CENTRE OP OKAYITY. 



It is often convenient to divide a solid into 

 nts. 



Let a series of planes be drawn ihnw^h the axis of - ; iho 

 solid is thus divided into wedge-sha]>< -1 slices such as COM/.. 

 Let a series of rijjht cones be described round the axis of z 

 having their vertices at 0\ thus each slice is divided into 

 pyramidal solids like OPQS. Lastly, let a scries of concentric 



spheres be described round as centre; thus each pyramid is 

 divided into elements similar to pqst. 



Let xOL = <f>, COP=0, Op = r, 

 LOM= A<, POQ = M, pt = Ar. 



Then pq is the arc of a circle of which the radius is r and 

 the angle A0 ; therefore pq = ?-A0. 



Also ps is the arc of a circle of which the radius is r sin 6 

 and the angle A ; therefore pa = r sin 0A<. 



1 1' nee, since the element pqst is ultimately a parallelepiped, 

 its volume is r 1 sin 0A0A</>Ar. 



Also the co-ordinates of its centre of gravity are ultimately 

 r cos <f> sin 0, r sin <f> sin 0, and r cos 0. Hence supposing its 

 density to be p, we Have 



