l"' Ci:- : , i:\vrrv. 



Thus the mass of the eighth part of the shell is - - 

 And the abscissa of the centre of gravity of the shell mi ;i-mv<l 

 along the semi-axis x is ^, by the preceding example. Thus 

 for the abscissa x of the centre of gravity we have 



. . x 



C" TTinii , , N 



J Tf-fw 



1 ["*>(*) <? 

 j 



where a is the semi-axis of the external surface corresponding 

 to the semi-axis x. When <j> (x) is given the integrations 

 may be completed; and when x is known, the other co-ordi- 

 nates of the centre of gravity maybe inferred from symmetry. 



(5) A chord of an ellipse cuts off a segment of constant 

 area; determine the locus of the centre of gravity of the 

 segment 



If a chord cuts off a segment of constant area from a ( 

 it is evident from the symmetry of the figure that the locus of 

 the centre of gravity of the segment is a concentric ci 

 Now if the circle be projected orthogonally upon a plane in- 

 clined to the plane of the circle the circle projects into an 

 ellip.se; and the segments of the circle of constant area project 

 into segments of the ellipse of constant area; al-o tin 



>'c circle projects into a second ellipse similar to the iiist 

 ellipse and similarly situated. 



Thus the required locus is an ellipse similar to the givm 

 ellipse and Similarly situated. 



This problem might have been solved without making use 

 of projections, in the manner shewn in the next example. 



(6) A plane cuts off from an ellipsoid a segment of con- 

 stant volume ; determine the locus of the centre of gravity of 

 the segment 



