64 TKEATISE ON ALTERNATING CURRENTS. 



and let the P.D. common to each circuit be e, the currents in the 

 respective circuits being i\, i%, i& . . . i m . 



Then, applying equation (11) to each circuit in succession, we 

 have the vector-current equations 



But the current i in the main circuit is the vector sum of the 

 currents in the several branches of the parallel circuit ; therefore 

 we have the vector equation 



i = ii 4- i' a + is + . . . + i m 



And by comparison of this with the equation for the combi- 

 nation 



we see that the equivalent resistance E and equivalent reactance 

 S are respectively given by 



