3(5 Mr. 0. Heaviside on Duplex Telegraphy. 



/, g, and I, the equation to determine it being of the sixth 



degree, viz. 



, (x>-f g)\ 2 V.//+ 2 s/xg + >/(*+/)(* +g) } == _ t 



(f+g + *+ ^)v(^+/)(^+^) + (2^+/+^)(vV+ Jxg) 

 This theoretical difficulty, however, is of no practical im- 

 portance, since the value of x can be determined as closely as 

 necessary in the very act of adjusting the instruments for du- 

 plex working for the first time. For long lines, a; may be 

 considered equal to 1+ \/fg. It is actually rather greater. 



The next thing to be considered is to what extent this ar- 

 rangement is liable to disturbance from variations in the resist^ 

 ance of the line. Now, as I have shown in a former paper 

 (Phil. Mag. Feb. 1873), the values of a, b, and c, which make 

 the most sensitive balance for measuring a resistance x with a 

 battery of resistance / and galvanometer of resistance g, are 

 precisely the same as those given in equations (7) above as 

 giving the maximum current in duplex working. We are 

 thus at once led to the conclusion that the arrangement of 

 Wheatstone's bridge for duplex telegraphy which gives the 

 maximum received current at both stations, is also the arrange- 

 ment which is most easily disturbed by variations in the resist- 

 ance of the line. We may show this otherwise. When 

 a : b : : c : x y station A, when sending alone, produces no cur- 

 rent in his instrument. Let now the external resistance x be 

 changed to a/ t then A sending alone will produce a current Ci 

 in his instrument, of strength 



Mc(x'-x) 



r _ b(c + x) + x(c + x') . 



-f b(c + x)(c + x>) \f <6 + *)(6 + .^~ TT 



\b(c + <c) + x{c + a/y y ) \b(c + x)-\-x(c + x') J J 



bc(x f -x) y 



b(c + x) + x(c + x / )) 



If another arrangement be made in which b and c are altered 

 to V and c', and the current be now C 2 , then C 2 may be found 

 by changing b into V and c into (/ in the expression for Oi ; 



and the ratio -^ when x r —x is small will express the relative sen- I 



sitiveness of the two arrangements. In the limit, when x' — x — 0, 

 we have 



__bc > 



c 2 ~ w ___/ 



{V{c f +x) +g(b' + x)\ {c\b f + x) +f{c' + x)\ 



-{ 



