NEWTON S PRINCIPIA. 139 



placed within a hollow spheroid, or in a solid, produced 

 by the revolution of an ellipsis, will not be attracted at all 

 by the portion of the solid between it and the surface, but 

 will be attracted towards the centre by a force proportioned 

 to its distance from that centre. 



It follows from these propositions, first, that any par 

 ticle placed within a sphere or spheroid, not being affected 

 by the portion of the sphere or spheroid beyond it, and 

 being attracted by the rest of the sphere, or spheroid in the 

 ratio of the diameter, the centripetal force within the solid 

 is directly as the distance from the centre ; secondly, that 

 a homogeneous sphere, being an infinite number of hollow 

 spaces taken together, its attraction upon any particle placed 

 without it is directly as the sphere, and inversely as the 

 square of the distance ; thirdly, that spheres attract one 

 another with forces proportional to their masses directly, 

 and the squares of the distances from their centres in 

 versely ; fourthly, that the attraction is in every case as 

 if the whole mass were placed in the central point ; 

 fifthly, that though the spheres be not homogeneous, yet if 

 the density of each varies so that it is the same at equal 

 distances from the centre of each, the spheres will attract 

 one another with forces inversely as the squares of the 

 distances of their centres. The law of attraction, however, 

 of the particles of the spheres being changed from the 

 inverse duplicate ratio of the distances to the simple law 

 of the distances directly, the attractions acting towards 

 the centres will be as the distances, and whether the spheres 

 are homogeneous or vary in density according to any law 

 connecting the force with the distance from the centre, 

 the attraction on a particle without will be the same as if 

 the whole mass were placed in the centre ; and the 

 attraction upon a particle within will be the same as if 

 the whole of the body comprised within the spherical 



