ON CALCULATION OF MATHEMATICAL TABLES. 291 
4y Jy(v) Jyi(v) | Ny(v) —Nya(v) | 
21 0-256957 0-383205 0-446340 : 0-193725 
22 0-253028 : 0-375708 0-439441 0-195047 
23 0-249329 0-368684 ; 0-432949 : 0-196148 
24 0-245837 0-362087 0-426825: | —-0-197061 
25 0-242532 0-355873 : 0-421034 : 0-197812 : 
26 0-239398 0-350007 : 0-415545 : 0-198425 
27 0-236419: |  0:344458 | + 0-410332 0-198918 
28 0-233584 | 0:339197 | 0-405371 0-199307 
29 0-230879 0-334199 ; 0-400641 : 0-199605 
30 0-228295 : 0-329445 0-396125 : 0-199824 : 
31 0-225823 : 0-324914 0-391806 : 0-199974 : 
32 0-223455 0-320589 0:387670 0-200064 
33 0-221183 0-316455 0:383703 0-200100 
34 0-219000 : 0-312498 | 0:379893 0-200089 
85 0-216901 : 0-308706 | 0-376231 0-200036 
36 0-214881 | 0305067 | 0:372705: | 0-199947 
a 0-212933 0-301571 : 0-369309 0-199825 
38 0-211054 | 0-298210 | 0366033 0-199674 
39 0-209239: | 0-294974 | 0-362870 0-199498 
40 0-207486 0-291856 | 0359814 0-199299 
(B). When the order of Jy(«) is not an integer, the function can be represented by 
T 
= 
i i ~ —xsinh@ —vé 
Jy (2) == | cos (vO — x sin 6)d0 — co 2a eaere ata) 
J 0 
Tw Jog 
as shown by Schilifli. 
Tables of the second integral, F(z) for both positive and negative values of v 
have been computed, for small values of the argument, from the ascending series, viz. 
Fy(z) = Sey(z) + V8) 7 4 FT yim, 
y sinvr 
h ; epee aa eT oP ats 4) 
where Sp-y(«) ae Y) (2— v2)(32— v?) ar (12 — v2)(32— v2)(52— v2) 
x xf 
PA 2 we) * Ee 
and for large ae of the argument, where x = y + «, from the asymptotic series, 
a= sl) G)-a(Z)T (g)taG)-s(e) 1 G)+ --] 
the functions o, . having the following values : 
and 8, <tr) 
o,=« 
poe ce ae 
a re 
taal 
6 15 
2 
a! 
a= * — =n, etc 
"94 ~24° 280° 
8 28, 24s, 
oe Ts @)=[ 55+ Gn)? * x\8 tes ‘ene! * | 
~ ‘The Lommel-Weber Q function and its application to the problem of electric 
Waves on a thin anchor ring.’ Proc. Roy. Soc. A., vol. 94, 1918, pp. 313-4. 
