< r.NTKi: or GKAYITY. 



small. Let X, y, Z be the C0-onlinatr< of any ]>"int of the 



. /,- tlu- thirkm-ss at that point, A# tl. 



an cK-iiu-iit of the surface thnv, thm k&S is ultimately the 

 volume of this clement, and a-, y, z the co-ordinates of its 

 re of gravity; hence 



_ fkxJS 



and similar expressions hold for y and z. 



It may be shewn (see Integral Calculus, Art. 17> that if 

 we take A/S such that its projection on the plane of (x, y) is 

 the rectangle Ax Ay, 



H 



Ex. The surface of the eiglith part of a sphere 

 Here 



[[ xds.l,, 



*~ 



dxdy 



First integrate with respect to y from y = to y = *J(a* a?) ; 

 we thus include all the elements that form the strip of 

 face of which LlmM is the projection on the plane of (x, y); 

 see fig. to Art. 128. 



., r - fbrxdx fxdx 



therefore x = f ^ = , 7 . 



/^TT c/-c / cto 



The limits of the integration for x are and a ; 

 therefore x = Ja. 



Similarly y = |a, 2 = ^a. 



