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II. The Roots of Bessel and Neumann Functions of High 

 Order. By John R. Airey, M.A., D.Sc* 



THE general formulae for the higher roots of J n (.!•), Gr n (#), 

 &c. have been given by McMahon f . If m = 4n 2 and 



fi=-j(2n + 4:S—l), and x\f represents the 5th root in order 

 of magnitude of the equation J n {x) = 0, then 



m- R m ~ 1 4(m-l)(7m-31) 





8/3 3(8/3) 3 



32 (m-l)(837tt 2 - 982m 4-3779) m 

 15(8/3) 5 "" { ) 



A similar formula has also been found J for the roots of 

 Y n {%), where 



ff=(2 W +4* + l)g-arc tan 2 ' 1082 "^ . 



4 7T 



When w is large, however, the above expressions are not 

 applicable to the calculation of the earlier roots. In calcu- 

 lating the roots of J„(#), for example, to two places of 

 decimals, the first root of Jioo(X) which can be found to this 

 degree of accuracy by (1) is the 50th, and in the case of 

 JioooM? ^ e fi^t r0 °t s0 obtained is approximately the 1000th. 



More suitable formulae for all the roots of J„(#) can be 

 derived from a result given by Debye §, of which the first 

 two terms are as follows : 



j »w = t^y [ c ° s { n(tan *-*>-!} 



+ ^,{1 +lt cot2< O cos {' l(tan *-*)-t} + -} (2) 



where cos d> = - . 



T x 



This result can be obtained directly from 



1 f" 

 J n (#) = — 1 cos {x sin (j> — n(f>) d<j> 



'""Jo 



* Communicated by the Author, 

 f Annals of Mathematics, vol. ix. (1895). 

 % Proc. Phys. Soc. vol. xxiii. (19ll). 

 § Math. Annalen, 67. Band (1909). 



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