62 DEFINITE INTEGRALS. [iNT. III. 



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

 the projection of the surface 8 on the F-f-plane. On the other 

 hand we cover the whole of the surface S, so that the volume 

 integral is transformed into a surface integral, 



taken all over the surface S. 



In like manner we may transform the two similar integrals 



cos 



Applying this lemma to the function 



where both U t V and their derivatives in any direction are uniform 

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



Similarly for FF= U 



dy 



and for W=U~- 



Adding these three, and performing the differentiations, 



rrrr /9 2 F 9 2 F 8 2 F\ dUdV difdV d_UdV~] , 

 JJJ L \W*W* W) + fafo ty ty fa ^\ 



= - II U f = cos (na&) + ^- cos (ny) + y- cos (f)J dS, 



