2 BELL SYSTEM TECHNICAL JOURNAL 



This led to my making a theoretical investigation of the mutual 

 inductance between direct-current grounded circuits which did, in 

 fact, show that the calculated numerical results are in reasonable 

 agreement with these actual experimental data. It is the purpose of 

 this paper to describe this work; the mathematical discussion of the 

 theoretical corrections for the incomplete penetration of alternating 

 currents into the earth will form the subject of another paper. 



2. Distribution of Current, Potential and Magnetic Force 

 with Direct-Current Earth Return Flow 



On the assumption of an infinite earth of uniform resistivity the 

 lines of flow and the equipotential surfaces for a direct current / 

 entering the earth at a point source at A and leaving the earth at a 

 point sink at B, both A and B being on the surface of the earth, 

 assumed flat, and the distance AB = 2b, are given by the equations, 1 



C 



-j = 5 (cos 0i — cos 2 ) 



= M7 1 -7j 



= sin§ (0i + 2 ) sin£ (0j-0i) 



, sin 2 ., b . .. 



= b , it — is small, 



(1) 



I 2ir\ri rj 



bp cos 6 . r b . ,, 



; — , it — is small, 



7r r r 



(2) 



where C is the total current flowing in the earth, from the source at A 

 to the sink at B, outside the current sheet of revolution defined by (1) ; 

 and V is the potential, with respect to the midplane, upon the equi- 

 potential surface of revolution defined by (2). 



These equipotential lines and stream lines are identical with the 

 equipotential lines and lines of force for a uniformly magnetized 

 filament. The formulas may be checked by regarding the return 

 flow from A to B as being due to the superposition of two flows, a 



1 The coordinates used in this paper (x u y, z), (x 2 , y, e), (x, y, e) and (r u d lt <f>), 

 (f2, 02, <t>), (r, 6, 4>) are rectangular and spherical coordinates with origins at A, B 

 and the midpoint of AB, respectively, the direction AB being the polar axis or posi- 

 tive x-axis in all cases, z being vertical, and <j> being measured from the earth's 

 surface in the plane perpendicular to AB. 



