ON THE TABULATION OF BESSEL AND OTHER FUNCTIONS. 115 
L Part II, BrssEL Funcrions. 
Mr. Atrey’s TABLE. 
Tables of the Neumann Functions G,(x) and G(x) or Bessel Fu-ctions 
of the Second Kind. 
Tables of the first solution y = J,(x) of Bessel’s differential equation 
io 
when n =0and n = 1 have been calculated by Meissel' from the ascending 
series 
Iz) =1- 
x 
eee APS PS 
Oo Bate ee 
to 12 places of decimals from 7 = 0:00 to # = 15°50 by the interval 0-01. 
For greater values of x than 15°50, these functions can be found from the 
semi-convergent expansions. 
It is possible, however, to use these semi-convergent series to deter- 
_ mine the values of J,(x), J,(x), etc., to 12 places of decimals” for values of 
gz as small as 8. 
Tables of the second solution® of Bessel’s equation—viz. 
Y,(z), Y,(x), G,(x), G(x) . ete— 
are much less complete, and as these functions—Bessel functions of 
the second kind or Neumann functions, as they are sometimes called— 
are of considerable importance in their application to many physical 
problems, tables of G,(x) and G,(z) have been calculated to seven places of 
decimals. 
Different writers have given different definitions of these second 
solutions. 
The Neumann cylinder function‘ defined by 
vio=2| (2)-a+4 sley+ — (log 2— y— loge) J s00 | 
are 
etc., differs from the G,(z) and G,(x) function only by the factor S 
1 Gray and Mathews, Treatise on Bessel Functions, 1895. 
2 Archiv. der Math. u. Physik. III. Reihe, XX., 1913. 
5 B. A. Smith, Messenger of Mathematics, 26, 1897; Smith, Phil. Mag., 45, 1898 ; 
Aldis, Proc. Royal Society, London, 64, 1898-9. 
4 Nielsen, Theorie der Zylinderfunktionen, p. 12. 
