REFLEX OSCILLATORS 641 



(c4) 



harmonic distribution. Thus 



ii =00+^1 cos (w/o -\- ip) -\- a2 cos 2(w/2 -\- <p) -\- • • • 



+ ^1 sin (a'/i + ^) + ^2 sin 2iwti + ^) + • • • 

 where 



1 r . 



On = - I h cos 11 (uk -\- (f) dwU 



TT J-ir 



1 r" 



bn = - I k sin n (cj/j + ^) do^h 



■W J-v 



Using (cl) to (c3) we change our variable to /i obtaining 



fln = - / /o cos «w I /i + ^ + ip\ dooti 



JM , fl 2co VvT o 0^F 



Let oj/i = Wi coTo = — - — =6 X = -^-— 



2Fn 



(c5) 



(c6) 



- / h cos n Idi + if + d\ I + ~ sin di 



X 



^ — sin' ^1 + 



^de, 



(c7) 



(c7) cannot be evaluated in closed form without further restriction. The 



first order theory may be obtained by assuming that ^ <<C -. It is 



6 2 



not suflficient to assume that — « A'. The latter assumes that the third 



6 



and higher terms of the expansion are small compared to the second. Let 

 the integrand be denoted as io cos nx. The quantity to be evaluated is the 

 argument of a trigonometric function where the total angle is of less impor- 

 tance than the difference nx — 2tmr where m is the largest integer for which 

 the difference is positive. The condition first expressed requires that the 

 contribution of the third and higher terms to the difference phase shall be 

 small. The restriction requires that 



n (^J wro « T (c8) 



This is a more stringent requirement than 



(c9) requires only a small modulation depth while (c8) imposes a restriction 

 on both the modulation depth and the drift time. 



