116 REPORTS ON THE STATE OF SCTENCE.—1913. 
On the other hand, the Neumann function’ Y,(z), etc., defined by 
\2 & 4 x : 
Y) (2) = J,(x) : log.¢+ (5) ae a+n(s) + a+4+ 4) a) ate ie etc., 
2? 
3 
are found readily from G, (x), ete. by the relation 
Y,,(2) a (log i y) J (x) a. G,(z). 
The following tables were calculated, with some slight corrections, from 
those already published,® by interpolation, first to fifths and then to 
halves. 
The values for «—0-01 to =0-40 were found from the ascending 
series, 
ge \? zy 
—G,(z) = (3 — (1+ 4) (3 fF... —(log 2— y—log,z) J,(x), etc. 
2!° 
To determine the value of G,(z) and G,(x) for intermediate values of the 
argument, interpolation formule may be used, such as 
G, (« +h) = [3 = e ie ‘A G, (a) -+ [= pe 5a G(x) 
and 
ae ht wie 2 IV? fi 
G,(¢@ +h) = E a (i-3) &: : G, (@) ++ [+a— a . |B. 
Neumann Functions or Bessel Functions of the Second Kind. Go(a) and G,(x). 
e Go(x) | Gila) a | Go(x) | Gi(x) 
0-01 | +4-7209587 +100-0261051 0-20  +1-6981963  +5-2210521 
002 | +4:0274517 | +50-0452769 0-21 | +1:6471663 | +-4-9887552 
0-03 +3:°6214494 | +33-3951624 0:22 +1-5983499 | -+4-7778488 
0:04 +3°3330736 | +-25-0766778 0:23) +-1:5515475 | -+-4°5855201 
0°05 | +3:1090945 +20:0902576 0-24 +1:5065855 | -|-4:-4094258 
0:06 | +2°9258067 | +16°7694905 0:25 +1:4633116 -+-4:2475986 
0-07 | +2°7705685 +14:4002597 0:26 +1:4215915 +4:0983739 
0:08 | +2°6358361 -+12°6255419 0:27 +1:3813067 | -+3-9603349 
0:09 | +2°5167454 -+-11-2470137 0:28 +-1°3423516 -|-3°8322673 
0-10 + 2:4099764 +-10°1456966 0:29 +1:3046317 | +3-7131248 | 
0-11 +2°3131625 | +9°-2458884 0°30 | +1:2680624 | -+3-6020011 
0-12 | +2:2245569 | +8-4971288 031 | +1-2325676 | -+3°4981072 — 
0-13 | +2°1428339 +-7:8644903 0°32  +1:1980784 +3°4007530 
0-14 | +2:0669638 _ +7:3230293 0°33) | +-1:1645327 -+-3-3093323 
0-15 +1:9961309 | -6°8544580 0°34 +1:1318738 +3°2233107 
016  +1-9296778 | +6:4450632 | 0-35 +1-:1000501 +3°1422149 
0-17 -+1:8670675 +6:0843612 0:36  +-1:0690145 | -+3-0656245 
O18 | +1:8078556 -+5:7642001 0:37 -+ 10387238 -++-2-9931649 
0-19 | +1-7516700 | -+-5-4781456 0:38 -+-1:0091385 | -+-2-9245007 | 
5 Gray and Mathews, Bessel Functions, p. 14. 
6 Report of the Mathematical Tables Committee: British Association, 1911, 
pp. 73-78. 
ae 
