172 BELL SYSTEM TECHNICAL JOURNAL 



The conductivities and dielectric constants are taken as: 



Xo, Co for s > 



Xi, ei for - ^ <2 < 

 X2, €2 for z < — b 



where b is the distance between the parallel planes of discontinuity. 



The primary field of a doublet in the direction of the x-axis at 

 z = h IS given by 



where 



Hox' = A —^ = A I uJo{ru)du i) > z > h, 



J< Jo «o 



(4) 



r^ = x^ -j- y"^, 

 R" = r^ + (z - h)\ 



ao^ = u"^ + 7o^ 



The constant A which is the moment of the doublet is as yet un- 

 determined. From (3): 



H,' = -A- To' curl Ho/. 



and for 70 = 0: 



, ic 



Ho' 



CO 



curl -^ . 



For this case the magnetic force due to unit current in an element 

 dS is given by Biot-Savart's law ^ as 



// = dS curl ^ , 

 K 



so that 



icodS 



A = 



70^ 



The secondary fields are derived from Hertzian vectors having 

 components in the direction of the x and z axes. Components in the 

 direction of the ^'-axis are eliminated due to symmetry with respect to 

 the X — z plane. The resulting fields are then composed as follows: 



Hox = nox' + Hox", Hq, = Ho," 2^0, 



TL2x, Il2z s ^ - b, (5) 



* Abraham and Foppl, "Thcorie der Elektrizitiit," 7Lh ed., Vol. I, § 55, page 189. 



