REFLECTIONS FROM CIRCULAR BENDS ^29 



By expanding the typical terms in partial fractions and using th6 fact that 

 the sums of (see, for example, page 238 of Reference^") 



£/, = 1 + 3-« + 5-' + . . . 



for 5 = 2, 4, 6, are ttVS, 7rV96, 7rV960, it may be shown that 

 53 = 7rV64, 54 = 7rV768 - t/U^, 

 55 = (IStt' - 7r')/3072. 



(4.1-7) 



(4.1-8) 



In (4.1-7) Bq denotes the q^^ BernoulU number. The values of Sp may also 

 be computed in succession from the two relations* 



Uip = 2-^2 CpJfi-l,2i-lSj:-\-i =2^2 Cp+f-l,2t-2'5'p4i 



t=l t=l 



where Cm,n is a binomial coefficient. Still another method is to make 

 use of the generating function 



E ^''^^+1 = (1 + Z {tn - 1 - t)-' = ^ - iirx cot TX 



p==0 m=2,4,6--- 



where Ax"^ = 1 -\- t. Note that by this definition Si is J in contrast to the 

 non-convergent series obtained by putting ^ = 1 in (4.1-6). 



Substituting the values for the F's given by (4.1-5) in the expression 

 (4.1-2) for 7? and using the sums (4.1-8) of the series which occur gives 



[ yl = Tl -f-Al+ aVUi - 6r~') + iaTio/T)\S - r'/S)] (4.1-9) 



When the dominant mode is propagated without attenuation both 71 and 

 Tio are negative. 



The general form of (4.1-9) has been obtained by both Buchholz^ and 

 Marshak^ by different methods. In our notation their result is 



7L = rL - £ [1 + ^'rL(i - 6.~'m-') + (^^J (5 - /mV3)] 



(4.1-10) 



where jmn is the propagation constant for the m, n^^ mode when the mag- 

 netic vector is in the plane of the bend. 



* I am indebted to John Riordan for these relations. 



