1916]
Thus and
WHEN n IS GREAT AND B HAS ANY VALUE
dz 1 , ,
ZtJ o
'397
-(19) .(20)
. For the present purpose A = 0, B = 1 ; so that for Pn, identified with u, we get
= J. 00
(21)
in which z-ad, a~ = n(n + I).
The functions J0, J0' = — Ji, are thoroughly tabulated*.
The Table annexed shows in the second column P20 calculated from (21) for values of 6 ranging from 0° to 35°. The third column gives the results from (11), (12), beginning with 6 = 10°. In the fourth column are the values of P20 calculated directly by A. Lodge. It will be seen that for 6 = 15° and 20° the discrepancies are small in the fifth place of decimals. For smaller values of 8, the formula involving the Bessel's functions gives the best results, and for larger values of 6 the extended form of Laplace's expression. When 6 exceeds about 35° the latter formula gives P2o correct to six places. For n greater than 20 the combined use of the two methods would of course allow a still closer approximation.
Table for PW
e Formula (21)
0 — . . ----- , ---------
0 1-000000
5 0-346521
10 -0-390581
15 -Q'052776
20 + 0-300174-
25 -0-078051
30 -0-216914
35 + 0-155472
. 40 —
45
11} From (11) and (12) Calculated by Lodge
0 1 '000000
1 — . 0-346521
1 -0-390420 -0-390588
6 -0-052753 -0-052772
4 + 0-300191 + 0-300203
1 -0-078086 -0-078085
4 -0-216997 -0-216999
2 + 0-155636 + 0-155636
+0-127328 + 0-127328
-0-193065 -0-193065
* See Gray and Mathew's Bessel's Functions._!_) Sin (im?r + 7)..........(9)