390 Prof. A. Gray on the Attraction of 



The resultant attraction due to such a shell exerted on a 

 particle at an external point is along the normal through the 

 point to the con focal surface on which the point lies. 



It can he seen at once without analysis that an elliptic 

 homceoid exerts no force at any point in the internal hollow r , 

 that is, that the potential has there a uniform value. For 

 the si' ell may he imagined constructed by homogeneously 

 straining a uniform thin spherical shell, within which, of 

 course, the potential is uniform. Such strain is that which 

 elongates all parallel dimensions of the shell in the same ratio, 

 and therefore leaves it of uniform density, though of varying 

 thickness, proportional to the length of the perpendicular 

 from the centre on the tangent plane at each point. If then 

 a cone of small solid anglw be drawn with its vertex at any 

 point in the hollow of the spherical shell, so as to intercept 

 two elements of the surface, these two elements exert equal 

 and opposite forces on a particle at the vertex. By the strain 

 the masses of these two elements are not altered, nor the ratio 

 of their distances from the vertex of the new r cone into 

 which the former one is changed. The elements, therefore, 

 still exert equal and opposite forces on a particle at the vertex. 

 Hence, as cones can he thus drawn so as to exhaust the shell 

 by pairs of elements, the shell as a whole exerts no force at 

 the common vertex. 



9. The same idea of division of a solid ellipsoid into 

 homoeoids had, however, occurred to 0. Rodrigues, and been 

 used by him in a " Memoire sur ^attraction des Spheroi'des/' 

 published in the Correspondance sur VEcole Polytechnique 

 (t. iii., 1816). The method adopted for the solution of the 

 problem of the attraction of a solid ellipsoid seems to have 

 been suggested by a previous paper by Gauss, and consists 

 in finding the variation, SW, say, in the ratio of the 

 potential of the ellipsoid at the given point, /?, k, I, say, to 

 the mass of the solid, when the squares of the semi-axes 

 a 2 , b 2 , (? are altered by the same small amount, &/>, say, that is 

 by the passage from the given ellipsoid to an adjacent confocal 

 ellipsoid. It is shown that for an external point SW = 0, 

 and for an internal point 



m= ^_ « + ~ 



£_.> . . - w 



3 o0 a 2 & 



4oabc\a 2 b 



From these results Rodrigues deduced the attraction of 

 the ellipsoid at the given point. It is easy to find from them 

 an expression for the potential. (See § 18 below.) 



The determination of SW depends on the evaluation of a 

 certain integral taken throughout the volume of the ellipsoid 



