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tion of ellipsoids. Having written out a simple demonstration of a 

 very elegant known theorem relating thereto, I shall subjoin it sepa- 

 rately. 



" The proposition above alluded to depends on the following theo- 

 rem, which may be very simply proved : — If from the extremities 

 A, B, C of the semiaxes of an ellipsoid whose centre is O, there be 

 drawn three parallel chords A*, B^, Cy, meeting the surface of the 

 ellipsoid in «, /3, y ; and if a perpendicular from u on OA meet it in r, 

 a perpendicular from /3 on OB meet OB in s, and a third from y on 

 OC meet OC in « ; then will 



Ar B^ C< _ 

 AA' "*" BB' "^ CC^ ~ 



AA', BB', CC are the whole axes. 



" The proposition itself is this : — If the particles of a homogeneous 

 ellipsoid attract inversely as the square of the distance, and if a, h, c 

 be the semiaxes, and a^, b„,Co its attractions on points placed at 

 their vertices, then will 



a ' b c 



" The attractions are here, as is usual, represented by lines ; the 

 attraction of an indefinitely small part of the solid being represented 

 by its volume divided by the square of its distance from the attracted 

 point. 



" The attraction being thus measured, it is evident that if from the 

 vertex of a pyramid, whose transverse sections are indefinitely small, 

 as a centre, with any radius, a sphere be described whose surface is 

 penetrated by that of the pyramid, the attraction of the pyramid on 

 a point at its vertex will be to its length, as the intercepted surface 

 of the sphere is to the square of its radius. 



" Let A, B, C be the three vertices of the axes, and from them let 

 parallel chords A«, B/3, Cy be drawn ; from whose extremities «,/8, y 

 let perpendiculars be let fall on the respective axes, meeting them 

 in r, s, t; then by the preceding theorem 



I 



