MUTUAL IMPEDANCES OF GROUNDED CIRCUITS 9 



for the radial flow towards b, we have finally for the complete integral 



N^ = (-A'a + A' b + B>a-B' b) -^ S { A 2Xf { B B ;X% (4a) 



which becomes, if 2 = 0, 



NjTA=A=Njr Je . 



The first line of Table I can now be filled in at once since the in- 

 tegrations have shown that the first two values of k are 1 and \\ the 

 next two entries are also \ since by symmetry NjTs = Njr<? = Njr a ; 

 Njfn = Njrx = since the nadir and zenith flows are perpendicular to 

 the -X" flow in the horizon plane. The remaining six entries in the 

 first row are for closed flows which are expressed as differences be- 

 tween the flows already considered, and the corresponding &'s are 

 the differences of the k's for the component flows. The first column 

 of &'s may also be filled in, the table being symmetrical, since inter- 

 changing capital and small script letters leaves N unchanged and >* 

 is a special case of ■*•. 



The second row of the table involves only special cases of Ns/ 

 and only the values \ and occur. 



From the flows included in the table nine pairs of closed flows may 

 be formed having zero mutual inductances, because one of the closed 

 flows of each pair has no magnetic field below the surface of the 

 earth and the other closed flow includes no current above the sur- 

 face of the earth, and thus there is no interlinkage of induction be- 

 tween the two closed flows. The portion of Table I referred to is 

 repeated in Table II, where the flows at the top are those for which 

 any difference such as (-*—«) has no magnetic field below the earth's 

 surface, just as (-*"— &) has no magnetic field above the earth's surface, 

 as was proved above, while no flow at the side penetrates above the 

 earth's surface. 



The top row of Table II includes only values of k already found, and 

 the remainder of the first column follows from symmetry and the 



