ATTRACTION OF BODIES. 273 



whole attraction of the circle whose radius is A M, being the 

 sum of all the rings, will be as bfydx,or the area of the 

 curve L N I, which is found by substituting for y its value 

 in x, that is x". This fluent or area is therefore = t> C x" d x 



bx n+l b n+3 



f- C ; and C = - . Also, making P b = P E in 



n -J- 1 ?i + 2 



order to have the whole area of L N I, which measures the 

 attraction of the whole circle whose radius is F A, we have 



c n+l ^n+Z 



(x being =Pb = c) for that attraction. Then 



n + 1 n + 2 , 



taking D X' in the same proportion to the circle D E in which 

 D X is to the circle A F, or as equal to the attraction of the 

 circle D E, we have the curve E 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 DE in- 

 creases, 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 at- 

 traction for P is to that of a sphere whose diameter is equal 



a . A 2 - D . L a 3 



to the spheroid s shorter axis, as to - , A 



d* + A* a 3d* 



and a being the two semi-axes of the ellipsoid, 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 sup- 

 posed inversely as the squares of the distances. But if the 

 particle is within the spheroid, the attraction is as the dis- 

 tance 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 fdf X f dfF, in which / is the 

 distance of the particle from the point where the ring cuts 



