TRANSMISSION OVER SUBMARINE CABLES 113 



1 Js ("■+0;) Jsoi 



We have ~ = '——- = { - \y '— ^ 



and 4^ = -^e'"(2T-<P-«i) = _el.,-.«(0+a,-, 



Therefore 



e^'-^y 1 ( - 1) 



~i{<p~aj(s-\)) 



:j L 1 C, ' 



^ 1.2 C^ ' 'J 



Equating the real and imaginary parts gives 



cos 50i (- 1)^ r s p , 



J— = — -s — cos saj - y — cos (0 - aj [s - 1]) + 



Pj ^ *- ^ O 



' ^\^^ ^^ I cos (2<^ - ay [5 - 2]) ] . 



sin 5<^j ( - 1)^ r . s S P ' fj. r UN . 



— -— ' = — - — sin saj + - - sin (</. - a,- [s - 1]) + 



^^|sin(2«-a,[.-2|) ]. 



Similarly 



log z,- = log (c; + z') 



= 1 r' 4-^-1 ^'-^l^' 



log Cy + ^^, 2 q2 + 3 Q^3 



= log c; + y ^ ~ -^^""^ ^e-e^-'^-'-). 



n = l •' 



Equating real and imaginary parts we have 



1 2 



log Pj = log Cj + ^ cos (0 - aj) - -y ^ cos 2 ((/)-aj) +...+, 



<^y = -7- sin (0 - aj) - -y ^ sin 2 (0 - aj) + . . . +. 



The following formulas may be derived in a similar manner: 

 cos 50j ( — 1)^ 



pj c^ 



[l+{^cos(e-7,)+^^^^cos2(e-7>) + ...+]. 



sin50j (-l)^ + ir5 r . ,. .. 5(5 + 1)^2 _ -, 



-^=^^L-^-sm(0-7.)+-3;2-7^^^"2^^-^^^ + --- + J' 



log py = log c cos {d - yj) - —-Y cos2{e - yj) . 



C Li 



