1216 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1957 



components by (8), we obtain: 



T (71) 



n 

 n 



n 



V<n 



(n) 



-r y [n] — r— 



e-zov . 

 dT[„] 



eidii 

 eodv 



— V[n 



-I 



dl\n 



(n) 



(») 



dT[n] 



eodv 



+ I[n 



(11) 



/ 



(n) 



dT(n) 



+ /[n 



[n] 



eiduj 



dT[n] 



e\du e2dv J 



For the longitudinal field components it is convenient to expand the 

 combinations esEy^ and ezHi, in the following series: 



esE^ = J^XMVu,,M(w)T^n)(u, v), 



" (12) 



eJH^o = 2lxMlw,[n]{w)Tin]{u, v). 



n 



It should be noted that the boundary conditions in the curved circular 

 guide are 



E, = E^ = (13) 



at the boundary of the guide, and that these conditions are satisfied by 

 the indi\'idual terms of the series for E^ and £"„, , on account of the 

 boundary conditions already imposed on T(n) and Tin] . Hence we do 

 not have the problem of nonuniform convergence which sometimes 

 arises in treating waveguides of varying cross section by the present 

 method. 



The procedure for transforming Maxwell's equations into generalized 

 telegraphist's equations is now straightforward, if a trifle tedious. One 

 substitutes the series (11) and (12) into e(iuations (5), and integrates cer- 

 tain combinations of the latter equations over the cross section of 

 the guide, taking account of the orthogonality properties of the T- 

 f unctions. For example, subtracting dT^rro/eidv times the first of equa- 

 tions (5) from dT(m)/eidu times the second equation, and integrating over 

 the cross section, yields 



dw 



X(m)V- 



w ,(,m) 



= —70) 



S [/(„) f m (grad Tm) ■ (grad T .r.)) dS (14) 



n |_ Js 



+ /[„, f ne, (grad 7'(.)) • (flux 7^,]) dS 

 Adding dT[m]/eidu times the first equation to dT[m]/e2dv times the second 



