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 + dg 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=Ve+ (BP — c*)cos*g, p'=V ec? + (a — c) cos? g, 
