66 TREATISE ON ALTERNATING CURRENTS. 



i = (A- l.-B)e 



I - U 



i--i \ _ j. I pL pG \ \ 



r./ + 



1 [ff+lW . , 1 I/ 6 



iWV \ ' 2 + ^( 



If r 2 = o, and p is large, ^ is small, and i% large, so that the 

 greater portion of the current passes through the condenser 

 circuit. 



If, in addition, i\ = o, the main current is given by 



If, further, f=pC, the main current is zero, while the 

 currents in the two branches are given by 



that is, a current will circulate round the branch circuits while 

 there is no current in the main circuit. It is to be noted that TI 

 and r<i can never be zero, although they may be very small, in 

 which case the current in the main circuit will be small compared 

 with those in the branch circuits. 



CASE 2. Mutual Induction taken into Con- 

 sideration. This case is somewhat complicated, since, if we 

 consider any particular branch of the parallel circuit, the E.M.F. 

 which drives the current against its ohmic resistance is the 

 resultant of m + I distinct E.M.F.s, viz. the applied P.D., the 

 E.M.F. due to the self-induction of that circuit, and the E.M.F.s 

 due to the mutual inductions between it and the remaining m 1 

 branch circuits. 



Let the mutual inductions of the several pairs of circuits be 

 Mifr M\ t z t . . . MW . . . where the suffixes denote the two cir- 

 cuits to which M refers. Since the mutual induction between two 

 circuits is reciprocal relation, M pq = M w , where p and q are any 

 different integers from o to m. Let the remaining notation be as 

 in Case 1. Then considering the circuit 1, the applied P.D. must 

 furnish m + 1 components, one equal to riii, to drive the current 

 against the ohmic resistance of the circuit ; a second given by 

 ksiii, to balance the reactive E.M.F. of the circuit ; a third given 

 by kpMi t zi% to balance the E.M.F. due to mutual induction of the 



