120, 121] POLARIZED DISTRIBUTIONS. 229 



so that the face dydz next the origin will have the charge 



(A +^ dx } dydz. 

 \ ox J 



This is superposed on the charge + Adydz, so that the whole 

 charge on that face is 



dA , 

 ^dxdydz. 



Similarly the faces dzdx and dxdy on the sides farthest from the 

 origin, have the charges 



dB, , , 



~ dxdydz. 



W i 

 and dxdydz. 



oz 



If we consider these faces as belonging to the element dr, while 

 the opposite faces belong to adjacent elements, the whole charge 

 belonging to the element dr is 



(dA dB 



Thus the charge per unit volume is 

 (f\ (dAdB 



' ts + * + 



Integrating this throughout the volume occupied by the body 



(7) = n{A cos (nx) + B cos (ny) + C cos (nz)} dS 



= [[/ cos (In) dS = - jf<rdS, 

 so that the total mass of the volume and surface charges is zero. 



121. Induction. Comparing the expression for the density 

 with the ordinary expression for the density as l/4?r times the 

 divergence of the force, we have 



(dA dB a 1 MX 8F dZ 

 p= " 



