of an Oscillatory Electrical Circuit. 559 



the condition that the roots of this cuhic should be all real is 



(d-tbc+2h z y 2 +4:(c-b' 2 y=Q 



or d 2 - Wc 2 - 6bcd + 4 W+ 4c 3 = 0, 



which, on transformation, reduces to the quartic in TLJIji 



V I - aK " W f + wi W ( b } + 6K "-LA • L 2 / 



+S{- 3K2 S + £*< wK, - 6K )Ek- 8K, (ws)*} 

 + £{S( 30K2 - l8K )Lk + & ( - 6K2 - 6K) (iip} 

 +i2K3 Lk-& :+ S (9 - 18K - 3K2) (w 



+4K (W= ' 



where 3K = - — =-~ . 



1 — kr 



We shall here confine our attention to the case 1^ = 0. 

 The condition that all the roots of the cubic (6) should be 

 real, or which is the same thing, the condition that n should 

 be imaginary, is given by the quadratic in (B 2 /Lc 



■*? 



;; : : (12 K=) + g(9-lSK-3K=) L ^, + 4K^, )2 =0. 



The condition that a range of values exists for R 2 /L 2 over. 

 which there are no oscillations is that the roots of the above 

 equation should be real, 



i.e. (9-18K-3K 2 ) 2 -192K 2 >0, 



or 3K 4 -28K 3 + 90K 2 -]08K + 27>0, 



or (K-3) 3 (K~l/3)>0, 



or K>3, 



or P>-889. 



Hence for all values of k* below '889, the oscillations are 

 real whatever be the value of R2/L2, whilst for values of P 



