464 ON METHODS FOE DETECTING SMALL OPTICAL RETARDATIONS, [415
But if we wish to avoid the infinity when 0 = 6, we must make some supposition as to the actual value of d%, or rather of 27r0£/A,. In some observations to be described later a= I inch, £ = | inch, I/A, = 40,000, and 6 — ^ajf. Also / was about 10 feet = 120 inches. For simplicity we may suppose /= 407T, so that 2?r0£/X = 500, or in our usual notation 6%= 500. Thus
(19) = Si {500(1 + 0/0)} + Si {500 (1 - 0/<9)j, .........(26)
and . (21) = Ci {500 (1 - 0/0)} - Ci {500 (1 + 0/0))
+ log(l+0/0)-log|l-0/0|......................(27)
For the purposes of a somewhat rough estimate we may neglect the second Ci in (27) and identify the first Si in (26) with \TT for all (positive) values of 0/0. Thus when 0 = 0,1 - -rr2; and when 0 = oo , / = 0.
When 0/0 = 1, we take
(26) = |TT = 1-571, (26)2 = 2'467.
In (27) Ci {500 (1 - 0/0)] = 7 + log 500 + log (1 - 0/0), so that (27) = 7 +log 1000 = 7-485, (27)2 = 56'03;
and . 7=58-50.
For the values of 0/0 in the neighbourhood of unity we may make similar calculations with the aid of Glaisher's Tables. For example, if 0/0 = 1 + '02,. we have
500 (1-0/0) =±10.
From the Tables
Si (± 10) = ±1-6583, Ci(±10) = --0455, and thence 7(-98) = 31-13, 7(1'02) = 20'89.
As regards values of the argument outside these units, we may remark that when x exceeds 10, Si(a) — |TT and Ci (x) are approximately periodic in period 2?r and of order ar1. It is hardly worth while to include these fluctuations, which would manifest themselves as rather feeble and narrow bands, superposed upon the general ground, and we may thus content ourselves with (25). If we apply this to ± 10, we get
7(-98) = 30-98, 7 (1-02) = 21-30;
and the smoothed values differ but little from those calculated for ± 10 more precisely. The Table (II) annexed shows the values of 7 for various values of 0/0. Those in the 2nd and 8th columns are smoothed values as explained, and they would remain undisturbed if tfee value of 0£ were increased. It will be seen that the maximum illumination near the edges is some 6 times that at the centre.n unity. Under the same conditions the Ci's in (21) may be omitted, so that