EEACTIVE CIRCUITS IN PARALLEL. 67 



circuits 1 and 2 (see Proposition 3) ; a fourth given by /pJ/i^ig, to 

 balance the E.M.F. due to the mutual induction of the circuits 1 

 and 3, and so on. 



Thus the vector equation of EM.F.s in circuit 1 is 



4- ksiii 4- kpMi t ^ 4- kpMi, 3 i 3 -f . . . -f kpMi m i m = e 

 Similarly for circuit 2 



i 4- r<fa 4- ks&t + kpM^is 4- ... 4- kpH* m i m = e 



Similarly for circuit 3 \ 



kpM 3i iii 4- kpMs^iz 4- r a is 4- ks 3 i 3 4- ... 4- kpM Sm i tn = e 



Similarly for circuit m 

 * 4- k 



These are m simultaneous equations from which to determine 

 the currents ii, i 2) is, . . . i m , whence, by substitution, the current 

 i in the main circuit can be obtained by means of the vector 

 equation 



i = ii + ia -f 4 4- . . . 4- *, M 



In this way we shall arrive, in any particular case, at an 

 equation of the form 



i = Pe + kQe 



where P and Q are independent of k and e. Having obtained this 

 equation, the equivalent resistance and reactance can be found in 

 the usual manner. 



There is no mathematical difficulty in the formation of the 

 vector equations (22), only great care should be taken in expressing 

 exactly the physical relations existing between the several circuits. 

 There is, however, a more extensive knowledge of mathematics 

 required for the general solution of the equations there given, so 

 we pass it over for the present, and content ourselves with a con- 

 sideration of the simple example of two mutually inductive circuits 

 connected in parallel. 



The reader with sufficient mathematical knowledge may refer 

 to the Appendix for the general solution. 



