is terminated 

 on each side by two parallel planes LG, IN, indefi- 

 nitely extended ; its attractive force is known, by 

 subducting from the attractive force of the whole 

 solid LGKO, indefinitely extended, the attractive 

 force of the more distant part NIKO, indefinitely 

 produced towards KO. 



Cor. '/. If the more distant part of this indefi- 

 nitely extended solid is rejected, when its attraction, 

 compared with the attraction of the nearer part, is 

 inconsiderable ; the attraction of that nearer part, by 

 increasing the distance, will decrease nearly in the ra- 

 tio of the power CG"~ 3 . 



Cor. 3. And hence, if any finite body, plane on 

 one side, attracts a particle placed opposite the mid- 

 dle of that plane ; and the distance between the par- 

 ticle and the plane, compared with the dimensions of 

 the attracting body, is very small ; and the attract- 

 ing body consists of homogeneous particles, whose 

 attractive forces decrease in the. ratip of any pow-cr 

 of the distances greater than the quadruplicate ; the 

 attractive force of the whole body will decrease near- 

 ly in the ratio of a power, whose side is that very 

 small distance, and whose index is less by three than 

 the index of the former power. This assertion does 

 not hold good of a body consisting of particles, 

 whose attractive forces decrease in the ratio of the 

 triplicate power of the distances. Because, in this 

 case, the attraction of the more distant part of the 

 indefinitely extended body, in the second Corollary, 

 is always indefinitely greater than the attraction of 

 the nearer part. 



The important investigations of Professor Playfair, On the so- 

 respecting the solids of greatest attraction, were sug- 1'dofgreat- 

 rrested by the experiments of Dr Maskelyne and Mr e . st attrac_ 

 Cavendish to ascertain the density or the earth. In 

 determining the figure which a given quantity of 

 matter ought to have, in order to attract a particle 

 in a given direction with the greatest possible force, 

 Mr Playfair has obtained results remarkable for their 

 simplicity, and highly interesting from their connec- 

 tion with experimental inquiries. In order to correct 

 the conclusions obtained by Dr Hutton from Dr 

 Maskelyne's observations, by taking into account the 

 unequal density of the mountain, the methods of Dr 

 Hutton could not always be pursued. This incon- 

 venience Mr Playfair has remedied by the proposi- 

 tions respecting the attraction of a half or quarter 

 cylinder on a particle placed in its axis. The ele- 

 gance of the solutions, and the address with which 

 Mr Playfair has conducted the whole of the investi- 

 gation, will appear from the following propositions, 

 which are selected from his paper on the Solids of 

 Greatest Attraction, with the kind permission of that 

 able mathematician. 



To find the solid into which a mass of homoge- Prop. I. 

 1 



