Triode Generator with Two Degrees of Freedom. 711 



amplitude either to return to or to depart further from its 

 initial stationary value, thus applying the usual method in 

 questions of dynamic stability. In this way we shall find a 

 certain conservatism of the system in that the particular 

 mode of vibration persists (when one parameter is varied 

 gradually) even when conditions have been reached which 

 are not favourable to it and which are such that, were this 

 mode of vibration not actually present, yet the other one 

 would exist. In other words, metastable oscillation con- 

 ditions may arise. 



Let these small changes of amplitude be represented by 

 the type 8. We thus substitute in (9) 



a 2 = a 2 + 8a 2 , 



b 2 =b 2 +.8b 2 , 



« 



and only retain first powers of the small quantities $a» 2 and 

 8b 2 . Hence we have 



l) = Ei(a s - 2a f - 2b s 2 )8a 2 ~ 2E I a s 2 8b s 2 , 



at 



d(8b s 2 ) 



= E n ( V - 2b, 2 - 2a, 2 )oV - 2Ei ib 2 8a s 2 . 



(12) 



dt 

 These linear equations (12) are solved by putting 



8a s 2 = A'e /d , 

 8b 2 =B'e kt , 



and we obtain as characteristic equation for k 



»+.*■{ E I (2a # » + 2 b 2 - a 2 ) t En(2« s 2 + 2/>, 2 - b 2 ) } 



4-EjEn {(2a 2 + 2b/-a 2 )(2a 2 +2b 2 -b 2 )-Wb s 2 } = 0. 



. . . (13) 



In order that a set of stationary values a s and b s should be 

 stable neither of the two roots k of (13) may be positive as 

 this would show the tendency of the system to depart from 

 the stationary solution in question. 



A set o£ stationary amplitudes a s and b s is therefore only 

 stable when 



E r (2i.» 4- 2b 2 -a 2 ) + E n (2a* + 26/- V) > ) 

 and (2aJ 1 + 2L 2 -a ( ?)(2a f 2 +2b 2 -b ( 2 ')-±a 2 b 2 > 0. J 



