Neumann Functions of Il'ujli Orde\ 

 Taking the first integral of (6) and putting 

 0=/j?+bfi 3 — cfi 4 ... 



we find, from 



1TX 



I 



COS 



COS 





\ d*- 1 cos 0d0 = T(v) cos — 

 Jo 4 



+ 2 r U) oos T 



=rg)co S f + Q + : Joo^ 1 )rg)co S ^ + ....(7) 



From the second integral, a similar expression is obtained 

 in which the sine occurs in place of the cosine in (7). 

 Finally, wo get from (6), 



*->-s(^[ r GM*-i) + 



The next four terms in the bracket have been calculated, 

 ihe coefficients of which are more conveniently expressed in 

 decimal form. 



K w .it^^«.{+-&±^} . ( 9) 



**- (fsinfc) L 4 J 



then the functions A s (fa) take the following values, & being 

 written for cot 2 ^ : 



Ao(<M = l. 



A^Js 0-12500 + 0-20833/fc. 



A,(0!) = 0-02344 + 0-012 1 5k t 0- 11140* 2 . 



A 3 (0 1 ) = 0*00488 + 0-05941& + 0-12310P + 0-06839A 3 . 



A 4 (0O = 0-00107 + 0-02249& + 0-06521P + 0-10673* 8 



+ 0-04443& 4 . 

 A 5 (fa) = 0-00024 + 0-00780* + 0-04503P+ 0-09702P 



+ 0-08960^ + 0-02986P. 



A first approximation to the roots of J n (x) can be found 

 by making the first term in (8) equal to zero 



cos 



Lvsmfa — ncfri— -) = cos\rt{t3Lnfa — fa) — ~\ =0. (10) 



