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



i = Tfr(v) cannot in general he represented by a simple 

 expression, but use may be made of a power series*. 

 If this series be written as 



ylr(v) = xv + /3v 2 + yv z + 8v* + ev 5 . . . , 



On substituting for %(y) in (6) and (8), we find that the 

 possible amplitudes are given by the real roots of 



1/ , CR\ , 3 , 5 , 



2\ a +-r) a +^ a+ Tt ea ° 



. 1.3.5 2n — 1 



+ " 2.4.6 2n "^- ia =0 ^ (9) 



while any amplitude a is stable for which 



( , CR \ . 3 2_L 15 4 



+ |-xiv;::S a2 "- ia2 "- 2>0 -^ 10 ) 



If the oscillation characteristic is of such a simple form 

 that the power series can be limited to the first three terms, 

 it may be shown that only one stable amplitude is possible 

 for any given set of triode and circuit parameters. Thus in 

 this case (9) reduces to 



*/ _i_ CR \ : _i_ 3 ^ n 



the solutions of which are given by 



«! = 0, 



4/ CR\ 



«2 2 = 



7 

 But from (10) we see that for an amplitude a to be stable 

 we must have 



v* + tt) * 2 7 



Thus in this simple case only one stable finite amplitude a 2 



* C/ 1 . van der Pol, Eadio Review, vol. i. p. 701, Nov. 1920; and 

 Tijdschrift van het Nederlandsch Radiogenootschap, Deel i. Oct. 1920. 



