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XXIII. The Numerical Calculation of the Roots of the Bessel 

 Function J n (x) and its first derivate J n '(x). By John 

 R. Airey. M.A.,D.Sc* 



FORMULA for the higher roots of J n (a>) and J n '(» 

 have been derived by McMahon from the asymptotic 

 expansions of these functions. From Debye's results, it has 

 been shown t'at p p , the pth root of J n (^)=0 can be found 

 by the method of successive approximations from 



p p = nsec(p, (1) 



n(tan *-<£)= fc^ +e , (2) 



, A,)— 15A 3 r> + 945A 5 r 5 -... .„, 



and {aae= A,- 3A> + 105V — . ' • (3) 



where 



n tan (j> ' 

 If k is written for cot 2 <£, then to five places of decimals, 



A = l, 



A 1 = 0-12500 4- 0-20833^, 



A 2 =0-02344 + 0'13368&+ 0-11140 k\ 



A 3 = 0'00488 + 0-05941 £+0-12310 k 2 + 0-06839 F, 



A 4 = 0'00107 + 0-02252 & + 008371& 2 + 0'10673P 

 + 0-01447 k\ 



A 5 = 0-00024 + 0-00780 ^ + 0-04501 F + O-09716/j 3 

 + 0-08956 ¥ + 0*02985 P. 



77 

 The second term in A 2 is -^r^cot 2 ^), the incorrect value 



rj O i O 



,^cot 2 (£ given by Debye | being due to the omission of 



the factor 3 in the third term 



3. (?i + l)(n + 3)(n + 5) c x 2 c 2 

 3 ! 2 3 ' c 3 



in the expression for a±(n). 



* Communicated by the Author. 



t Math. Annalen, vol. lxvii. p. 545 (1909). 



