1S75.] Theory of Duplex Telegraphy. 53 



to zero, not even approximately. Besides the E. M. F. requisite for duplex 

 working being necessarily comparatively large, (3 will always be a quantity 

 which cannot be neglected against the other resistances of the system, even 

 if the single cells were of small l'esistance. 



But supposing it were practicable to construct a battery of exceedingly 

 low internal resistance, then, as f = b -f- d, it would be necessary to make 

 b = o and d = o another physical impossibility, as b must consist of con- 

 volutions to produce magnetism, and d must be variable to produce balance. 



This solution f = b -{■ d = to -f /3 = o, or even each of these three 

 branches of an only exceedingly small resistance, must therefore be rejected. 



(b.) Adjustment of balance by moving the coils or armatures. 



This, it will be clear, is the solution for immediate balance, for such a 

 mode of adjustment would involve no relation between the resistances of the 

 three branches, leaving their determination free for other purposes. In order 

 that the slightest movement of the two coils, or their armatures, may pro- 

 duce the required balance, it will be best to move both the coils or armatures 

 simultaneously in the same direction. In fact to be able to produce 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 sea sons, 

 and the armatures for the daily changes. 



It is clear that the differential method, when balance is adjusted by the 

 movement of the coils or armatures, can alone be compared in efficiency with 

 the double balance, and the superiority 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 pre-supposing complicated mechanical arrangements requir- 

 ing delicate workmanship and being liable to get out of order. 



Hap id approximation of the two functions D and S towards zero. 

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

 tial conditions, we know that 



p = S for each station invariably. 

 Now for Station I we have 



where 



A' = (&' -f d') m' ~( a ' + h' + c') n' 



N'=f {V + d' + a' + K + C) + (b' + d>) (a' -f h> + C ') 

 If we call c' that value of the measured circuit, which for any o-iven 

 values of the two branches V + d' and a' -f- h' produces balance in Station 



