﻿Straight Wire on Electric Waves. 55 



decrease rapidly in importance, and we get a close approxi- 

 mation by taking only the first two, namely, 



Neglecting ("a) 2 , we get from (19) 



g l (ica) = 1 cob 2 6 d0 = —iKa y 



7T Jo 



whence 



<7,'M = -*' (21) 



Formula (18) does not hold for * = 0: the corresponding 

 formula is 



1 C 2n 1 

 g Q (icr) = 9—1 2 — 1 1 ( — for cos 0) n d0, 



whence approximately 



f/ M=l-i(/ca) 2 , 

 and g '(Ka) = —±Ka (22) 



ThuS yi = H o [ iK a^P- + i^0^l . . (23^ 



' °L 2 h (Ktl) lh(KCL)J 



It remains to evaluate the h functions which, unlike the 

 corresponding g functions, need not be precisely defined as 

 to constant multipliers. Now A (p) satisfies the equation 



d 2 /i , 1 dh . A ^ 

 ap- p a/a 



differentiation of this gives 



dp"\dp ) p dp\dp ) \ p") dp 



which proves that/* '(/j) satisfies the same differential equation 

 as /t,(/o). Both are finite for p=co , and so we have a right 

 to assume 



I'i(p)=K'(p) (24) 



The scattered wave is now given by 



Tj I~i h( Kr ) , ■ /jVWl / n 



^^"b-'^'N+^^CMJ' • • (2o) 



We may take h to be the h function discussed in 

 Article 2 above, and for values of r great compared with the 

 wave-length we may use the approximations of formula (2), 

 but with — ix instead of t.r, since ,v, (=«r), is real and we 



