DIFFKACTIOX OF RADIO WAVKS U\ A I'AKAHOI.IC CVLlNDKIt 171 



b = -log iS = log tan (^+t)='^ + "^'/^ + 



7-1/3 



= h g, 

 tanh b = sin (p. 



(8.9) 



Replacing c(r) exp (?7i sin v?) by c(p) plus a correction term then con- 

 verts (8.8) into 



ie-'^ + S,)r + Sn = ic-^' + S,) 



+ 



• c(p)[l + exp (26p)]' 



21/2 



1 exp (?7i6 — ^■/i tanh b) 



1 - e' b 



+ o(/r^//-^-) + oCr^'V^'^''), 



(8.10) 



where the subscript p on the square brackets indicates that h is to be 

 replaced by bp defined by 



bp = log tan (xP/2 + x/4) = ^ + ,^76 + 



(8.11) 



The ciuantity within the square brackets in (8.10) is continuous at 

 6 = where it behaves like (neglecting 0(6) terms but retaining 0{hh^)) 



y^ + m^l^ = 3^ + z7i^'V/3. 



(8.12) 



Expression (8.10) is to be used with (Soo + ^^2.3) and (.S32 + -S^aa) ob- 

 tained from (7.17) and (7.46) (with d = 7r/2 and w = 1), respectively. 

 When ^l/ is small, expression (8.10) becomes 



+ c{p) 



11 h'^' exp (7ffV3) " 

 2 lA S' . 



+ 



(8.13) 



The subscript p on the square bracket indicates that g is to be replaced 

 by g^ defined by 



g, = h"%, = h''' (^ + ,/^70 +•••)• 



(8.13) 



When, in accordance with (8.1) and (8.2), we add to (8.13) the ap- 

 proximations (7.26) and (7.49), namely 



S32 + *S33 ~ iMi^vig), 



(8.14) 



