Neumann Functions of fllgli Order, 



when n is large is 



\ 2 . 3\ 4 



13 



Pp 



where 

 and 



-*L 1+ 2 + 40 "J' 



Thus the first root o£ J , 1000 («) = is 1008*84. 



Lord Rayleigh has given an approximate expression * for 

 the first root of J„'(#) =0, viz. 



Pl = n + 0-5134?i*. 



The Bessel function J_„(#) of negative order is represented 

 by a formula similar to (9), yfr in this case being equal to 

 7i (tan cf) l — (/>! + 7r). Hence the first approximations of the 

 roots of J_„(#) can be derived from 



rc(tan<k-<k + 7r)= l***^ 1 ) * 



and 



p p = u sec (pi, 



(pi for the first root being found from the least positive value 

 of tan (p l — (/>!. Thus, for the first root of J_ 5 _(#),£> = 3 and 



77 



tan fa — (/>! = Ypi- The small quantity e can be determined 



as before, leading to closer values of fa and p. The first two 

 approximations of Pl for J_a(#) are 1*802 and 1*868. 

 The Neumann function G n (x) is given by 



1 , w(, + ,i) i 



(2«+l)7T 



}• (19) 



Hence it follows that the roots of Gc n (%) can be calculated 

 from formulae similar to (12), (13), and (14). In this case, 

 (13) is replaced by 



w (tan <f> x - </>!) = / -^ — J 7T4- 



(20) 



A difficulty arises when p = l in this expression, but the 

 first roots of G„(a?) of any order are' found from values of 



* Phil. Ma-?. Dec. 1910. 



