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Full text of "Scientific Papers - Vi"

1914]
SPHERES  OF SMALL RELATIVE  INDEX
225
For values of n much greater, (22) is sufficiently represented by nV2/16, or m"; simply. It appears that there is no tendency to a falling-off in the scattering, such as would allow an increased transmission.
In order to make sure that the special choice of values for <m has not masked a periodicity, I have calculated also the results when n is even. Here sin 2m = 0 and cos 2m = + ], so that (21) reduces to
112 (1 or 0)
n~7r~
WIT"
RA.
    _ 4) [7 + log (nvr/2) _ Ci (rwr/2)].   (23)
The following are required:
n	Ci (ttTr/2)	n	Ci (717T/2)
2 4 6	+ 0-0738 -0-0224 + 0-0106 -	8 10	-0-0061 +0-0040
of which the first is obtained by interpolation from Glaisher's Table VI, and the remainder directly from (19).    Thus:
	1		
n.	(28)             !	n	(23)
2	0-7007	8	32-336
4       i          6-1077		10	53-477
6       i        16-156			
The better to exhibit the course of the calculation, the actual values of the several terms of (23) when n = 10 may be given.    We have
1 12                                fi27T2
_ -^ = _ 0-11348,    -v- = 61-685,
n 7T~                                 10
,      64      ,
7 + log (7T/2) + log n - Ci (7i7r/2) = 0'57722 + 0-45158 + 2'30259 - 0'0040
= 13-094,
so that
4 
{7 + log (n7r/2) - Ci (n7r/2)} = 13'094.
It will be seen that from this onwards the term nV/16, viz., m2, greatly preponderates; and this is the term which leads to the limiting form (20).   ,
The values of 2JS/X concerned in the above are very moderate.    Thus, n = 10, making m = 4,7rR/\ = IQ-irfi, gives 2R/\ = 5/4 only.    Neither below
E. VI.                                                                                                                         15 - du ............................ (17)