of the Attraction of Spheroids. 283 



and Poisson*. I hope the following remarks will make the 

 matter clear. 



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 Lapiace 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 <p 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 Qr^dr; di- 

 viding, therefore, by r, and integrating, we have £ $r 2 , or <p 

 multiplied into half the square of the length, for the value 

 ofV. 



Again, supposing the force to vary inversely as the square 

 of the distance, — the only hypothesis which 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 ma- 

 nifestly equal to <pr. 



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

 gular, 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'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'jp', ~P"p", the first two 

 terminated by the surfaces of the solid and of the sphere, the 

 third P'y in the same direction with them and equal to their 

 difference, without regarding which of them is the greater, and 

 suppose all the points p u taken according to the same law, to 

 trace the surface of a third solid. Let Pp, F'p', V'p", be edges 

 of three small pyramids with their other edges proceeding 

 from P, F, P", parallel, and having of cqurse the same solid 

 angle which we shall call <p, denoting by r, r', r", their respec- 

 tive lengths, and by V, V, V", the values of the function V 

 for each of them. Drawing pK perpendicular to PQ, the 

 attraction of the pyramid Pp in the direction of PQ will be 



* See Pontecoulant, vol. ii. p. 380 ; Foreign Quarterly Review, vol. v. 

 P .248> 



202 



