of Duplex Telegraphy. 115 



dc ! 

 "° r 7T Can ^ ecome infinite, it follows that 8c 1 must be always 



very small in proportion to c r itself, and more so as compared 



with p' + c'. 



Thus we have at last 



n, dc [ ~ dc' ~. dc f ttr 



^ = — ^ + ^T a? + -7iT 6L - 



/o' + c/ p' + c' p' + c' 



and therefore to make #' for independent variations S.2?, Si, and 

 SL as small as possible, each term should be made as small as 

 possible. Now, taking p' and p" as independent variables, it will 

 be seen that the total differential of each term is negative. Thus 

 6' becomes smaller the larger p' and p ,! are selected ; and the 

 same of course is the case for n (station II.). 



Now the complex resistance of any one station can be ex- 

 pressed as follows : — 



fl . = («+/)(y+<») 



H a+g+d+f 



(«d-gf)* 



{a + d+g+f)\b{a + d+g+f) + {a+g)(S+i)Y 



Thus for any given sum of resistances (i.e.a-{-f+d+g = const.), 

 p will be largest if 



ad-gf=0, (VI.) 



which is the " immediate-balance condition." 



The fulfilment of the immediate-balance condition is therefore 

 no longer an assumption made to afford convenient and quick 

 means of adjustment when balance is disturbed, but, as has been 

 proved, is necessary in order to reduce the effect of any disturb- 

 ance whatever to a minimum. 



Supposing now the fulfilment of the immediate balance, we 

 have 



( g + d)(a+f ) 

 P~ a + d+f+g ' 

 which again has a relative maximum for 



g+d=a+f; 

 whence it follows, in consequence of equation (VI.), that 



u = d=f=g (VIII.) 



represents the general solution of the problem. ) 



* This expression is nothing else but the resistance of a Wheatstoue's 

 bridge between the two battery electrodes. It is most easily obtained by 

 the application of Kirchhoff's rules. 



12 



