CONFORM A L TRANSFOR\fATION 133 



(v, d) and dS for the element of area dvdd so that the integral equation 

 (3-5) for Q{v, 6) may be written as 



Q(ro) = e-''"'" + k'dT)-' I g(r)Q(r)G(ro , r) dS (A4-3) 



where the integration extends over the interior of the guide and G(ro , r) 

 denotes the dreen's function (3-3). 



If the number of equations in the set (A4-1) were increased from two to a 

 large number .V, the set of .v's would correspond, say, to the values of Q{r) 

 or of g(r)Q(r), and the 6's would correspond to the values of exp{—ikvo). 

 In any event, we take the analogue of / to be 



Je= f g(r)Q{r)[Q(r) - 2e-'''] dS 



(A4-4) 

 ■- k-K2irr' ff g{r)Q{r)g(ro)Q(ro)Giro , r) dSo dS 



where the subscript E indicates tiiat we are dealing with an electric corner. 

 It may be verified,* by giving Q{r) a small variation 8Q(r), that the function 

 Q{r) which makes Je stationary is the one which satisfies the integral 

 equation (A4-3). Furthermore, when we assume Q(r) to satisfy the integral 

 equation, the expression for Je reduces to an integral which is proportional 

 to the integral (3-6) for the reflection coefficient Re . More precisely, 

 Re ^^ given by 



ik 

 Re ^ ■;r- [Stationary value of Je] (A4-5) 



2ir 



It follows that if, by some means, we have obtained a fairly good approxi- 

 mation to Q, we may obtain a better approximation to Re by computing 

 Je and using the formula 



Re = ik{2ir)-'JE 



When we use the first approximation exp(— i^ti) for Q to compute Je it 

 turns out that the above formula gives the third approximation, R^e\ to 

 the reflection coefficient. 



The magnetic corner may be treated in much the same way. The 

 integral equation (5-6) for P{v, d) becomes, in the notation of this appendix, 



P{h) = g-'^'" sin ^0 + Kilir)-' j g{r)P{r)G{ro , r) dS (A4-6) 



in which the v in dS = dvdd is integrated from — qo to -j- oo and d from to tt, 



* See Courant and Hilbert, Methoden der Mathematischen Physik, Julius Springer, 

 Berlin (1931), page 176, where a similar problem is treated. 



