290 REPORTS ON THE STATE OF SCIENCE, ETC. 
Part II. 
Bessel and other related functions of equal order and argument. 
(A). Tables of the Bessel function J,,(x) and the Neumann functions G,,(z) and Y,(z), 
where the order and argument are equal or differ by unity, have been published in 
the 1916 Report, the values of n and x increasing by unit intervals in the earlier part 
of the tables. There does not appear to be any record of the computation of these 
functions for fractional values of the order and argument. The values of the argument 
of the Bessel functions J,(x), Jy_;(7) and the Neumann functions Ny(2) and Ny_,(x), 
tabulated below, range from 0 to 10 by intervals of 0-25, the order of the functions 
being equal to the argument or less by unity. 
For small values of the argument, the functions J1(2) and J43(x) were calcu- 
lated from the ascending series, J}(x) and J—3(x) together, the terms of these series 
being simply related. 
Oe 2 ig a 3). a } ‘ 
SP (1/2)! (2 5 ome igeoae ttt! ) 
4)\2, 
J 3(4)— SEN. (1 — x? +- m0. Rati ate den aie ) etc. 
ane): BIN 2:5 2:3-5-9 
4/\2 
5 ; 
The use of the recurrence formula, Jy+,(%) = —JSy(a) —Jy_,(z). gave the values 
entered in the following table. 
The Neumann functions were easily obtained from the relations 
J_y(z) = Jy(x) . cos vz — Ny(x) . sin vz 
N_y(«) = Jy(z) . sinva + Ny(x) . cos vz. 
For large values of the order and argument, the functions were directly calculated 
from the asymptotic series, viz. : 
Iwo) = seal) T (3) aaa) (3) atmos) TG) + | 
with similar series for Jy_,(v), Ny(v) and Ny_,(v). The results were checked by 
2 
Nyy . Jy —Ny . Jya= =n 
ae r G ar t 
| 4y | Jy(v) Jy_a(v) | = —Ny(v) —Ny_(v) 
0 | — 1-000000 0-000000 | co ott | 
1 | 0647832 1-230518 | 1252815 ; —1-551122 
2 0540974 =| = 0990245: || 0990245 —0-540974 
3 0480568 0855013 || —«0-861905 ; —0-232820 : 
4 0-440051 0-765198 ||: 0*781213 —0-088257 
ag 0-410288 : 0-699974: | 0724086 «=| —0-006067 
| 6 0387142 | 0649838 | ——-0680560 0-046083, 
| 7 | 0368420 | 0-609742 0644714 : 0-081608 
8 |  0-352834 0-576525 0617408 0-107032 : 
9 |  0-339572 : 0-548919 0:593338 : 0-125900 
10 0-328091 0:525080 0572630 : 0-140293 : 
| 1 0-318012 ; 0-504345 0:554541 0-151510 
12 0-309063 0-486091 0-538541 : 0-160400 : 
13 0-301038 0-469858 : 0-524243 0-167544 
14 0-293783 : 0-455298 : 0-511351 : 0-173345 : 
15 0-287179 0-442140 | 0499642 0-178099 : 
16 0281129 0-430171 | 0488937 0-182022 
17 | 0-275557 | 0-419222: || + —-0-479094: 0-185276 
18 0-270401: = «0-409155. =|, ~S(0-470000 0-187986 
19  0-265610 | 0:399856: | 0-461559: | 0-190249: 
20 | 0-261141 0391232 |S s«0-453695 = s«é- 192142: 
