POLAR FORMULA 



tdr 

 fwaeddVdr ' 





l. Apply the preceding forraul; 1 the 



centre of gravity of a hemisphere whose density varies as the 

 a* power of the distance from the centre. 



Take the axis of t perpendicular to the piano base of the 

 ^phere. Let a be the radius of the hemisphere, and 

 p - /ir*, where /A is a constant. First integrate with respect to 

 it to a ; we thus include all the elements Yikepqsi com- 

 prise pyramid OPQ8. Next integrate with res|>- 

 from to MT, we thus include all the pyramids in the slice 

 COM! . 1 mull v. integrate from = to 2ir; we thu* 

 include all the slices. Thus 



i and y each 0. 



right cone has its vertex on the surface of a 



e and its axis coincident with a diameter of the sphere, 



fiii-1 the centre of gravity of the solid included between the 



and sphere. Take the axis of * coincident with that 



of the cone ; suppose a the radius of the sphere, the sexni- 



ycrtical angle of the cone. The polar equation to the sphere 



is r2acos0, and to the cone = 0. Hence we hare 



_ 



n'f 

 * J 



x and y each = 0. 



