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 



A B C 



- + - + -- 4^* (5) 



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.p, B<?, Cr, 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', CO', 

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



Now conceive three chords Ap, A>', A^", 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 o> ; 

 and from B and C let two systems of chords Bg, B/, B<?", and 

 Cr, Cr', Cr", be drawn, each system forming a very small pyra- 

 mid whose three edges are parallel to the three edges AJO, 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 

 /u/jow.Aa, ju/pw.B/3, JU//QW . Cy respectively ; and, therefore, the 

 sum of those attractions divided respectively by the lengths of 

 the axes will be 



Aa B/3 C 7 



Proceedings of the Royal Irish Academy, VOL. u. p. 525. 



