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Full text of "Scientific Papers - Vi"

224
ON THE DIFFEACTION   OF  LIGHT BY
[381
which is always convergent.    When x is great, we have the semi-convergent series
(1     1.2     1.2.3.4
r / x             f1     l ' 2 , i 0) = sin as \ ---- r- +
^   '                  (X         X*
(1     1.2.3    1.2.3.4.5
- cos x \n -- - -- 1 -- -. -he2         a;4                  #6
.(19)
Fairly complete tables of Ci (x), as well as of related integrals, have been given by Glaisher*.
When m is large, Ci (2m) tends to vanish, so that ultimately
nn 1  cos 2m 'o          m
' dm = 7 -}- log (2m).
Hence, when TsR is large, (13) tends to the form
(13) = 7rM4(.fir-l)2.......................-.(20)
Glaisher's Table XII gives the maxima and minima values of the cosine-integral, which occur when the argument is an odd multiple of ^TT.    Thus:
n   I    Ci(nr/2)     l[   n 1              il			Ci (H7T/2)
1	+ 0-4720007	7	- 0-0895640
3	-0-1984076	9	+ 0-0700653
5	+ 0-1237723	11	-0-0575011
These values allow us to calculate the {  ] in (13), viz.,
7(1  cos 2m)    sin 2m    e ,         / 4       \r      ,              n.       X1    ,    .
-- -  TT-* - -- + 5 + m2 + -- - 4  [7 + log 2m - Ci (2m)],  (21)
2m2                m                     \m2      y L '                     v     /J   x    x
when 2m = mr/2, and n is an  odd iribeger.    In this case cos 2m = 0 and sin 2m = + 1, so that (21) reduces to
56   .   4
nnr/2) - Ci (mr/2)]. (22)
We find
		1	
71	(22)	n  ,     (22)	
1	0-0530	7	23-440
3	2-718	9	42-382
5	10-534	11       65-958	
Phil. Trans. Vol. CLX. p. 367 (1870).e 7 is  Euler's  constant  (Q'5772156) and  Ci is   the   cosine-integral, defined by