On the Attraction of Ellipsoids, 355 



angles are 9 and + dO respectively, since it is made up of the 

 components in the direction CO' of the attractions of all the 

 elementary pyramids CP, is 



cos 2 6 . S (OR 2 <) = - cos 2 6 dO . S (ORV0), 

 c 



d(jt being the angle between two consecutive sides of the cone 

 generated by the revolution of OR. 



But S (OR 2 d<j>) is equal to twice the superficial area of the 

 part of this cone which is enclosed within the ellipsoid. More- 

 over, the projection on the plane ab of this portion of the sur- 

 face of the cone is an ellipse, whose semi-axes are >i sin 9, 1\ sin 0, 

 and whose area is trr\ r 2 sin 2 9, r, and r 2 being the maximum and 

 minimum values of OR : the superficial area of the portion of 

 the cone within the ellipsoid is therefore 7m r 2 sin 9. 



Hence it follows that 



The attraction on the point C of the part of the ellipsoid con- 

 tained between the two cones of revolution, whose common 

 vertex is at C, and whose semi-angles are 9 and 9 + d9 respec- 

 spectively, is therefore 



cos i 9d9.r l r 2 sin 0, 

 where 



1 If cos 2 9 sin 2 0\ , 1 / 



- = + , and - = / + 



>'i >/\ c 2 a* j r 2 >/ 



/cos 2 sin 2 

 

 \ c 



On substituting these values, the expression given above 



becomes 



abc cos 2 sin 0f/0 



" Ar/r^ / / 9 9/ 9 *9/l\ / l 1 '> *>"/! .1 t> /\\ * 



r </(a* cos 2 + c 2 sm 2 0) v^(?^ 008*0 + c 1 sm 2 0) 



Hence the attraction of the solid ellipsoid on the point C at the 

 extremity of the least axis is 



7T 



2 abc cos 2 si 



( 2 cos 2 + c 2 sin 2 0) v/(6 2 cos 2 + c 2 sin 2 0)' 

 2 A2 



