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Full text of "Scientific Papers - Vi"

166                             ON  THE  PASSAGE  OF WAVES THROUGH                          [375
We will now verify that (19) is independent of the special value of 77. Writing y = b cos 0, ij = b cos or, we have
0                                              - cs a
r*"
4-      log 2 (cos a  cos 0) dd = TT log (&)
.' a
4- I ""log 12sin ^41(20 4- I   log j 2 sin ^ 1(204- |log J2sin--^> d0
.'O             1                   ^     J                .'O             (                   &     }                .'a            (                   ^     j
log (2 sin <) dd> 4- 2 /    log (2 sin <) d(f)
J o
4-2            log (2 sin d>) dd>
J o
log (2 sin 0) <i(jb 4- 2 I         log (2 sin <^>) d<b
J       i
4-2fr   aiog(2sin-0)d<^
J o
rjrr
= TT log %b + 4      log (2 sin </>) d^, .' o
as we see by changing 0 into -TT  ^> in the second integral.    Since  has disappeared, the original integral is independent of 17.    In fact*
/*
log (2 sin <) d$ - 0,
j o
/"*"^ log 7"   rf'2/
and we have                                fl _- = 7rlog|63 ........................ (23)
as in the particular case of 77 = 0.
The required condition (17) can thus be satisfied by the proposed form, of "fy, provided that Jcb be small enough. When kb is greater, the resulting value of ^ in (15) will no longer be constant over the aperture, but we may find what the actual value is as a function of ij by carrying out the integration 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
h0 = f " log (2 sin 6} de = f "" log (2 cos 6) dO =  f '"log (2 sin 20) dO Jo                             Jo                               Jo
r^Tr
log (2 sin <) dtj> = $      log(2sin^) drf> = J/i,0.
'
o See below.ng at different parts of the width will arrive in various phases, of which due account must !>< takem The disturbance is no longer circularly symmetrical as in (16). But if, as is usual in observations with the microscope, we restrict ourselves to