1875.] Theory of Duplex Telegraph/. 57 



Substituting therefore in the expression for P 

 h = d = o 



we get 



E" 

 2 {a" + 7') + 5 

 and 



P' = ^ , „ . — „. . ,„ ft' A' for Station I. 



P " = 9 / . f \ J. w ? X " " Stati ° n IL 



2 (a + c ) + b 



These two expressions do not as yet contain the balance conditions. 



The factors ** and , ,** ■ 



2 (a ' + c ) + o 2 (a + c ) + o 



are identical, namely : — 



(*! //' i 



2 («" + c") + b" 2 (a' + O + 6' Q 



7 / 7// 



Where Q = i [ 2 («' + a" + V + 2") + &' + &"} + ~- 



+ (a" + I") (a' + l' + V) + (V + Z') 0" + I" + 6") 



as can be easily calculated by sustituting for /a and c their known values. 



In the second investigation it has been stated why P' and P" cannot 

 be made maxima separately, and that we could do nothing else but make 

 their sum a maximum. In this case we have to do the same. Hence the 

 question to be solved is reduced to the following : 



I> = P' + P" = i. E " X ' + E ' X " 



is to be made a maximum with respect to the variables a, b, q and r, while 

 they are linked together by two condition equations, namely : — 



r' (of + c' ) — q'\/a' V = o balance in Station I 



and r" {a" + c") — q"\/a" b" = o „ „ II 



This general problem can be solved in exactly the same way as it was 

 in the second investigation. It is however not needed to do this again, 

 since the general solution can be written down from inference, after having 

 solved the special problem for a line which is perfect in insulation. 



Suppose that i = go , or at least very large as compared with V + V 

 = L, then obviously P' and P' become identical without condition, 

 namely : — 



V = P" = P = E ^q^/a+_r\/b_ 

 4 L + 2a + b 



while the two balance equations become also identical namely : — 

 2 2 \/« b — r(4<a + b + 2L)=o 



