of Duplex Telegraphy. 539 



(for a perfect line, i. e. i=co ), and that, further, 



a=h 

 and 



f=w+{3, 

 we have 



d + a=a + 2w + 2P + L. 

 Now in this equation suppose every thing constant except a 

 and /3, the internal resistance of the two batteries E and e 

 respectively. Hence, if we could achieve that 



8ot = 2$B invariably, 

 the variation of the internal resistance of the two batteries 

 would not disturb the equation R=K, and therefore also not 

 affect the balance. With absolute certainty we cannot fulfil 

 this desirable relation between the two variations ; but with 

 some probability we may. For it is well known that the in- 

 ternal resistance of a galvanic battery decreases in time by the 

 current passing through the battery. Hence, if we suppose 

 that the two batteries consist of identical cells (equal in nature, 

 size, and internal resistance), we may say that the variation of 

 the internal resistance of a single cell by the unit current in 

 the unit of time is the same for both the batteries. Further, 

 if we make the other not improbable supposition that the va- 

 riation at any one time is proportional to the current which 

 passes at that time, we have 



"P 1 TT 2 



and 



where e is the variation of the internal resistance of a single 

 cell in unit of time by unit of current, <f>^ a certain unknown 

 function of the time which, as the two batteries are working 

 simultaneously, is not required to be known. 

 Hence, from 



Sa==2S/3 



and 



it follows that 



K=E, 

 E 



and 



q V 2 



These values of X and v bring the compensation method, 



