1S75.] Theory of Duplex Telegraphy. 55 



To have S' therefore for any variation as small as possible, we must 

 makey= b -j- d. Substituting this value of/" we get an expression for S' 

 which shows that it has an absolute maximum for b but no minimum, from 

 which we conclude that b should be made either very much smaller or very 

 much larger than the value which corresponds to a maximum of S, but no fixed 

 relation between b and d or a can be found. 



In order to prove that b -f- d = f is the solution, we must now show 

 that it also makes D as small as possible. 



But as D = — 



we have only to show that the regularity condition b -f d = f, makes P 

 either as large as possible, or, which would be still better, a maximum. 



Now 



P' = A" ft X 

 where A" is the current which enters the line at point 2 (Fig. 2) when 

 Station II is sending alone, while /*' is the factor which determines the loss 

 through leakage of the line, and A.' is the factor to which the magnetic force, 

 exerted by the current A' fi in Station I, is proportional. 



fxf as well as A' are functions of the resistances in Station I only* but 

 not of those in Station II. 



Now for constant values of \j! and A' (i. e. leaving everything in Station I 

 constant) P' becomes larger the larger A" is : 



N" 



Substituting its value for N", and dividing numerator and denominator 

 by b" + d", we get 



E" 



A" = 



J + b" + d" + a + ll + c <- L + wiri') 



Supposing balance in Station II rigidly fulfilled, we have 

 \b"+ d") m" — {a" + h" + c") n" = o. 



.-. c" -= (b" + d") ™, — (a" + h"). 



Substituting this value of c" in the expression for A" and reducing, 

 we get 



E" r" </b" 



A" = 



f" r" \/b" + q" {b n + d"+f") s/d' 



i x/ __ , . r 



+ V + P " /'+ V + d> 



