296 iv I:M. 



It will be seen that the demonstration establishes ft 

 thing more than Ivory's theorem enunciates, nainelv tl, 

 lowing: take any elementary prism of the first ellipsoid the 

 edges of which are chords parallel to an axis, and take the, 

 :;ientary prism of the second ellipsoid; then 

 the attractions of these prisms resolved parallel t< the axis on 

 the corresponding points are as the products of the 

 axes: and Ivory s theorem follows from the fact that the 

 ellipsoids may be supposed to be formed of corresponding 

 elementary prisms. 



We observe that one of these ellipsoids lies entirely within 

 the other. For if not the points at which they intersect would 

 lie on the curve of which the equations are 



the co-ordinates of the points of intersection must therefore 

 satisfy the equation 



Since the ellipsoids are confocal this becomes 

 a? ?/* ^ 



and this equation can only be satisfied by supposing x, y, 

 and z to vanish; and these values do not satisfy the 

 tions to the ellipsoids. Thus the ellipsoids do not im 

 at any point. 



II' nee to find the attraction of an ellipsoid of which the 

 semi-axes are a, 5, c on an external particle of which the co- 

 ordinates are /", #', h\ we must first calculate the attraction, 

 resolved parallel to the axes, of an ellipsoid of which the 

 axes are a', b', c on an internal particle of which the 

 co-ordinates are/,//, h; these six quantities being determined 

 hv the equations 



