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 - c°s a r*" 4- log 2 (cos a — cos 0) dd = TT log (£&) .' a 4- I ""log 12sin ^4—1(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