Self-induction of Branched Circuits, 355 



where 



Now, as before, we have 



i= sin pt'Zx — cos pt %y, 

 and, therefore, 



E sinpJ =R/(sinjt>£ "%x — cos piZy) +pL/(cos pt%x + sin pt 2?/) , 



whence 



E^R'2^+^1/%, 



0=pL / ta:-Wty, 

 and 



w _ ES# T ,_ ESy 



"-(S^ + W iJ "(2^) 2 +(%) 2? 



or 



11 - (2Q) 2 +p 2 (SU) 2 ' ^ -(2Q) 2 +p 2 (2U) 2 * 



In the simple case of two branches of resistances R and S, 

 coefficients of self-induction L and N, and coefficient of 

 mutual induction M, we easily find 



R 2 S 2 +/(2M 2 RS + L 2 S 2 + N 2 R 2 ) + p 4 (LN-M 2 ) 2 

 p 4 RS > 



^ n _RS(R + S)+p 2 S(L-M) 2 +^R(N-M) 2 



^ ^RS > 



and 



VTT _2MRS + LS 2 + NR 2 +p s (LN-M 2 )(L + N-2M) 

 ZU ~ PES ' 



The values of 1/ and R' are obtained by substitution. If 

 p=0, we see by inspection that 



„, RS ,'_# 2MRS + LS 2 + NR 2 

 E " RTS and L = (R + S? ' 



and if p=<x> , that 



S(L-M) 2 + R(N-M) 2 LN-M 2 



K - (L + N-2M) 2 ' L ~L+N-2]vr 



