On the Attraction of Ellipsoids. 359 



means of arcs of the focal ellipse and hyperbola respectively. In 

 consequence of the third focal conic of the ellipsoid being ima- 

 ginary, no direct geometrical representation can be given for the 

 attraction A on a point placed at the extremity of its greatest 

 axis. It will, however, be found, as was intimated above, that 

 a simple relation exists between the three attractions, which 

 enables us to represent this last by means of arcs of both focal 

 conies. 



The relation alluded to is 



This can be easily proved by the help of the following geo- 

 metrical theorem : 



If from the extremities A, B, C of the three axes of an ellip- 

 soid three parallel chords A/?, B<?, O, be drawn, and if these 

 chords be projected each on the axis from whose extremity it is 

 drawn, then the sum of these three projections, Aa, B/3, Cy, 

 divided respectively by the lengths of the axes AA', BB', CC', 

 on which they are measured, will be equal to unity. 



Now conceive three chords A/>, A//, Ap", to be drawn from 

 A, making each with the other two very small angles, and so 

 forming a pyramid with a very small vertical solid angle w ; 

 and from B and C let two systems of chords B^, B/, Bj", and 

 Or, C/'', Or", be drawn, each system forming a very small pyra- 

 mid whose three edges are parallel to the three edges A.p, A//, 

 Ap", of the pyramid which has its vertex at A. 



The attractions of the three pyramids, reduced each to the 

 direction of the axis passing through its vertex, will be equal to 

 /4//o(u.Aer, ju//3w.B/3, ////ow.Cy respectively; and, therefore, the 

 sum of those attractions divided respectively by the lengths of 

 the axes will be 



An B/3 C 7 \ 



* Proceedings of the Royal Irish Academy, VOL. n. p. 525. 



