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Full text of "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