REACTIVE CIRCUITS. 63 



Let the resistances of the circuits be respectively r b r- 2 , r 3 , . . . 

 / and their respective reactances s^ s 2 , % . . . s m ; and let the 

 potential differences between the terminals of the respective cir- 

 cuits be GI, c>2, c 3 , . . . e m , and i the current common to all the 

 circuits. 



We then have, by applying the equation (10) to each circuit 

 in succession 



r'ii + ksii = 0i 



r^i 4- 



r$i 4- 



r m i 4- ks m i = c m 

 Therefore 



c = c\ + c 2 4- e 3 4- . . . 4- e m 

 = (n + ^2 + r a -J- . . . 4- rji 



4- /^(i 4- s 2 4- s 3 4- ... 4- s, H )i . . . (13) 



But, applying equation (10) to the combination, we have 



e = Ri 4- kSi (14) 



Thus, by comparison of (13) and (14), we see that 

 R = n + r 2 + r a + 



and 



4- 



(15) 



that is, the resistance and reactance of the series combination are, 

 respectively, the sums (algebraic) of the resistances and reactances 

 of the constituent circuits. 



PAEALLEL CIRCUITS. 



47. Suppose that m reactive circuits are connected in parallel, 

 and that it is required to determine the equivalent resistance, 

 reactance, and impedance of the combination. It is better to sub- 

 divide this problem into two distinct cases, according as mutual 

 induction between the circuits is not, or is, taken into con- 

 sideration. 



CASE 1. Mutual Induction neglected. 



Let the resistances and reactances of the individual circuits be 

 PI, ''a, ?*& r ,n respectively, and si, s 2 , s$, . . . s, tl respectively ; 



