10 Dr. J. R. Airey on the Roots of Bessel and 



w(tan fa - fa) - j = * 2 



or 



»(tan</, 1 -<^ l )= ( ^ T ^. • • (H> 



The errors in the corresponding values of fa for the first 

 four roots of J^) are 9' 8", V 19", 0' 25", and 0' 11". 

 The roots p p are then given by 



p p = nsecfa (12) 



Putting 



n(t*nfa-fa)= {4ip - 1)ir +6, . . . (13) 



it can be shown without difficulty that, when r is written 

 for n tan fa, 



Air - 1 . 3 . 5 . A 3 r 3 + 1.3.5.7.9. A 5 r 5 - . . . „ . , 

 tane== ■-A^T^TA^TiT3.5.7.A 4 ^r:— * < U > 



The quantity e is in general very small and decreases as p, 

 the number of the root, increases. For the first four roots of 

 JxOb), € has the following values : 0*03596, 0*01839, 0*01248, 

 and 0-00947. 



Any root of J n (a?) can thus be calculated from (13) by 

 successive approximations. Only in extreme cases is it 

 necessary to go beyond the second approximation. In 

 calculating the first root of Ji(#), for example, the following 

 values of fa and e were successively obtained : 



<£ 1 = 74°43' 10"-4 : 6=003631. 

 74° 52' 23"*5 : e = 0*03587. 

 74° 52' 16"-9, 



the last value of &i giving 3'8316 as the first root of Ji(a'). 

 A number of errors have been discovered in Bourget's Table * 

 of the roots of J x to J t5 , seven of these occurring in the 

 values of the roots of J 5 . The first root of J 5 (#) is 8*7715,. 

 not 8*780, and the ninth root 34*989, not 34*983. Other 

 methods f give 8*77148 as the first root of J 5 (#). 



The first five roots of Jio(#), Jioo(#)» a11 ^ JioooOO have 

 been calculated from the formulae (12), (13), and (14j : the 



* Ann. de TEcole Xormale, 3 (1866) ; Lord Ravleigh, < Theorv of 

 Sound,' vol. i. Table B. p. 330. 



t Lord Ravleigh, Collected Works, vol. i. p. 190; Phil. Mag. June 

 1916, p. 520. 



ii 



