382 Mr. Geokge J. Allman's Account of 



OR^cos^e 

 fifpw . CQ= 2 fifpw . . 



Now suppose the radius vector OR to revolve around the axis OC, then 

 the attraction on the point C of the portion of the elHpsoid bounded by the 

 two cones of revohition, whose serai-angles are 6 and 6 + dO respectively, 

 since it is made up of the components in the direction CC of the attractions of 

 all the elementary pyramids CP, is 



^ coshes (OK^o.) = ^ cos-edes. (OR^ dtp), 



c c 



d(j) being the angle between two consecutive sides of the cone generated by the 

 revolution of OR. 



But S {OR-d<p) is equal to twice the superficial area of the part of this cone 

 which is enclosed within the ellipsoid ; moreover, the projection on the plane 

 ah of this portion of the surface of the cone is an ellipse whose semi-axes are 

 ri sin e, r, sin 6, and whose area is m-ir. sin' 6, I'l and rj being the maximum 

 and minimum values of OR : the superficial area of the portion of the cone within 

 the ellipsoid is therefore Trr, r, sin 6. 



Hence it follows that 



2 (OR-fZ0) = 27rrir2sin0. 



The attraction on the point C of the part of the ellipsoid contained between 

 the two cones of revolution, whose common vertex is at C, and whose semi- 

 angles are 6 and + dO respectively, is therefore 



— ^^ cos'' eder^r^ sine, 

 c 



where 



1_ /fcosH sWe \ J^_ /f coa'e sWe \ 



tT ~ V \ 0- a' J ' ^° r7 ~ V \ c' 6^ y ■ 



On substituting these values, the expression given above becomes 



abc cos^ 6 sin BdO 

 ^'^^■^P /(a- cos^e + c' sin' 6) V{b- cos' 6 + c' sin2 6i)" 



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

 of the least axis is 



