# Full text of "Scientific Papers - Vi"

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```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)
```