232 BELL SYSTEM TECHNICAL JOURNAL 



This expression gives the amplitude of the typical component of 

 frequency {mp ± nq)/2iT. The remaining steps are concerned merely 

 with the calculation of the integral (9) for particular values of m and n. 



It will suffice to work through one example in detail and give the 

 results in tabular form for the other products up to the fourth order. 

 The second order side frequencies, {p ± q)/2ir, will be taken as a 

 typical case. 



By direct substitution 



op f" /^nTC COS (— I' ens v) 



^11 = —7 1 COS ydv I (cos x -\- k cos y) cos xdx. (10) 



^" Jo ' Jo 



Performing the inner integration and substituting the limits for x, we 

 obtain: 



p r^ 



An = ^c, I cos A'[arccos (— ^cos^O + ^cosv Vl — y^^cos^ vlJ-v. (11) 

 ^"Jo 



Considering separately the integral, 



cos y arc cos {— k cos y)dy, 



r 



integrate once by parts, letting 



u = arc cos (— k cos 3'), 

 dv = cos ydy. 



The result is, after combining with the remainder of the integral foryln, 



. kP r sin2 y + cos2 y{\ - k"" cos^ t) ^ 



^11 = -7 '- , ,, „ '~dy. (12) 



^ Jo \ 1 — ^^ cos^ y 



Now substituting 



cos y = z, 

 we obtain 



. 2kP r' 1 - Fs" 



An = —^ \ dz. (13) 



^- Jo \(1 - s2)(i - k^z'') ^ 



This is a standard elliptic form.^ It is convenient here to let 



- It may be remarked that a large number of the integrals required in the evaluation 

 of the coefficients are listed by D. Bierens de Haan, " Xouvelles Tables d'Integrales 

 Defuiies." See in particular Tables 8 and 12, pages 34 and 39. 



