1 80 Mr. Appleton and Dr. B. van der Pol on a Type of 



an approximate solution for conditions which closely resemble 



those of a practical case is possible. Thus in the case of a 



triode we may consider v as approximately sinusoidal, and 



equal to a sin wt since %(v) may be regarded as small compared 



with iv v. In this way the higher harmonics may be neglected, 



and an approximate value for the fundamental amplitude in 



the Fourier expansion for v obtained, where a is regarded as 



da da 



a function of the time 1, but such that — ^ wa, and that -y- ^ 



is negligible. In accordance with our assumption made 

 above regarding harmonies, we shall retain only the terms 

 involving the angular frequency to. 

 We thus have 



t* = a sin ivt, 



dv da . 



— - = — - sin wt + aw cos wt, 



dt at 



d 2 v ^da 2 . 



-ru = 1 —r w cos wt — aw 1 sin wt, 

 dt 2 dt 



X[v) = sin wt . r 7, l %(a sin wt) . sin wt . dt 



"° 2 f T 



+ cos wt . „ 1 ^(a sin wtf) cos wt . rtf£, 



— ^{v) = w cos wt . ™ I ^(a sin wtf) . sin ^^ . dt 



dt y J-Jo 



2 r T 



— it' sin w£ . ™ 1 %(a sin wrt) cos wtf . dt, 



2ir da 



where T is equal to : — , and — has been neglected ^in 



comparison with wa. 



d 2 v d 



On substituting in (3) for v, -^ and -j- %(v), we have 



„ . o da 



aiv Q 2 sin wt — aw z sin wt + Z ~ w cos wt 



and 



2w cos 

 + 



30S tot i T - ,n • . i, 



I p^(a sin wt) sin we . ar 



Jo 



^(a sin i<;c) cos wt .dt = 0. (3 a) 



2w sin w£ 



T ^o 



We may here note that the last term in the left-hand side 

 of (3 a) is small compared with the first two. Thus, on 



