26 INTRODUCTION TO DYNAMICS. [48. 



48. Homogeneous spherical solids can be treated by a method 

 analogous to that used for circular areas (see Art. 37). Thus 

 a homogeneous spherical sector can be resolved into infinitesimal 

 elements, each of which is a pyramid whose vertex lies at the 



centre of the sphere and whose base is 

 an infinitesimal element of the spherical 

 surface area of the sector. Such an 

 element, regarded as a pyramid (Art. 

 46), has its centroid at the distance | a 

 from the centre, if a be the radius of 

 the sphere. We may regard its mass as 

 concentrated at its centroid and have 

 thus the solid sector replaced by a homo- 

 geneous segment of a spherical area, of 



radius \a. It has been shown in Art. 41 that the centroid of 



such a segment bisects its height. 



Let 2 a be the angle at the vertex of the given sector (Fig. 1 2) ; 



then the height of the segment of radius \ a is \a(\ cos a) ; 



hence the distance He of the centroid of the solid spherical sector 



from the centre is 



x=\a cos a + ftf (i cos ) = f a (i +cos)=f a cos 2 -. 



49. In a homogeneous solid of revolution the centroid lies on 

 the axis of .revolution, since this line is an axis of symmetry 

 (Art. 47 ()). Taking this line as the axis of x, the equation 

 of the surface of the solid is determined by that of the curve 

 bounding the generating area, say y=f(x). 



We select as element the circular or ring-shaped plate of 

 thickness dx contained between two sections of the solid at 

 right angles to the axis of revolution (Fig. n, p. 22). The 

 centroid of each such element lies on the axis, and the volume 

 of the element is ir(yy?}dx, if y lt jj/ 2 , are the ordinates of 

 the curve corresponding to the same value of x. ^ ( 



We have, therefore, 



