132 BELL SYSTEM TECHNICAL JOURNAL 



which, when we choose g{v) to be zero for w < and kik~ — 1 for i) > 0, 

 become 



n'^ = 2c{c, + c)-\ R^J^ = {c- ci)(ci + c)-i (A3-14) 



which again agrees with results obtained from transmission Hne considera- 

 tions. WTien the entering and leaving guides are the same size we may use 



M"(oo) = 1 + u\2c)~' ( g(i) dv (A3-15) 



J— 00 



It seems difBcult to give any general rules for the choice of g{v). Since 

 for Rh and Th , the factor sin d reduces the effect of the singularities on the 

 walls of the transformed guide, the choice g(v) = g(y, 7r/2) suggests itself. 

 The factor sin d is not present in the formulas for Re and Te and regions 

 near the walls are more important. In this case the selection 



i(v) = T-' [ giv, e) 



dd 



may be useful, especially since it allows us to use the result (A3-7) when k 

 and ki become small. 



APPENDIX IV 



Variational Expressions for Reflection Coefficients 



The reflection coefficients are proportional to the stationary values of 

 certain forms associated with the integral equations. In order to obtain 

 these forms we proceed as follows. It is readily seen that the values of 

 Xi and Xi which satisfy the symmetrical set of equations 



(A4-.1) 



cinXi -\- a22X2 = ^2 

 are the ones which make 



/ = aiiX"i -j- 2auXiX2 + 022^:2 — 2biXi — 262.T2 (A4-2) 



stationary when xi and X2 are given small arbitrary increments. This 

 stationary value of J is 



Js = —biXi — b2X2 



If we take the integral equation to be the analogue of the set of linear 

 equations, the reflection coefficient turns out to be proportional to /, . 

 In order to set down the actual expressions it is convenient to write r for 



