of Duplex Telegraphy. 123 



when we get 

 G'—Wik - ■ a9 



''(a»+g'k)-\-a"(d»+g'k)\ \q'{a' + _/)+«'(</ + d')\' 



nil-. j?l; a 9 



Now it will be seen that G' has clearly a maximum with 

 respect to g 1 , while G" has a maximum with respect to g" ; thus> 

 if we take _/ as the only variable in G' (k constant) and differen- 

 tiate with respect to g\ we get 



and if we take g n as the only variable in G" and differentiate, 

 we get 



These two equations must be fulfilled simultaneously in order 

 to have the simultaneous maxima of the two currents in question. 



Executing the differentiation, and resubstituting for k its value 

 of" 

 —, we get, after reduction, 



a' a" (c" + _") {q' + d') -g'g"{a' + §') (_" + c") 



and 



while 



a'a'\c' + <* 'j (g" + a 7 ") -_// («" + _") (a' + c') 



-g»(a< +g<) \q" (a»+g») + a"(g" + d") } ** =-. 0, 



d#_ J?_ dja'-d') 

 dc' , i* a"{a"-d") 



Now the terms in the two equations which have -j-j and 



dg" {q" + p")°~ {a"+g") 2 



dc" , dc J 



¥ an( V 



for factors become independently zero — the first for a' = d', and 

 the second for a" = d" ; and substituting these values for d' and 

 d" in the other two terms, both become zero for 



a'a"-g f g" = 0; 

 whence it follows that 



