NEWTON S PRINCIPIA. 149 



that attraction. Then taking D N in the same proportion 

 to the circle D E in which D N is to the circle A F, or as 

 equal to the attraction of the circle D E, we have the 

 curve R N T, whose area is equal to the attraction of the 

 solid L H C F. 



To find an equation to this curve, then, and from 

 thence to obtain its area, we must know the law by which 

 D E increases, that is, the proportion of D E to A D ; in 

 other words, the figure of the section A F E C B, whose 

 revolution generates the solid. 



Thus if the given solid be a spheroid, we find that its 

 attraction for P is to that of a sphere whose diameter is 



equal to the spheroid s shorter axis, as 7^ 7-5 - to 



d 2 + A 2 a 



-^-j-, A and a being the two semi-axes of the ellipsoid, d 



o d~ 



the distance of the particle attracted, and L a constant 

 conic area which may be found in each case ; the force of 

 attraction being supposed inversely as the squares of the 

 distances. But if the particle is within the spheroid, the 

 attraction is as the distance from the centre, according to 

 what we have already seen. 



Laplace s general formula for the attraction of a spherical 

 surface, or layer, on a particle situated (as any particle 



must be) in its axis, is f f d f x f d f J? 9 in 



which /is the distance of the particle from the point where 

 the ring cuts the sphere, r its distance from the centre of 

 the sphere, or the distance of the ring from that centre, 

 d u consequently the thickness of the ring, TT the semicircle 

 whose radius is unity, and F the function of/ representing 

 the attracting force. The whole attraction of the sphere, 

 therefore, is the integral taken from / = r u to f 



L 3 



