REFLECTIONS PROM CIRCULAR BENDS 



337 



give 



/ci2 — .0465 K32 = .6524 

 Ku = .0165 K23 = —.4555 

 Thus, the numbers entering (5.1-7) are 



r2C = .4712 (8.284)'^' 



yic = .4712 (-8.886)'^' = i 1.404 

 7ictanh7ic = —8.382 

 7ic coth 7ic = .2354 



,1/2 



1.356 



73C = .4712 (47.06)^ 

 73C tanh 73C = 3.228 

 jzc coth 73C = 3.243 



3.233 



72C tanh 'Y2C = 1.187 

 72C coth 72C = 1.549 



1 



For the purpose of calculation it is convenient to transform (2.3-3) by 

 inserting (5.1-7) and premultiplying by kV~^. We obtain 



([7c tSinhyc]dK + kV~^Tqc) x = kV ^cToe °h 

 {[yc coth yclciK + kV~'Toc) y = Kl^'cToe'^^h 



(5.1-12) 



in which 



where the elements of V are obtained from the formulas and tables of Ap- 

 pendix I. 



The i equation of the set obtained by writing out the first of equations 

 (5.1-12) is 

 3 

 2 [K^^iC tamhyiC + (KV-^)iiT]oc]xj = (kV-^JucT oie"^'' (5.1-13) 



where (KF~^)ty denotes the element in the i* row andy* column of kV~^, 

 Kji is the element in the i'^ row and /^ column (note the reversal of the 

 usual convention regarding the order of subscripts) of k, kh = 1, and h 

 has disappeared because it is a column matrix whose top element is unity 

 while the remaining elements are zero. It will be noted that the only 



