344 



BELL SYSTEM TECHNICAL JOURNAL 



B = hho -[- hi cos pt + hi cos 2pt + hz cos Zpt 



-\- ai sin pt + a2 sin 2/)/ + as sin 3/?/ (49) 



defines the induction everywhere. 



The coefficients of equation (49) are given by the expressions 



bn = ^ [6/ + h,/'-\ + - [h,/ - hj'-j Z 



sin (2m + l)<7r 



(2m + 1) 



a. = ^ [a,/ + a,/'.] +- [a,/ - a,/'] E 



sin (2m + 1)^7 



(50) 



(2m + 1) 



After putting the values of the primed coefficients into the respective 

 terms, changing the arguments of functions of qr to qt by using equa- 

 tions (46), and expanding the result into multiple angles, there remains 

 when terms beyond the third order are dropped 



B = lfx, + IvP^P cos pt + ~ vlP + ZkQ']P sin pt 



OTT 



+ 

 + 



M0 + 2W 2 + yfe -|k- j 



^ + ^^ + 2iT-^. 



Q cos qt 



PQ sin qt 



- -^ j^[l - 6^]/cP2 cos (2p + q)t 



40 

 97r 



1 



'+T0* 



kP^ sin (2/> + q)t 



+ ^v[l - 6^]/cP2 cos {2p - q)t 



40 

 32 



10 



kP^ sin {2p -q)t 



+ ^2 ^>^PQ cos (/> + 25) / 



2 r- 



- 3^ [mo + 2^2 + 3k)P']KP sin (/? + 2(/)/ 



32 



- ^2 '"^'^<2 cos {p - 2q)t 



2 r- 



- 3^ [/io + 2K2 + 3k)P']KP sin (p - 2q)t 



4_ 

 37r 



Mo« ~ c "-P 



P sin 



3/j/ 



+ ^ "Ki'C cos 2,qt - -~- 1/(32 sin 3qt. 



(51) 



( 



