/>, = -/" sin (^1 + X sin 0,) 



REFLEX OSCILLA TORS 

 IX' 



645 



sin^ 01 + • • • U^i 

 + ^° f cos (01 + X sin di) 



TT J-ir 



r 1 ^' • 2 . , 1 ^' • :i . . 



(e4) 



(/^i 



Now of these terms, not all give contributions; some integrate to zero since 

 the integrand is an odd function of di. Rewriting with those terms omitted, 



(e5) 



ax^^^ \ cos (01 + X sin dM\ - ^^ sin' 0i + • • • J rf0, 



- ^^ I sin (01 + X sin 0i) [5^ sin' 01 + • • • J rf0i 



bx = -^f cos (01 + X sin 0,) I -^ y sin' 0i + ■ • • 1 rf0i . (e6) 



Evaluation of these terms is formally simplified by the following relation- 

 ships, each obtained by differentiation of the previous one: 



-2Ji{X) = 1 f cos (01 + A' sin 0i) dOi 



IT J-w 



-2/i(X) - -- f sin (01 + A' sin 0i) sin 0i ddi 



IT J-TT 



-2Ji(X) = -- [ COS (01 + A sin 0i) sin' 0i ddi . 



(e7) 



(e8) 

 (e9) 



Continuation of this process gives all the terms of interest in (e5) and (e6). 

 Hence 





(elO) 



(ell) 



Therefore the expression for the fundamental component of the beam 

 current may be written as follows, passing to complex notation: 



(^'2)/ ■^ Oi cos (w/o — 0) + ^1 sin (aj/2 



r'^e-*/o('-2/i +iy // + ^, {x'j[" + IX' jn^ + 



(el2) 



