6 BELL SYSTEM TECHNICAL JOURNAL 



4. Neumann Integrals for Return Flows 



The required mutual inductances of grounded circuits will be found 

 by means of the Neumann integral 



N= ffjf(cose/r)dIdSdids 



extended over every current filament in both flows. Since the earth 

 return portions of the two flows are independent of the flows in the 

 arbitrarily located conductors on the earth's surface, it is convenient 

 to divide the Neumann integral into four partial integrals which in- 

 volve either no return flow, one return flow or both return flows ac- 

 cording to the following formula 3 



N(jr- £) (*-#) = Njr* - Njfe - #<£*■+ Ne. 



= iVjKr-(i + *-l)A, by Table I, 



= A r JKr. (4) 



Checking the entries of Table I may be accomplished without per- 

 forming more than two integrations. It will be convenient to make 

 the integrals somewhat more general than is required in checking 

 the table and find Njr s and NjT'a where & is any flow in space from 

 A to B, which need not be coplanar points with the terminals a and b 

 of *, and •%"' is any flow in a plane parallel to the horizon plane be- 

 tween terminal points A' and B' . 



Consider first the part of a space return flow * which is radial from 

 a in connection with an element dS on any filament of current dl 

 of a flow & from A to B. The component dx of dS along the line x 

 from a to ds is the only component which need be considered, since by 

 symmetry the normal component contributes nothing to the Neumann 

 integral. As the total radial flow is to be taken equal to unity, the 

 amount flowing out through a ring, taken as the volume element, 

 lying between the spheres of radii 5 and s-\-ds and between the cones 

 making angles 8 and 8-\-d8 with x will be \ sin 8 dd. If this ring lies 

 at a distance r from dS the Neumann integral will be 



r.Ba r°° r* cos 8 sin 8 dd , , . , „ 



N= I dx I ds I = , r 2 = x 2 -\-s z — 2xs cos 8, 



Jao. »A) *A) Iy 



4J A a X 2 J S 2 J\ X - S \ 



3 Each term indicates the Neumann integral for the pair of flows designated by 

 the script letter subscripts, as explained in the note accompanying Table I. Both 

 (Jf— £) and (je—o) are arbitrary flows on the earth's surfa.e closed by earth return 

 flows from A to B and from a to b, respectively. 



