202 Dr. J. R. Airey on Bessel Functions 



For small values of re, the approximate value of 

 p 2 =ir{2?i + 7)/4: 

 only is required to give the value of p 1 to five or six places 

 of decimals. The first root p Y of J n [ary==0 for n=— i, 

 ~f ■■• + 85+1 by intervals of J, thirteen values in all, were 

 computed in this manner and intermediate values obtained 

 by central difference interpolation. The second root, p 2} is 

 correctly given to four places by MacMahon's formula * 

 derived from Jacobi's asymptotic expansion | of J n (^). 

 The values of p. 2 of J n (v) for the foregoing values of n were 

 calculated to six places of decimals from the expansion of 

 the function in ascending powers of the variable. In the 

 following tables, the colon : indicates that the fifth place 

 of decimals is approximately 5 ; thus p 1 of J_5(.t') may 

 be written 1*79465, the exact value to five places being 

 1-79463. 



n. 



Pi- 



p 2 . 



n. 



Pi- 



p 2 . 



i 



2 * * 



. 1-5708 



4-7124 



i 



4 •• 



. 2-7809 



5-9061 





1-6167 : 



4-7541 : 





2-8175 : 



5-9442 





1-66201 



4-7958 





2-8541 



5-9822 





1-7068 



4-8372 





2-8905 



6-0201 





1-7509 : 



4-8785 





2-9267 : 



6-0579 



"I - 



. 1-7946 : 



49196 : 



1 - 



. 2-9628 : 



6-0957 





1-8378 



49606 





2-9988 : 



6-1333 



• 



1-8805 : 



5-0014 : 





3-0347 



6-1709 





1-9228 : 



5-0421 





3-0704 :, 



6-2084 





1-9647 : 



5-0826 : 





3-1061 



6-2458 



1 



. 2-0063 



51230 : 



I 



2 • • 



. 3 1416 



6-2832 





2-0475 



5-1633 





3-1770 



6-3204 





2-0883 



5-2034 : 





3-2123 



6-3576 





2-1288 : 



5-2434 : 





3-2474 : 



6-3947 





2-1690 : 



5-2833 : 





3-2825«: 



6-4318 



1 

 ~* 8 •• 



. 2-2090 



5-3231 



1 " 



. 3-3175 



6-4688 





2-2486 : 



53627 





3-3524 



6-5057 





2-2880 : 



5-4022 





3-3872 



6-5425 





23272 



5-4416 





3-4219 



6-5793 





2-3661 : 



5-4809 





3-4565 



6 6160 



.. 



. 2-4048 



5-5200 : 



1 " 



. 3-4910 



6-6526 





2-4433 



5-5591 : 





3-5254 



6-6892 





2-4815 : 



5-5981 





3-5597 : 



6-7257 





2-5196 



5 6369 





3-5940 



6-7621 





2-5574 : 



5-6757 





36282 



6-7985 



i ■■ 



. 25951 



5-7143 : 



i ■• 



. 3-6623 



6-8348 





2-6326 



5-7529 





3-6963 



6-8711 





2-6699 



5-7913 : 





3-7302 : 



6-9073 





2-7070 : 



5-8297 





3-7641 : 



6-9435 





2-7440 : 



5-8679 : 





3-7979 : 



69795 



i 



4- .. 



. 2-7809 



5-9061 : 



1 .. 



. 3-8317 



7-0156 



* " The Roots of Bessel and other related functions." 

 Mathews, ' Bessel Functions,' p. 241. 



t Astron. NachricMen, Band 28, p. 94 (1848). 



Gray and 



