12 



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



Taking an extreme case, the second root of Ja(V) and 

 applying (15), the following values were obtained : — 



K. 



*i- 



^2. 



f t 



0-0553582 



3°10'41"-90 



6-0121 



0-02065 



0-0551690 



3° 10' 2"-64 



6-0328 





the second approximation agreeing with the value of p 2 from 

 equation (1). For p 50 of Ji(#), (17) gives at once the value 

 157-8626554. 



A simple expression for the roots of J 2l (.f) when n is large, 

 can be found from (11) and (12). 



From (11) 



'here 



tan ^ =X+ 5+175 ' 



Therefore, from (12), since 



tan 2 c/>! 

 sec0!=lH ^~ 



tan 4 <£] 



8 



r 1 v J 3V 1 



(18) 



Even for the small value ?i = 5, this gives p 1 = 8*744... 

 instead of 8*7715, and for n= 100, ^=108-77, p,= 115*72, 

 and p 3 = 121*58. From (18) it is seen that for very large 



values of n, — approximates to a ratio of equality. 



The equation — -~^ =0 is nearly as important in mathe- 



matical and physical problems as J n ( t i«) = 0. 

 If 



o / 9 ^ \ 1 



*=£,( 2 » + */'- 3 > and ' x = X -8/?X~fe~12&?)x 1 -' 

 . 1 2 19 



cosec ^ = "-^""V~«V'" 



■and the higher roots of J n '(«) = are given by 

 Pp = n cosec $i, 

 A simple and convenient formula for the roots of J n (a-) = 



