of Duplex Telegraphy. 465 



balance no matter how great the variation in the resistance 

 of the line may become, it will be necessary to make the coils 

 movable for the changes of seasons, and the armatures for the 

 daily changes. 



It is clear that the differential method, when balance is ad- 

 justed by the movement of the coils or armatures, can alone 

 be compared in efficiency with the double balance-, and the supe- 

 riority of the latter is most striking. While immediate balance 

 and the fulfilment of the other two essential conditions can be 

 obtained with the double-balance method within any given range 

 by a variation of the resistance in one single branch (b branch), 

 this same result with the differential method can only be arrived 

 at by either supposing four branches simultaneously variable, or 

 by supposing the coils and armatures movable — both presuppo- 

 sing complicated mechanical arrangements requiring delicate 

 workmanship and being liable to get out of order. 



Rapid approximation of the two functions D and S towards zero. 

 Supposing the fulfilment of the key-equation as one of the 

 most essential conditions, we know that 



p = S for each station invariably. 

 Now for station I. we have 



i>'=S' = E'A', 



where 



A' = (b 1 + d')m> - ( ffl ' 4 h' 4 d) ri, 



N' =/'(# + d' + a f 4 h! + d) 4 {b' + d')(a' 4 h' + d). 



If we call c' that value of the measured circuit which for any 

 given values of the two branches b 1 + d ! and a! + h' produces ba- 

 lance in station I. (i. e. for which A' = 0), then, if c' varies oY, 

 we have A' = n'Bc' } while N r becomes N' + 8N'. 



Thus we have 



n'Sd 



c . E'»' M 



O = m—n—fi' 



f' + » + d j + H+ fW + d ') +c< + 8c'' 

 a + h +f + i + d f + c+oc 



fib' + d 1 ) 

 but as a' 4 h! + - ;, ,, — 77 =p' the complex resistance in station 

 j'-\-b'-\-d' 



I., and as further 8c' can be neglected against d, we have finally 

 G /__-p/ n ' . Bc ' . 

 Phil. Mag. S. 4. Vol. 50. No. 333. Dec. 1875. 2 H 



