342 VIII. NEWTONIAN POTENTIAL FUNCTION. 



and in integrating with respect to y and z we cover the whole of 

 the projection of the surface 8 on the YZ- plane. At the same time 

 we cover the whole of the surface 8, so that the volume integral is 

 transformed into a surface integral, 



taken all over the surface 8. 



In like manner we may transform the two similar integrals 



Applying this lemma to the function 



dx* 



where both C7 ; V and their derivatives in any direction are uniform 

 and continuous point -functions in the space r, we have 



/ \ -j o 



Similarly putting for W, 



dy' 

 and integrating with respect to y } 



and for 



Adding these three equations, and performing the differentiations, 



icn C C C[TJ(^ V . &V . d*V\ JL dUd 



19) J J J [ u (l^ + W + '^) + ^z* 



Jj U (^x C S ( nx ) + If C 



dV . , 



dr 



