Triode Generator ivith Two Degrees of Freedom. 707 



The general solution of (3) seems not to be possible, but 



since we consider — and — as small compared with unity, 



an approximate solution can be obtained. 



From the general nature of the problem two modes of 

 vibration may be expected to be possible, and to express this 

 mathematically we are thus led to a trial solution, 



v = a sin (Ojt -f b sin (co n t + A) , 



where a and b are certain unknown functions of the time and 

 ftjj and (o n are unknown frequencies. X is an arbitrary phase 

 constant. 



As 



«i 



^1, 



*2 



«a, 



and g>! and w 2 are of the same order of magnitude, we may 

 expect the possible building up or deca}^ of the amplitudes 

 to occur slowly compared with the oscillations themselves, 

 that is, 



da 



dt^^ a > 



db 



Hence the second and higher differential coefficients of a 

 and b with respect to time will be neglected. "We thus have 



v = a sin co^ + b sin (co n t -f X), 



v = co^i cos Wjt + a sin ay^ -f (D n b cos (o) n t + X) 



-h 6 sin (ay n t + X), 



6 = — w T 2 asin cojt + 2 (o L 'i cos w^ — o) n 2 6 sin (g) j;[ £ + \) 



+ 2o) II 6cos(ft) II ^ + \). }■ (4) 



tj = — co/a cos ©j^ — 3ft)j 2 tt sin tOjt— (Ojjb cos (ft> n £ -f- \) 



— 3&) n 2 /> sin (a) n i + \). 

 y — ft)j 4 a sin ©^ — 4a» 1 3 a cos o^i + o> n 4 & sin (ft> n £ + A,) 



— 4ft) n 3 6 cos (ft)n^ + A.). ^ 



We shall further have to consider the terms involving v 2 

 and y 3 . These non-linear terms obviously suggest the pre- 

 sence of higher harmonics and combination tones, but as the 

 increment and decrement are small, the main effect of 

 the non-linear terms is in their influence on the amplitudes. 



2 Z2 



