10 BELL SYSTEM TECHNICAL JOURNAL 



fact that >* is a special case of the general flow *. Any other entry in 

 Table II is now found in terms of three of the entries in this border, 

 thus 



= N(jtr-s)(A-a)=NjrA-Njfa-NsA-\-Ns a = (l-h-h)^ + Nsa, 



and we have the interesting result NSa — ^', the remainder of the table 

 follows. 



The result iVj^= —A may be readily checked directly since it 

 involves only the mutual Neumann integral between straight parallel 

 filaments, and by using the expanded form of the integral for equal 

 filaments beginning at a common perpendicular with opposite positive 

 directions 4 the result can be written down at once. 



The important difference A which is utilized in Table I may be 

 expressed in the following useful forms : 



A = (-Aa+Ab+Ba-Bb) 

 d 2 R 



dSds 



-If, 

 -If 



= 2 /sin i (0 2 -0i) sin [hidi+dj-e] ds, 



dSds 



dVdv+dWdw 

 R 



(5) 



where the notation is the same as for formula (3) above. The third 

 form of (5) shows that when the separation R is great the mutual 

 inductance varies inversely as the first power of the separation. 



5. Mutual Inductance Between Grounded Circuits Lying on 

 the Surface of the Earth 



It has now been shown that for direct currents the mutual inductance 

 between grounded circuits consisting of conductors lying on the surface 

 of the earth and grounded at their terminals is equal to the mutual Neu- 

 mann integral between the conductors alone, since in the complete Neu- 

 mann integral for the closed flows the total contribution of those parts 

 which involve the ground returns is zero. For low frequencies the 

 effective inductance can differ but little from the direct-current in- 

 ductance, and it is therefore of practical importance to investigate 



4 Mutual Inductances of Circuits Composed of Straight Wires. Physical Review, 

 5, pp. 452-458, June, 1915, formula (6). 



