298 BELL SYSTEM TECHNICAL JOURNAL 



We still have a volume-integral in the formula; but there is a very 

 noted theorem whereby it may be transformed with sign reversed into 

 a surface-integral over the two surfaces, S and the sphere, which bound 

 the region of integration. According to Gauss' Theorem, any vector 

 function satisfying certain simple conditions of continuity throughout 

 a region enclosed by a surface enjoys this property: its volume- 

 integral through the region is equal to the area-integral, over the 

 enclosing surface, of the projection of the vector upon the direction of 

 the oMtoar J-pointing normal. This latter is the same in magnitude 

 and opposite in sign to the projection upon the direction of the inward- 

 pointing normal, which it is traditional to prefer. The theorem is 

 not valid, if the vector should exhibit certain singularities within the 

 volume; one of the reasons for introducing the infinitesimal sphere is 

 that the vector W has a singularity at the origin, which point must 

 therefore be excluded from the volume of integration. 



Remembering the definition of W (equation 9), we see that its pro- 

 jection upon the direction of the inward-pomt'mg normal at any place 

 upon either surface is 



W„ = Wcos {n, r) = W^ cos (n, x) + Wy cos (w, 3;) + W^ cos (w, 2) 

 = - lid U/dx) cos {n, x) + (d U/dy) cos (w, y) -f (d U/dz) cos (n, s)] 



= ~{dU/dn), 



in which (dU/dn) stands for the rate at which the function U, owing to 

 its explicit dependence upon x, y, and z, varies as one moves imvard 

 along the normal to the surface. The distinction drawn in the fore- 

 going pages between partial and total derivatives must be remembered. 

 The partial derivative dU/dn existing at any point P and moment / 

 is equal to the value which the corresponding derivative ds/dn of the 

 function 5 possessed at that same point at the earlier moment (/ — r/c). 

 The quantity Wn is to be integrated over the surface of the sphere 

 and over the surface 5. However, the integral over the^sphere vanishes 

 as the radius of this latter approaches zero, for the same reason — and 

 under the same restriction — as caused the integral of the first term in 

 (25) to vanish. This leaves us with nothing but the integral of Wn 

 over the surface S, so that eventually: 



fdiv H^^F = - [ ^{dU/dn)dS. 

 [irSo = I dS cos (w, r) -— i 



r I r On 



(28) 



(29) 



