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Full text of "Scientific Papers - Vi"

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)