672 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1953 



Integration of (35) now leads to 



S = [2A (cosh ^ + B)f" (36) 



where B is another integration constant. Substitution of (36) into (28) 

 in the form 



yields after another integration 



where C is the fourth integration constant. 



In order to express in terms of tabulated functions the relationship 

 between ^ and y defined by (37) we shall consider two cases: — 1 < 

 B < I and B > 1. (It is not necessary to consider B < —I because A 

 is essentially positive [see (33)] so that B < —1 would imply an imagi- 

 nary field strength [see (36)] at the plane in the intrinsic region where 



The changes of variable of integration 



s = 2 sinh"' cot X for - 1 < B < 1, 



s =. 2 sinh"^ tan ^ for B > 1 



permit the carrying out of the integration indicated on the left side of 

 (37). This gives 



.,.(4(L-)-]_,[(i^«J»,.,„-.^.|]) 



= i2Ar\C - y) 



(38a) 



or 



— {(^7'4(^T 



for -1 < ^ < I 

 and 



- A"\C - J)) 



- A"\C 



y)) 



(38b) 



(grrTP^Ciri)" ■ ^'"" ^"^"^l = (2^)"'^^ - ^-'^ (39.) 



