( 349 ) 



I. ON A DIFFICULTY IN THE THEORY OF THE ATTRAC- 

 TION OF SPHEROIDS. 



[Transactions of the Royal Irish Academy, VOL. xvi. p. 237. Read May 28, 1832.] 



AN approximate theorem, discovered by Laplace, 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. 



Let 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 Laplace has done, by the letter V. 

 It is necessary to find the value of Y for a pyramid of inde- 

 finitely 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 r from the vertex is 0r 2 c?r; dividing 

 therefore by r, and integrating, we have ^<t>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 <f>r. 



Let us now consider a solid of any shape, regular or irregu- 

 lar, terminated at one end by a plane to which the straight 

 line PQ is perpendicular at the point P; and let there be a 

 sphere of any magnitude, whose diameter P f Q f is parallel to 

 PQ. Let P" be a fixed point, and from the points P, P 7 , P", 

 draw three parallel straight lines Pp, Pjt/, P"p", the first two 



* See Pontecoulant, Theorie analytique du systeme du monde, tome ii. p. 380 ; with 

 the references there given. 



