TRANSMISSION OVER SUBMARINE CABLES 103 



The series (34) can, therefore, be written 



^-1 N-i ^ . . 



E = AiK- logr) + BoiNK - ^logp,)+ ^^ ^ Bs S£^i£^, (35) 



j = O 1= o s - 1 ■' 



in which B^ has absorbed the constant quantities. From this, the 



magnetic intensity in the sea water can be obtained by differentiation. 



Inside any armor wire, at the surface, the field intensities are 



E = CoJo{^) + CiJi(^) cos </, + ... + C„J„(^) cos n0 + . . . +, (36) 

 H^ = ^[_CoJo'{0 + CJi'a)cosct>-\- . . . +C„j;(^)cosn0+ . . . +1(37) 



where ^ = ai\/^Tz\yLpi, 



X and y. being the electrical conductivity and the magnetic perme- 

 ability, respectively, of the material of the armor wire. The quan- 

 tities a and 4> are centered on the axis of the wire. 



In order to determine the coefficients A, B^, Bi, — , Co, Ci, — we 

 make use of the fact that the electric and the magnetic field intensi- 

 ties are continuous at the surface of the wire. It is obvious, however, 

 that nothing can be learned by equating (35) and (36) since they 

 are formally dissimilar. We therefore transform ■* the various terms 

 of (35) to a common axis which coincides with the axis of one of the 

 armor wires, hereafter called wire " zero," and the electric field 

 intensity in the sea water, close to the surface of the armor wire, is 



E = {A -\- NBo) K - A\ogc - Bo log (aciCz . . . c„_i) - So 

 + [gi/f - f (^ + Sn Bo) + ^ 2i1 cos 



+ [ g2/f 2 + 4 ("^ + ^'' ^°^ + i^ ^0 """^ ^*^ ^^^^ 



where 



+ [5"/^+ ^-^ (^ + S„„ Bo) - ^—^ 2„1 cos n4>, 



2o = -311^1 — 022^2 -T ^33^3- 



2i = So2qi — 2 Sisq2 + 3 52493- • • • , (39) 



22 = 1.2 SuQi — 2.3 Soiq2 + 3.4 S^qa- ■ . . , 



23 = 1.2.3 52451 - 2.3.4 5i592 + 3.4.5 5o6g3 



* See Note II. 



