1913] FINE SLITS IN THIN OPAQUE SCREENS 175
At the centre of the aperture (77 = 0),
(61) = 26 {7-1 +log^'6}, and at the edges (77 = + 6),
(61) = 26 [7-l + log-i6}.
It may be remarked that in (61), the real part varies with 77, although the imaginary part is independent of that variable.
The complete expression (60) naturally assumes specially simple forms at the centre and edges of the aperture. Thus, when 77 = 0,
b" 5"
¥
22.42.5V 2 and, similarly, when 77 = + b,
i i r
.....'(62)
i i
22.42.5 .
......(63)
To restore k we have merely to write kb for b in the right-hand members of (62), (63).
The calculation is straightforward. For the same values as before of kb and of cos2 a, equal to 7f/62, we get for (60) -f- 26
TABLE III.
w M=4 M = l /c& = 2
0 i 1 -1-7649 + 1-5384 i - 1-4510 + 1 '4912 i - 1-0007 + 1 -4447 i - 1 -0007 + 1 '4447 1 -0-6740 + 1 -2771 i -0-2217 + 1'1 198 i - 0-2167 + 1 '1198 i -0-1079 + 0-7166 i + 0-1394+0-4024 i
We now proceed to combine the two solutions, so as to secure a better satisfaction of (17) over the width of the aperture. For this purpose we
determine A and B in
y = A(b*-f)-* + B, ..........................(64)
so that (17) may be exactly satisfied at the centre and edges (77 = 0, 77 = + &). The departure from (17) when if/IP = •£• can then be found. If for any value of kb and 77 = 0 the first tabular (complex) number is p and the second q, and for 77 = + b the first is r and the second s, the equations of •condition from (17) are
= -l..........(65)ading term in (60) is thus kb = 2