of Duplex Telegraphy. 117 



Multiplying now G f by */g' and G" by \/g v , we get 



the magnetic moments of the two instruments in Nos I. and II. 

 stations respectively ; and considering that 



where Q = {g f + V) (g n + 1") + i(g' +g" + V + 1"), we may write the 

 two above expressions as 



The first expression has clearly an absolute maximum with 

 respect to g', and the second with respect to g n ; but these two 

 maxima cannot be simultaneously fulfilled, and do not therefore 

 represent a solution in this particular case. But if we consider 

 that during a duplex signal both the instruments g l and g" are 

 in circuit, while during a single signal, though not both the in- 

 struments yet certainly their equivalent in resistances is in 

 circuit, it will be clear why simultaneous maxima of the two 

 single expressions are not possible. It represents simply the 

 more general case to which the question belongs of making the 

 magnetic moments of two instruments, connected up in the same 

 single circuit, maxima. In this case it is well known we can do 

 nothing more than make the sum of the magnetic moments a 

 maximum ; and here therefore we must do the very same. 



Adding, then, we get 



r - r+r- 4 Q 



which expression has a maximum with respect to both g' and#" 

 considered as independent variables ; and such indeed, according 

 to the nature of the problem, they really are. 



Thus, differentiating P with respect to g ] and g", we get 



* This can be easily shown by substituting for fi', ft", c f , and e" their 

 actual values. 



