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