ON CALCULATION OF MATHEMATICAL TABLES. 



271 



Part II. 

 Zeros ot Bessel Functions of High Order. 



The roots of Bessel functions J„(x) where the order n is large are of importance 

 m the solution of physical problems. The table calcnlated by Bourget' gives the 

 first nine roots of the six functions J|i(xj to J.-,{x). 



For large values of n, p,„ the pth root of J„{x) is approximately found^ from 



p,=n(l + ^^+^'-...) where 



Ln [ 4 I8(4j;-1)7T I J 



For the first three roots, 



Pi = w + l-8579ii + l-OMn-i 



p.2 = n + 3-245n\ + 3-l58n-!s 



p3 = w + 4-382wi + 5-760M-i 



The table below was computed to six significant figures over a wide range of 

 values of n by the method of successive approximation,' The colon is approxi- 

 mately equivalent to 5 : e.g. 11 -0363(5) is the first root of .^^{x), nearly 



First ten roots of Jn(x). 



Normale, 3 (1866). Lord Rayleigh, Theory of Sound, 



' Annales de VEcole 

 Vol. i., Table B, p. 330. 



2 ' Calculation of the Roots of Bessel Functions. ' Phil. Mag., Sept. 1917, p. 193. 

 ■'•' The Roots of Bessel and Neumann Functions of High Order.' Phil. Mag., 



July 1916, pp 

 1922 



10 and 11. 



