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Full text of "Scientific Papers - Vi"

1913]                        FINE  SLITS IN THIN  OPAQUE SCREENS                             181
the denominator.    In (30) we are to introduce under the integral sign the additional factor M>2sin2#.    As regards the second term of (87) we have
dDj  _ [+by(y-r,)dyl dD dy dy   y ~] _6 V(&2 - 2/2)   r ~fo '
where in - -7- we are to replace r by + (y — tj).    We then assume as before
y — b cos 0, rj = 1} cos a, and the same definite integrals hn suffice ;  but the calculations are more complicated.
We have seen already that the leading term in (87) is TT.    For the next term we have
n          .     ikr            I dD    k*    k* f       ,     ikr
£ = 7 + logT,         ?^ = 4-
and thus
% cos20 + |cosacos#)log±2(cos0-cosa). ...(88)
o
The latter integral may be transformed into
/V 2     cZ</){l-fcos2(2<j!)~cO + .£cosa cos (2<j> - a)
Jo
4- 1 - f cos2 (20 + a) + £ cos a cos (2^E> + a)} log (2 sin 0), and this by means of the definite integrals h is found to be
-~ (1 + 2 sin2 a). o
To this order of approximation the complete value is
-^ = ,r + iwM>%-sina.a + logiM?&) ............. (89)
CLX
For the next two terms I find -f --^- [(1 -I- 4 cos2 a) (1 - 47 - 4 log J ikb)
oL2
-|- 3 sin4 a 4- ^ cos4 a + 6 sin2 a cos2 a]
7rM>°     T
+ oriT-fi   (^ + f ^s2 a + 1 cos4 a) (7 + log ^B - f )
Zl   . IS   . U   [_
+ Q-^-YK cos8 a - 5^-5 cos4 a sin2 « - -T cos2 a sin4 a - ^r-^ sin8 a ... .(90) cr . lo             o . o                     o                       o^ . o        J
When cos a = 0, or ±1, the calculation is simpler.    Thus, when cos a = 0,t departure from (86) is entailed when kb is no longer very small.