184 Mr. Appleton and Dr. B. van der Pol on a Type of 



If we now consider all possible combinations of: signs o£ 

 a, 7, and e, the possibility and stability of the three ampli- 

 tudes can be tabulated as follows, where it is assumed that 

 A 2 >B. When this is not the case, stationary amplitudes 

 a 2 and a z are obviously impossible. 





Sign 



Of 6. 



Sign 

 of y. 



Sign of 

 , CE, 



Sign 

 of a 2 2 . 



Sign 

 of a 2 . 



1 



a 3 2 . 



1 



2 



3 



4 



5 



6 



7 



8 



+ 



+ 

 + 



"t- 



+ 



+ 

 + 



+ 

 + 

 + 



+ 



+ 

 4- 

 + 



+ 

 + 



+ 

 + 





unstable 

 | stable 1 

 1 stable J 

 | stable 1 



stable 

 stable 



unstable 

 stable 



unstable 



| stable | 

 unstable 



| stable 1 



unstable 



I stable 1 

 unstable 

 | stable 1 



unstable 



unstable 



unstable 



stable 



1 stable J 

 unstable 

 unstable 



unstable 



(In this table, amplitudes which are both possible and stable are 

 emphasized by thick rectangles.) 



We thus see that the only combination giving two possible 

 and stable amplitudes is (3), in which case e is positive, 



-j — I- a) positive, For such a combination 



of oscillation characteristic and circuit constants the system 

 is stable when not oscillating (a 1 = 0), and also when oscil- 

 lating with an amplitude a 2 . In this case the circuit possesses 

 a positive initial damping coefficient, so that oscillations will 

 not build up automatically, although a stationary amplitude 

 of value a 2 , when once produced, will be maintained. 



These analytical considerations can further be elucidated 



d / y(v) 

 In (3) "y can be written as 



, where "y may be regarded as representing 



Oil Ou V 



in the following way. 



dv 



a kind of damping factor which may be either positive or 

 negative, and which, unlike the damping factor used in the 

 ordinary linear treatment of oscillations, is a function of the 



amplitude. The relation between -~ — - and v for cases 2 



