I 



1220 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1957 



where 



^[m][n] = X[m]X[n] / ^T [„,] T [n] chS, 



^{m)(n) = X(m)X(n) I ^T (^,n) T (r>,) dS, (28) 



5(m)(n) = X(,m)Xin) I 5T(,„) T(„) dS. 



The quantity 8mn is the Kronecker delta, and is not to be confused with 

 5(,«)(n) , which is defined by the last of equations (28). Note that ^[m][n] 

 and ^(m)M are zero unless the angular indices of the two modes in- 

 volved differ by exactly unity. 



It is not difficult to obtain approximate solutions of (20) in the forms 

 (22), since the off-diagonal elements of the coefficient matrices of (20) 

 are small compared to the diagonal elements. Using the expression de- 

 rived by Rice'^ for the inverse of an almost-diagonal matrix (we shall 

 not attempt to prove this result for infinite matrices), we find the first- 

 order approximations 



■17 X[m]X[n] fs r^ V 1 



1 w.[m]ln] — : [Omn "T ^[m][n]i, 



iwHo , , 



(29) 



'7 — X(m)X{n) \^ _\ f ;■ 1 



^w,{m)in) — : \Omn "T C,{m^{n) 0(m)(n)J. 



tweo 



Approximate expressions for the impedance and admittance coeffi- 

 cients appearing in the generalized telegraphist's equations (23) are: 



Z(m)(n) = ?COyUo[5m,j + H(m)(nU + Za',(m)(n) , 



Z{m)[n] — 'ZW/XoH(m)[rt] , 



Z[m]{n) = io:iJ.o'E[m]{n) , 



Z[m][n] = ta))Lio[5,„,j + H[m][n]], (30) ■ 



^' (m)(.n) = io:eo[8mn + H(m)(n) + '^(»i)(n)], 



Y(m)[n] = ?'coeo[H(m)[„] + A(m)[«l], 



Y[m](,n) — ^''*^€o[H[m](n) + ^[mlC'oJ) 



y[m][n] = io)to[8,nn. + H[ml[«] + •^['«]["]] + ^ w,["']\ti] . 



