High-Frequency Induction- Coil . 315 



From (3) and (5), disregarding 4«y as small, we get 



g , +/3 , = B + VB*-4D y2 + g , 2= B-VB»-4D ; (g) 



and from these and (2) and (4) 



= A AB-2C = A AB-2C 



4 + 4 V'B 2 -4D ) 7 4 4 VB 2 ^4T)' ' 

 By making the proper substitutions, these become 



9 _ L 1 K 1 + L 2 K 2 + V(L 1 K 1 -L ;] K 2 ) 2 + 4M 2 K 1 K 2 " 



a + P " 2K 1 K 2 (L 1 L 2 -M 2 ) ; 



(8) 



,- L^ + L 2 K 2 - /( LA- L,K,)« + 4M«K 1 K 8 

 ^ + * = SK^L^-M*) ; .... (9) 



(10) 



R 1 [L 2 (L 1 K 1 + L 2 K 2 )-2K 1 (L 1 L 2 -M 2 )] 

 vi ^t>t +R 2 [L 1 (L 1 K 1 + L 2 K 2 )-2K 2 (L 1 L 2 -M 2 )1 



4(L 1 L 2 -M 2 ) 

 E 1 [L 2 (L 1 K 1 + L 2 K 2 )-2K 1 (L 1 L -M 2 )] 



y- 4(1^- M 2 ) " " -' t 11 ) 



The general solution may now be written in exponential 

 form 



Q x = E^i' + Fe^ + Ge^ + Re^, 



Q 3 = yEW* + * g F«** + A 8 Ge^ + * 4 H^, 

 which reduces to the trigonometrical form 

 Qx^-^E + F) cos^H-z(E-F) sin /ft] 



+ *-*[(6r + H) coso7 + *(G-H) sin&], 



Q 2 = e -«<[(A: 1 E + & 2 F) cos /ft + i(k 1 E-k 2 F) sin/3/] 



+ «-*[ (& 3 G + k±R) cos 3/ + i(k 3 G - kji) sin &] ; 

 or otherwise 



Q^e-"^ cos fit + B x sin /ft) + ^(Ql cos & + B 1 sin &), 

 Q 2 = e -°-\ A 2 cos /ft + B 2 sin £0 + *-* (C 2 cos It + E> 2 sin 8/) . 

 Equating the coefficients of corresponding terms, elimi- 



