of Duplex Telegraphy. 461 



P, and Q, we have to fix the signs of the two terms of which they 

 consist. This is best done by considering the forces m and n as 

 absolute numbers, and determining the direction in which they 

 act with respect to one and the same magnetic pole by the direc- 

 tion of the currents passing through the coils a and b. 



To fix the signs of the currents, we shall call, arbitrarily, that 

 current positive which passes through the coil a in the sending- 

 station when the negative pole of the signalling-battery is joined 

 to earth. 



Further, if we suppose at the outset that the movement of the 

 key k does not alter the complex resistance p of its own station, 

 i. e. the fulfilment of the key-equation 



a condition which is essential, it is clear that the currents y' and 

 g y are the algebraical sums of the currents A', %! and B', 23' re- 

 spectively ; whence it follows that 



Q'=(A'+a'X+(B'+i&'K 

 where the currents contain the signs. 



Now, with respect to the manner of connecting up the two 

 signalling-batteries E' and E", we have the following two differ- 

 ent cases : — 



1st. The same pole of the signalling-battery is connected to 

 earth in each station, thus : 



p' = ±A!m' + B'n!, 



Q'= (± A'+aV+ (+b'+b'K 



where the upper signs are to be used when the negative poles of 

 the signalling-batteries are connected to earth in both stations, 

 and the lower signs when the positive poles of the signalling- 

 batteries are connected to earth in both stations. 



2ndly. Opposite poles of the signalling-batteries are connected 

 to earth in the two stations, thus : 



p' = ±A'm' + Wn', 



V'=±<R'm'±m'ri, 



Q!=(±A.'±®')m , + { + B'±®')n', 



where the upper signs are to be used when the negative pole in 

 station I. and the positive pole in station II. are connected to 

 earth, and the lower signs when the reverse is the case. 



Subtracting in either of these two cases Q' from P', it will be 

 seen that invariably 



S' = Q'-P'=//; 



or that, on account of having fulfilled the key-equation w + /3=f, 



