Oscillations in Three Coupled Electric Circuits. 37 



Following the usual practice, the coefficients of coupling 

 may be written : 



M 2 , v 

 For coupling between circuits 1 and 2, a 2 = ^— f~> I 



„ 2and3,/3 2 =^, I . (3) 



M 2 

 3 and 1, 7 2 = r ^-. 



Let us assume a solution of (2) in the form 



<> 2 = B**«, I (4) 



e 3 = Ce^. J 

 Substituting these values in (2), we obtain 



^(1 + RAp - L Ap 2 ) + < 2 M 12 (V 



+ e 3 M 13 C lP 2 = 0, 



^M 21 C 2 j> 2 +^(l + R 2 C 2 pi-L 2 C 2 ? 2 ) 



+ e 3 M 23 C 2 p 2 = 0, 



^M 31 C 8 i? 2 +^ 2 M 23 C 3 ^ 2 



+ ^(l + R3e 3 ^-L 3 G 3i 9 2 ) = 0. 



Eliminating e^ e 2 , and 6 3 from these equations by means 

 of the determinant 



»j (l + Rxdp-Lidp 8 ), M 12 C ljP 2 , M 13 C lP 2 , 



M 21 C 2i > 2 , (1 + R 2 C 2 ^-L 3 2 /), M 23 C,p 2 , 



M 31 <V, M 32 C tf 2 , (l + R 3 C 3j oi-L 3 C 3 /), 



we obtain an equation of the sixth degree in p, viz. : 

 p* [2M 12 M 23 M 31 - L,L 2 L 8 + L^ 2 + L 2 M 31 2 + L 3 M 21 2 ] 

 +pH [R, L 2 L 3 + L X R 2 L 3 + LiL 2 « » - RiM 23 2 - R 2 M 31 2 - R 3 M ]2 2 ] 



L . \J3 vvi Kj2 -I 



3 .rR 1 L 2 + L 1 R 2 R 2 L 3 + L 2 R 3 . R 3 L 1 + L 3 R l . -p -d -d 1 



~ p l L — 5 — + — c^ — + — or ' + RlR2Rs J 

 -f r _k_ + _k. + . l__ + » A + BA + BA-j 



LC 2 C 3 ^i^3 C-iC'2 C 3 Gi G 2 J 



0. 



