of Duplex Telegraphy. 135 



Further, substituting 



rof f.,/ 



and 



n' 

 the expression for S' may be written as follows : — 



r f 'i 

 I _/ — ^ | 



L 1 " y j 



which is the best form of S' for our purpose. 



The function S' consists of two factors — namely, of G', which 

 at or near balance is proportional to the current by which duplex 

 and single signals in station I. are produced, and of F', which 

 at balance =0. 



Therefore to make S f as small as possible when balance is 



disturbed, we can only do so by making F' as small as possible, 



771 ylf 

 which is evidently the case for y ] — —~ a maximum. Further, 



SJ — pf — jj/> 



S'=G'F'; 

 and since at or near balance 



F a G', 



it follows that D' = F' • 



i. e, the first condition is also fulfilled by 





y ■= — 



Our problem for station I. would therefore be most generally 

 solved if we make the function y' a maximum, remembering that 

 the variables contained in y 1 have to fulfil two condition equa- 

 tions, namely the immediate balance (equation VI.) and the ba- 

 lance (equation V.). 



Substituting for m' its value, and remembering that 



on account of the immediate-balance condition (equation IV.), we 

 get 



,__ a'(g' + d') aW 



