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Full text of "Scientific Papers - Vi"

1913]                        FINE  SLITS IN THIN  OPAQUE SCREENS                             183
With the same notation as was employed in the treatment of (82) we have
__ cfy _  f w cos 0 (cos 6 - cos a) cW _     ' " cos3 0 (cos 6 - cos a) doc     'Jo   (cos 6  cos a)2 + '2         . 0 (cos 6  cos *)- + a'2
The first of these integrals is that already considered in (83). It yields STT. In the second integral we replace cos3 9 by {(cos 6  cos a) + cos a}3, and we find, much as before, that when x' = 0
[w cos* 8 (cos 0  cosa)dd   .    .,              .                      ,nn.
 - ^ - -  >~ = TT (i. + cos2 a) .............. (96)
Jo    (cos (9  cos a)2 + * -         \*            t                   \    t
Thus altogether for the leading term we get
This is the complete solution for a fluid regarded as incompressible. We have now to pursue the approximation, using a more accurate value of D than that (log r} hitherto employed.
In calculating the next term, we have the same values of D and r~1dD/dr as for (88) ; and in place of that equation we now have
+      d8[% sin4 0-% sin2 0 + % sin2 0 cos 0 cos a] log ( 2 (cos 0 - cos a)}, (98)
The integral may be transformed as before, and it becomes
4 f ff d<j> log (2 sin <) [| (sin4 20 cos4 a + 6 sin2 20 cos2 2</> sin2 a cos2 a .'o
+ cos4 20 sin4 a) - f (sin2 20 cos2 a + cos2 20 sin2 a)
+ f cos a cos 20 |sin2 a cos a + sin2 20 (cos3 a - 3 sin2 a cos a)}]. (99)
The evaluation could be effected by expressing the square bracket in terms of powers of sin2 0, but it may be much facilitated by use of two lemmas.
If/(sin 20, cos2 20) denote an integral function of sin 20, cos2 20,
[ "" d0 log (2 sin 0)/(sin 20, cos2 20) = f "" d<j> log (2 cos 0)/(sin 20, cos2 20) Jo                                                      ' o
" d0 log (2 sin 20)/(sin 20, cos2 20) =  |    cZ0 log (2 sin 0)/(sin 0, cos2 0), o                                                          ' o
'..................(100)
in which the doubled angles are got rid of.ond we have to consider