ATTRACTION OF BODIES. 271 



the masses are as the cubes of these sides, S and s, the attrac- 

 tions are as S 3 . s" : s 3 . S", or as s"~ 3 : S n ~ 3 . Therefore, if 

 n = 1, the attraction is as S 2 : s 8 ; if the proportion is that of 

 the inverse square of the distance, the attraction is as S : s ; if 

 that of the cube, the attraction is as 1 : 1, or equal ; if as the 

 biquadrate, the attraction is as s : S ; and so on : and thus the 

 law of the attractive force may be ascertained from finding 

 the action of bodies upon particles similarly placed. 



Let us now consider the attraction of any body, of what 

 form soever, attracting with force proportioned to the distance 

 towards a particle situated beyond it. Any two of its particles 

 A B attract P, with forces as A x A P and B x B P, and if G 

 is their common centre of gravity, their joint attraction is as 

 (A + B) x G P, because B P, being resolved into B G and 

 GP, and AP into AP and GP, 

 and (by the property of the 

 centre of gravity) GP x A = 

 A P x G, therefore the forces in 

 the line AP destroy each other, 

 and there remain only P G x B 

 and P G x A to draw P, that is 

 (A -}- B) x P G ; and the same 

 may be shown of any other par- 

 ticles C and the centre G' of gravity, of A, C, and B, the 

 attraction of the three being (A + B + C) x G' P. Therefore 

 the whole body, whatever be its form, attracts P in the line 

 P S, S being the body's centre of gravity, and with a force 

 proportional to the whole mass of the body multiplied by the 

 distance P S. But as the mutual attractions of spherical bodies, 

 the attraction of whose particles is as their distance from one 

 another, are as the distances between the centres of those 

 bodies, the attraction of the whole body A B C is the same with 

 that of a sphere of equal mass whose centre is in S, the body's 

 centre of gravity. In like manner it may be demonstrated that 

 the attraction of several bodies A, B,C, towards any particle P, 

 is directed to their common centre of gravity S, and is equal 

 to that of a sphere placed there, and of a mass equal to the 



