82 T. H. Havelock. 
and then values of the P functions were found from (12). With increasing 
values of p, the multipliers in (12) become large and this method loses accuracy 
unless the Bessel functions are known to a large number of places. It is then 
preferable, and sufficiently accurate for the present purpose, to calculate from 
a few terms of asymptotic expansions. These can be found independently, 
or derived from those of the Bessel Functions ; they are 
a ff, CSS IL ze, Der 
P; VE{ (1 128 pe 7 sin (p 7) 
ae 1 42-964 | zoey “i (p— a 
De p 
r 1569 1, 527-3 7 
Eas V E(t 128 pp! ) eos (p— 4) 
21.1 73-608 on) = 
| —_— ——— —_ —_ 
C2 a 
5 nf Sly AD 2 983°2) sin (p—2) 
ES Veale me pe an) esa 
@ i Hee 3858) aes (»- a 
iP p 
Although systematic computation of these functions has not been attempted 
(14) 
to any high degree of accuracy, it was found necessary to calculate a large 
number of values from p zero up to p equal to 40. Some of these are recorded 
in Table I. 
Table I. 
p P; ID, 12. D 125 Iho P;. 
0 +0-6666 0 —0-5333 3:6 —0-3515 | —0-3606 | +0-3457 
0-4 --0-5880 | --0-2563 | —0-4795 4-0 —0-1517 | —0-4624 | +0-1784 
0-8 --0-3876 | +0-4551 | —0-3361 4-4 +0-0597 | —0-4828 | +0-0150 
1-0 --0-2569 | +0-5198 | —0-2381 4-8 -+-0-2580 | —0-4220 | —0-1800 
1-2 --0-1171 | +0-5573 | —0-1300 5-9 --0-3317 | —0-3570 | —0-2721 
1-6 —0-1590 | +0-5480 | +0-0903 6 +-0-4478 | +0-0741 | —0-4230 
2-0 —0-3867 | +0-4366 | +0-2940 7 +0-1380 | +0-3932 | —0-1623 
2-4 —0-5106 | +0-2509 | +0-4405 8 —0-2659 | +0-3195 | +0-2291 
2-8 —0-5651 | +0-0281 | +0-4902 9 —0-3884 | —0-0406 | +-0-3799 
3-0 —0-5436 | —0-0828 | +0-4849 || 10 —0-1475 | —0-3348 | +0-1669 
3-2 —0-4998 | —0-1880 | 4-0-4579 bes = = pe 
Many intermediate values of the functions were required, and the only 
practicable plan was to construct graphs from which these could be taken 
with an accuracy of three figures. This was obtained by drawing graphs 
219 
