44-] DETERMINATION OF CENTROIDS. 23 



43. To find the centroid of a portion of any curved surface 



F(x y y, z) = o, we have only to substitute dM=pdS in the 

 general formulae of Art. 15, and then express dS by the 

 ordinary methods of analytic geometry. 



Denoting by /, m, n the direction cosines of the normal 

 to the surface at the point (x, y, z), and putting for shortness 

 dF/dx=F x , dF/dy = F y , dF/dz=F z , we have 



jc__ dydz _ dzdx _ dxdy 

 I m n 



F x ~ F~ F 

 Hence, substituting 



F. 



in the formulae of Art. 15, we find 



F. 



where the integration is to be extended over the projection of 

 the portion of surface under consideration on the plane xy. 

 The equation of the curve bounding this projection must be 

 given : it determines the limits of integration. It is obvious 

 how the formula has to be modified when the projection of the 

 area on either of the other co-ordinate planes be given. 



The expressions for M-x, M ' y, M-z differ from the above 

 expression for M only in containing the additional factor 

 jr, y, z, respectively, under the integral sign. 



44. If the equation of the surface be given in the form 

 z=f(x,y\ as is frequently the case, we have 



F(x,y,z)=z-f(x,y); 



