On a difficulty in the Theory of the Attraction of Spheroids. 
By James M‘Cutxacn, A.B. 
Read May 28, 1882. 
AN approximate theorem, discovered by Laprace, and relating to the attraction of 
a solid slightly differing from a sphere, on a point placed at its surface, has given rise 
to many disputes among mathematicians.* I hope the question will be set in a clear 
light by the following remarks. 
Left us consider the function which expresses the sum of every element of a solid 
divided by its distance from a fixed point, and let us denote it, as Laptace has done, 
by the letter V. It is necessary to find the value of V for a pyramid of indefinitely 
small angle, the fixed point being at its vertex. Calling ¢ the small solid angle of the 
pyramid (or the area which it intercepts on the surface of a sphere whose radius is 
unity and centre at the vertex), it is manifest that the element of the pyramid at the 
distance 7 from the vertex is gr*dr ; dividing therefore by 7, and integrating, we have 
4@r°, or » multiplied into half the square of the length, for the value of V. 
Again, supposing the force to vary inversely as the square of the distance—the 
only hypothesis that can be of use in the present inquiry—the attraction of the same 
pyramid on a point at its vertex, and in the direction of its length, is manifestly equal 
to 9r. : 
_ Let us now consider a solid of any shape, regular or irregular, terminated at one 
end by a plane to which the straight line PQ (Fig. 1,) is perpendicular at 
ie the point P ; and let there be a sphere of any magnitude, whose diameter P’ Q’ is 
parallel to PQ. Let P” be a fixed point, and from the points P, P’, P”, draw three 
parallel straight lines Pp, P’p', Pp", the first two terminated by the surfaces of the 
solid and of the sphere, the third Pp’ in the same direction with them and equal to 
their difference, without regarding which of them is the greater, and suppose all the 
. 
_* See Pontécoulant, Théorie analytique du systéme du monde, Tome II. p. 880; with the references 
there given. 
