1218 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1957 



Solving (20) for /^.[^i and V^^,(m) in terms of F[„] and /(„) respectively, 

 we may write 



Iw.lm] — — Z^ l^tn], 



" "-' (22) 



Vw,(m) — —2^ -'(»)> 



n X(m) 



where Yw,[m\[H] and Zy,,{m)(n) may be defined as the ratios of certain de- 

 terminants involving Zy,,[m][n] and F^, („)(„) respectively.* 



We are now able to eHminate Vw,(m) and Iw,[m] from (14) and (17), 

 and to write down the generalized telegraphist's equations for the 

 curved circular waveguide filled with an inhomogeneous dielectric in 

 the following form : 



dw 



= — 2-( [Z {m){n)I (n) + Z i^m)[n]I [n]\ , 



r^ = —/^ [Z[m\(n)I(n) + -^[m] [n]/[n]], 



aw n 



-^ = —Z^ [F(m)(n)F(„) + F(,„)[„]y[„]], 



(23) 



dw 



—r^ = — ZJ [Ylm]in)Vin) + Y [m] [n]V [n]] . 

 dw n 



The impedance and admittance coefficients are defined by : 



Z(^;n)(n) = 10} / UBiigYSid T'(„o)-(grad T(„)) dS + Z^a.(m)(n), 



Z^,n)[n] = ioi \ M<?3(grad T(„o) • (flux T[„]) dS, 

 Js 



Z^mMn) = ioi \ /xe3(flux T[,„]) " (grad T^rC)) dS, 

 Js 



Z[n,.][n] = tco / iue3(grad T[„,]) • (grad T^) dS, 

 Js 



F(„,)(») = -iw / ee3(grad T(„o) • (gi'ad T^) dS, 



Js 



(24) 



* If (20) are actually regarded as infinite sets of equations, one has to pay some 

 attention to defining the meaning of the statement that their solutions are given 

 iiy (22). In practice we shall deal with a finite number of modes, and shall by- 

 pass all questions about infinite processes. 



i 



