Scientific Papers - Vi

1913] FINE SLITS IN THIN OPAQUE SCREENS 167
Accordingly, h0 = 0. More generally we set, n being an even integer,
Jo or, on integration by parts,
hn = I ' cos 8 {(n ~ 1) sin71"2 8 cos 8 log (2 sin 8) + sinw~2 8 cos 6} dd
= (n-l) (An_2 - hn) + (sin"-8 0 - sin71 0) d0. J o
Thus
, _
»n —
In — 3, n — 5 . . . 1 TT
H "n --- ~
?i - - w» ft - 2, n - 4 ... 2 2 ' by which the integrals 7in can be calculated in turn. Thus
hz = 7T/8,
7 3, . 1 1 7T 7T3.1 / 1 1
.(25)
24.211.2 3.4
A -5-3_ilz:fJ_4. a
fl 6.4.22 Vl.2 + 3^
7T 5 . 3 . 1 / 1 1
. 62 4,'. 2 2
1
7r7.5.3.1 /I 1 1 1 \ ,
«— _ ________________L._jnL [ ______ _i_ _,,,r.-,.J,_-^-r „! f_______ _L________ QTin Qn nn
"2 8.6.4.2ll.2 + 3.4 + 5.6 + 7.8j' ana so on-
Similarly
It may be remarked that the series within brackets, being equal to
approaches ultimately the limit log 2. A tabulation of the earlier members of the series of integrals will be convenient : —
TABLE I
2fco/7T =0
2/i2/7r = 1/4 2A4/7r =7/32 2Vw = 37/192 2V-7T = 533/3072 2/i10/7r = 1627/10240 fir = 18107/122880
2/i14/7r= 2/ilfl/7r= 2/i18/7r=
0-25
0-21875
0-19271
017350
0-15889
0-14736
013798
013018
012356
0-11784
The last four have been calculated in sequence by means of (25).tegration with inclusion of more terms in the series representing D. As a preliminary, it will be convenient to discuss certain definite integrals which present themselves. The first of the series, which has already occurred, we will call /?„, so that