24 INTRODUCTION TO DYNAMICS. [45. 



hence with the usual Gaussian notation 



3* df dz df 

 =/=/, =^-=4, 

 dx dx By dy 



F*=-j>> F y =-g, F g =I, 

 which gives M=\ J x V ^ l +P* + f dxdy, 



px V i +/ 2 + g 2 dxdy, 

 M-y= ] \ *py V i +/ 2 + q* dxdy, 



JVl J*,. 



M- z= ] )pz^/i +p* + q 2 dxdy. 



*/V */! 



In the case of a homogeneous spherical surface 

 =.a 2 , we have / = dz/dx= x/z, qds/dy=y/z\ hence 

 / 2 + ^ 2 = ^, so that the last of the above formulae gives 



5 . z = a J dxdy = a S z , 



where 5 is the area of the surface and S z the area of its pro- 

 jection on the plane xy. The formula shows that the distance z 

 of the centroid of any spherical area 5 from a plane passing 

 through the centre is equal to the radius a multiplied by the 

 ratio of the projection 5, of the area on the plane to the area 

 itself. 



45. We proceed to the methods of finding the centroids of 

 volumes or solids. 



Considerations of symmetry make it clear that the centroid 

 of a homogeneous parallelepiped lies at the intersection of its 

 diagonals ; similarly, that of a homogeneous prism or cylinder 

 coincides with the centroid of the area of its middle section (i.e. 

 a plane section parallel to, and equally distant from, the bases). 



