﻿1874!.] L. Schwciullur— O^i iJic General Theory of Bi^plex Telcp-ap],^. 231 



These two equations must be fuirillcd simultaneously in order to 

 have the simultaneous maxima of the two currents in question. 



Executing the difFerentiation, and re-substituting for k its value '— , 

 we o-et after reduction 



and 



diile 



a' a'' {c" + d") {(/ + cV) — c/ f («' + q') {a" + c") 



- 9' («' ' + g") { W {a + g') ^- a' {c/ + d') j ~ = ^ 

 ^ Sd ff 



a' a" {c' + d') {(/ + d") — (/ <j" {a" + ^") («' ^ c') 



— g" («' + g') { 9." («" + gl + a" {g" + d") yJL^^o 



dc" e a' {a' — d') 



^V (/ + p'f («' + g'T 



dc' _ t^ a" (a" — d") 



¥' ^ (9." + p'T ' i^" + g'T 



Now the terms in the two equations which have — -; and — for factors 



dg d,j" 



become independently zero, the first for a' = d', and the second ibr a" = d'' ■ 



and, substituting these values for d' and d" in the other two terms, both 



become zero for 



a a" — g^ g' = o 



whence it follows that 



a' — d' z= 



a" — d" = 



a' a" — g' g" = o * 



is one of the simultaneous solutions of the two equations.* 



Thus, substituting for d' its value a\ and for d" its value a", we get 



a g 



a' = E" i 



{c" + a") [a" -f- (/") {a + y'} [a + q) 



G'' = JE'i "^ 



The first equation has clearly a maximum 'with respect to a', and the second 

 with respect to a', namely 



— — = 0, which gives a = g ^ 

 and -— = 0, which gives a == g , 



* The other solutions which arc possible from a mathematical point of view arc 

 however hnpossible with respect to the physical problem, for the quantities being all 

 electrical resistances must be taken with the same sign, say positive. 

 30 



