367 



The following note by Professor Mac Cullagh, on the 

 attraction of ellipsoids, was read. 



" The object of the present note is to show how the final 

 integrations by which the attraction of a homogeneous ellipsoid 

 is found, when the force varies inversely as the square of the 

 distance, may be performed geometrically ; and thus to com- 

 plete the synthetic solution of a celebrated problem. It has 

 been always supposed that the utmost geometry can do is 

 to arrive in a simple way at the differential expressions on 

 which the attraction depends, leaving the further treatment of 

 the question to the integral calculus ; but we shall see that, by 

 putting the differential of the attraction under a certain form, 

 the integral is at once obtained, and that in a very elegant 

 shape, by geometry. 



" To avoid useless generality, we shall suppose the attracted 

 point to be at the extremity of an axis of the ellipsoid, as it is 

 well known that the solution of this particular case enables us 

 find the attraction wherever the point is placed. Let O be the 

 centre of the ellipsoid, and A, B, C the extremities of its semi- 

 axes, the lengths of which are denoted by a, b, c respectively, 

 a being the greatest, and c the least. And first, suppose the 

 attracted point to be at C, the extremity of the least semiaxis. 

 Let two right lines passing through C, and making respec- 

 tively the angles ^ and (^ -\- d<p with OC, revolve within the 

 ellipsoid, describing two right cones, of which OC is the com- 

 mon axis, and which include between their surfaces a differen- 

 tial portion dM of the volume of the ellipsoid. The attrac- 

 tion of the matter contained in dM is evidently in the direction 

 of OC. 



" Now let us consider the focal ellipse having its centre at 

 O, and lying in the principal plane which is at right angles 

 to OC. Let E be the extremity of the major axis of this 

 ellipse ; and putting 



p — \/c^ + {b' — c^) cos '^(j), p' = \/c^ -f (a^ — c^) cos^ ^^ 



