8 BELL SYSTEM TECHNICAL JOURNAL 



To this is to be added the corresponding expression \{Ab — Bb) 

 for the radial flow converging on b, giving the result |A. As the path 

 of the line of flow between A and B does not enter into the result, 

 it is immaterial whether the flow is confined to a single filament or 

 is spread out in any way whatsoever in space, provided only all 

 stream lines extend from A to B, as assumed for &. Thus 



Njr, = ^A = JVjy. 



To find Njf'A let r and s be the distances of any element dS of a 

 line of flow forming a part of .#"' from a and from any element of a 

 plane radial flow from a, respectively, the projections of r and 5 on 

 either of the planes being x and y, which include the angle <£; s 2 = y 2 -\-z 2 , 

 r 2 =x 2 -\-z 2 . The component of dS parallel to x will be dx and this is 

 the only component which need be considered, since, on account of 

 the symmetry of the radial flow, the normal component in the plane 

 of flow contributes nothing to the integral. 



r Ba r x r 27r x-ycos<f) ydyd<t> 

 Jag ' Jo «/o x 2 -\-y 2 — 2xy cos </> 2-ws 



= \_ r Ba dx r~ ds r f x 2 -y 2 \ 



2-kJao. xJ z Jo \ x 2 +y 2 — 2xycos<f>) 



_ JL_ r Ba dx r°° r _ . _ 1 (* 2 +y 2 )cos0-2xy ~l 7r 

 2-ttJao x J z L x 2 -\-y 2 — 2xy cos J u 



= r * fds, 



J a a x "z 



since inspection shows that the two values of the definite integral 

 2-w and are to be used for 5>r respectively, and therefore 



N= I ' — - dx 

 x 



pBa r . 

 = / - 



= f 



J A' a 



B'a r d r 



r+z 

 = \ r-z log (r + s) 



= (B'a-A'a)-z\og^±^. 

 This is for the radial flow from a. Adding the corresponding expression 



