On the Attraction of Ellipsoids. 355 



angles are 9 and 9 + d9 respectively, since it is made up of the 

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

 elementary pyramids CP, is 



.S(OE 2 w) = - ^cos 2 0t/0.S(OE 2 6fy), 



i/ C 



dQ being the angle between two consecutive sides of the cone 

 generated by the revolution of OE. 



But S (OE 2 d(f>) 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 n sin 0, r z sin 0, 

 and whose area is TTT^TZ sin 2 0, r l and r 2 being the maximum and 

 minimum values of OE : the superficial area o4ke portion of 

 the cone within the ellipsoid is therefore 7rnj<sm& L 



Hence it follows that 4*A 



**?**& 



The attraction on the point C of the part of the 

 tained between the two cones of revolution, whose common 

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

 spectively, is therefore 



c 

 where 



2 sin 2 



On substituting these values, the expression given above 



becomes 



abc cos 2 sin 06/0 



^ 



(a 2 cos 2 + c 2 sin 2 0) ^/ (6 2 cos 2 + c 2 sin 2 0) ' 



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

 extremity of the least axis is 



7T 



abc cos 2 sin 



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

 2 A2 



