TRANSMISSION OVER SUBMARINE CABLES 105 



which expresses C„ in terms of </„. Multiplying (43) by f/„' (f) and 

 (45) by /„ (f), and subtracting gives 



g„ = ( - 1) "\„r-" \- {A + SnnBo) - ^ 1\1 . n = 1 , 2. . . . 00 (47) 

 where 



, _ nnJn (^) - ^Jn' {Q , .„, 



^" - ny.Jn iO + ^/«' iO- ^^^^ 



From the infinite set of simultaneous equations (47) the infinitely 

 many variables q„ may be determined in terms of A and B^.^ 



We have thus determined the arbitrary constants C^ ■ . . C„ and 

 qi . . . g„ (or Bi . . . B„) as functions of A and Bg. It remains to 

 express the latter quantities in terms of physical quantities. If /i 



is the current in the armor then -— is the current in a single wire. In- 



N 



tegrating (41) completely around the armor wire " zero " gives, 

 therefore, 



2pi^^-Bo. (49) 



Similarly, if /„ is the current in the core conductor, we find 



2piIo = - A. (50) 



We can, therefore, express all the arbitrary constants as linear, homo- 

 geneous functions of !„ and Zi. 



To determine the relation between these currents, we have from 



(49) and (44), 



CJo (^) = ^, (51) 



where 



2txip Jo iX) 



Z = 



^ Jo'iO' 



Substituting (49), (50) and (51) in (42) gives 



~ Ii= -2ip {Io + Ii) K-\-2iplo\ogci-2ip^\og(ac, . . . c„) (52) 



— (Siiqi — Siiqi + S^^qy, — . . . ), 



from which, since qi . . . q„ are functions of /i and !„, the ratio /o//i 

 can be obtained. 



"See Note III. 



