TRANSMISSION OVER SUBMARINE CABLES 95 



Substituting the values of Ej' and ^'V+i from (7) in (15) gives 



- [ ^. ^]p^ , - ip Lj^ (/i + /2 + . . . + /.) = (Z/+.,, - z;:o /i + . . . 



~ ^j+l<j+l^j+l- 



(16) 



An equation of this sort may be obtained for each layer of dielectric 

 and these combined with equations (9) and the condition that the 

 electric field intensity in the sea water must vanish at infinity, 



E„' = Z'n'i 7i + . . . + Z'n'n I„ = 0, 



give n relations between /i . . . /„. In order that these shall be con- 

 sistent, the determinant of the coef^cients must vanish. 



Z'>\ — Zu, 

 Z31 — Z21, 



Zi2, 

 Z32 — Z22, 







Z.{3, — — — 



Zj+1,1 — Z"i + Zj, Zj-\-\o — Zj'2 -\- Zj, 



7" 



where 



Zj = 



Zn2, 



r2 



7" 



(17) 







Gj + ipCj 



ipLj. 



This is an equation in F^ of degree equal to the number of dielectric 

 layers; consequently, there are as many independent modes of propaga- 

 tion in the system as there are branches in the network of conductors. 

 From this point the method of determining the behavior of the 

 system depends upon conditions in the particular problem. For the 

 case where there are k dielectric layers separating the conductors 

 into ^ + 1 groups the current on the j'^ group may be written in the 

 form 



Ij = Ajy exp{-TiZ -\- ipt) + . . . + Ajk exp {-TkZ -{- ipt) 

 -\- Bj, exp (Fi 2 -f ipt) -\- . . . + Bjk exp {TkZ + ipl), 



where T^ . . . T^k are the k roots of the determinant (17) and Aji 

 , . . A^',, Bji . . . Bp are constants. These constants are not all in- 



