Resistance and Self-induction of Branched Circuits. 353 



JSow, denoting the effective resistance of the system by W 

 and the inductance by I/, we have 



di 

 Esin^ = EV + L'-r, 



or 



sin pt= (AR' -t-p 2 BI/) smpt +^(L'A— R'B) cos pt ; 

 and, therefore, 



L'A-R'B=0, 



so that 



B 



R'= ; ., OT ,o , L'= 



A 2 +p 2 B 2 ' ^-A 2 +i? 2 B 2? 



which are the results quoted. 



We may employ the same method to find the effective 

 resistance and inductance in the case where the conductors 

 have mutual induction. Let M rs , or M sr , denote the coefficient 

 of mutual induction between the conductors r and s. It will 

 now be convenient to discontinue the use of the symbol L, 

 and to denote the coefficient of self-induction of the con- 

 ductor r by M rr . We have, in this case, 



E s'm pt^U^ sin (pt-0 l )+p[M ll I l cos {pt—OJ 



+ M 12 J 2 cos (pt~0 2 ) + . .. + M ln I B cos(p*-0J] 



= R 2 T 2 sin {pt-0 x ) + j p[M 21 l 1 cos (pt-0^ 



+ M 22 r 2 cos(.^-0 2 ) + ... +M 2 J w cos(p*-0J] 



Hence 

 E = Eili cos 0! + p [Mulx sin X + M 12 I 2 sin 2 + . . . + M x J B sin J , 

 = RJi sin 0, — pfMnIx cos X + M 12 I 2 cos 2 + . . . + Mi J„ cos 0J . 



Denoting 



Ij cos 1? I 2 cos 2 , . . . by a'j, # 2 , . . ., 

 and 



I x sin 1? I 2 sin 2 , . . . by y l3 ?/ 2 , . . ., 



we have the equations 



E = R^ +/iM lr y r =R 2 a? 2 +i?2M 2r y r = ... = ..., 



= Ri3/i-p2M lr ^=B 2 y 2 -p2k^= ... = ... 



