Oscillations in Three Coupled Electric Circuits. 43- 



To determine the relations between A, B, and 0, insert 

 equation (4) in equation (2) and obtain 



A(l + RA^z-LAp 2 ) + EMxAp 2 



4 CM 13 C lP 2 = 0, 



AM 21 (V + B(l + B 2 C 2 p-L 2 C 2 ^ 2 ) 



+ CM 23 C 2 p 2 = 0, 



AMmCsp" + BM 32 dp 2 



+ C(l + K 3 C3pi-L 3 C3^ 2 ; = 0. 



From these equations we find 



= p* (M 12 M 23 C 1 C 2 - M 12 L 3 A - M 3 iM 23 dd + M 3 iL 2 C A) 

 +^(M ]2 R 3 C 1 C 3 - M 31 RAC 2 ) +p 2 (M 12 C 1 -M 13 C 1 ) 



^(LiMadUj- LiLgUA + M 13 M 2 Ad + M 3 iAd) 

 -/z(R 1 L 3 C 1 C 3 -L 1 R 3 C 1 C 3 + M 23 Il 1 C 1 C 2 ) 

 +i> 2 (LA + L 3 C 3 + R^C A-M 23 C 2 ) -^(R 3 C 3 + RA) - 1 



C ^ />4(M J2 M 23 dd- M 21 L 3 C 1 C 3 -M 31 M 23 CA + M 13 L 2 C 1 C 2 ) 

 A + /^(M 12 R 3 d C 8 - M 31 RAC 2 ) +p 2 (M 12 C 1 -M 13 C 1 ) 



^(LxMaaCA - L t L Ad + M i2 M 13 1 3 +M 2 7C 1 C 2 ) 



-pH (R x LA C 2 - LACA + M 23 RAd) 



+^ 2 (LA + LA + K 1 R 2 dd-M 2 A)-^(Rid + R2d)-l 



Let us rewrite the above relations in the form 



B 



(16) 



B = EA, 



C = FA, 



in which each of E and F have three values, which we will 

 term E„ E 2 , E 3 , F x , F 2 , F 3 , corresponding to the three 

 values o>i, o) 2 , o) 3 of p. 



Then for the initial conditions considered above we tind 

 that the constants of integration have the following values : 



ft)! \ 



a2 = jo / 



o) 2 \E 2 F 3 



" . . jo / 



o) 3 \E 2 F 3 



&! SS 1V 1} 



EsFi-E^a + EiF^EaFi 



E 2 F 3 - E 3 F 2 + E 3 F 1 - EjF 3 + E X F 2 - E 2 Fj 



E 3 F! — EiF 3 



EsFa + EsFi-EiFg + EiFa-EaF 



E L F 2 -EF! 



E 3 F 2 + E3F1-E1F3 + EiFa-EaF 



b 2 = E 2 « 2 , b z = E 8 a 3 , 



c 2 = F 2 a 2 , c 3 = Fs«3- 





> (17 



