A PARTICULAR METHOD IN CENTROIDS 229 



9, // two equal spheres intersect to find the Center of Mass 

 of homogeneous matter filling either of the enclosed spaces not com- 

 mon to hoth, when the spheres are about to coincide. 



Let the whole enclosed space be supposed filled with homoge- 

 neous matter and let Fig. 5 represent a section of it through the 

 centers C and C of the spheres, g and g' now representing the 

 required Centers of Mass, namely of the portions of which DAD' A' 

 and DB'D'B are sections. The volume of DAD' A' in the limiting 

 position, is by reasoning similar to that used in the case of the over- 

 lapping lamina?, =CC'X area of the circle of intersection of the 

 spheres in the limiting position, = CC'XTrr-. If now DAD 'A be 

 supposed transferred to the position, of DB'D'B, carrying its Center 

 of Mass from g to ^', the Center of Mass of the sphere DAD'B 

 travels from C to C ' . We have then — 



CC CC'Xtt/"^ 



.'. the distance of the required Center of Mass from the center 



2r 

 of the sphere is — •• (17) 



If the spheres intersect in any position^ we can, as in the case 

 of the circular lamince just preceding, find the Center of Mass of 

 the part of either not common to the other. 



Let Fig. 5 represent a section of the spheres through their cen- 

 ters, g and g' being now Centers of Mass of the solids DAD 'A', 

 DB'D'B, and zCDC'=2^. 



The volume of D AD ' A ' = ~'^'^' "''' ^ (2+cos^ ^)(') 

 ' ' 2/- sin e 27rr' sin (9(2+cos' 0) 



(1^ Vol. DAD' A'= Sphere DAD'B-2 Segment DA'D'= Sphere-2 Sector C'DA'D'+2 Cone 

 C'DD'. 



