332 BELL SYSTEM TECBNICAL JOURNAL 



By using the asymptotic expressions (Al-19) for the Ts and U's and sum- 

 ming the series with the value of (4.1-7) for ^ =4 given in Section 4.1 w© 

 obtain 



Fom = -2^[2Tl^~'nr' + oT'] 



Fmo= -8^ro'i7r~w"' (4.3-5) 



Foo = frli/12 



In these expressions m is supposed to have the values 1, 3, 5, • • • . When 

 m is even Ffy^ and Fmo are 0(^^). 



Substituting (4.3-5) in (4.3-2) and summing the series with the heln 

 of the values of (4.1-7) given in Section 4.1 gives 



70 = r^i - ^'ro'i(5 + 2aVoi)/60 (4.3-6) 



A result equivalent to (4.3-6) has been given by Buchholz who also gives 

 the approximation to the propagation constant when m > (and the elec- 

 tric vector in the plane of the bend). In our notation his. approximation 

 is 



+ i (^i=) (21 + r'm') 

 \irm / J 



In writing (4.3-7) we have corrected a misprint in Buchholz's expression. 

 In order to agree with Buchholz's equation (5.30a) the leading term within 

 the square brackets would have to be changed from 3 to —3. This change 

 was indicated by the results obtained when our equation (3.2-2) was used 

 to obtain special cases of (4.3-7). Probably the best way of obtaining 

 (4.3-7) is furnished by Marshak's method (WKB approximation, out to 

 second order terms, applied to Bessel's differential equation). If one wishes 

 to verify (4.3-7) by using Marshak's report^ as a guide, he should correct 

 the misprint in Marshak's equation (12a). 



4.4 Reflection Due to Dominant Mode Incident upon Gentle Bend — E in Plane 

 of Bend 



The problem here is the same as that treated in Section 4.2 except that 

 now the electric vector lies in the plane of the bend. In line with equation 

 (1.1-4), the reflection coefficient /r for the dominant mode will be denoted 

 by d^i . As in Section 4.3 the subscripts indicating the position of matrix 

 elements will be adjusted so as to start with instead of 1. The square 

 matrix W given by (1.4-5), and associated with the junction conditions for 



