SI.] DETERMINATION OF CENTROIDS. 27 



It is easy to see how the formula has to be modified when 

 only one value or more than two values of y correspond to a 

 given value of x. 



50. In the most general case of any solid whatever the for- 

 mulae of Art. 15 assume different forms according to the system 

 of co-ordinates used. Thus for rectangular Cartesian co-ordi- 

 nates the element of volume is dv = dxdydz, and we have : 



M= j* J J/o dxdydz, M- ~x = J J J px dxdydz, 



M-y = J J J py dxdydz, M- z = j j j pz dxdydz. 



51. In polar co-ordinates, i.e. for the radius vector r y the 

 co-latitude 6 and the longitude </> (Fig. 10, p. 21), the element 

 of volume is an infinitesimal rectangular parallelepiped having 

 the concurrent edges dr, rdd, r sin Qd$ ; hence 



As ;r=rcos0, jj/ = rsin 0cos </>, ^=rsin^sin^>, the centroid is 

 determined by the equations : 



sn 



>x= | J Jpr 3 sin (9 co& 

 * sin 2 (9 cos 

 3 sin 2 ^ sin 



