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Full text of "Scientific Papers - Vi"

1913]                        FINE  SLITS IN  THIN  OPAQUE SCREENS                             109
Thus, if n = 2, we get TT [|- 4- cos2 a].    If n = 4,
f3.1     4.31      9                1           ,
TT   -: — ^ + :. — 77 H cos2 a + cos4 a   ,    and so on.
The coefficient of (31), or (32), in (30) is
At the centre of the aperture where t] — 0, cos a = 0, (32) reduces to its first term. At the edges where cos a = + 1, we may obtain a simpler form directly from (31). Thus
(31) = T (1 + cos 6)n dO = 2'V27*    l'   n n 3 '-v    y    J o               y                    2n. 2w - 2 ... 2
............ (34)
For example, if n = 6,
11.9.7.5.3.1     231 TT
6.5.4.32              6'
We have also in (30) to consider (n even)
dd (cos (9 - cos a)n log {+ 2 (cos 6 - cos a)}
fa7/i   •„# + «   • „ ^ - a ,      ( .   .    e + a   .   a -6} =     dv smn —  • sinn —= — log 44, sm — „•— sin — - - — y
f"   7/1     -«^+«     '    «^~«!       (,      '       0 +  «     '       ^
+  1      /Vw cjivi'*        _ _ . din" ___ . _   no* ) At cnvi _ cain CtC' bill         _•      bill         ^       J.Ug   < •* HJ.11       g       (S1U
Jo.                     ^                  £              (               &
M sin« f + ' sin" iz« log {2 sin 9+?l ^            ^         (.          A   )
+ I   c^^ sin71 —5-^ sinw —J^ log j 2 sin ^r- [ ^o                &             A          [           /   J
+ f* *9 sin" * + i sin" ^ iog (2 d
Jo.                 "              A
[                    "
sinn <i sinw (6 - a) log (2 sin </>)
"   " o
+ 2 f "   " dd> sinw $ sin n (<6 + a) log (2 sin <j>)
Jo
= 2 f " d<j> sinn <j> (sinn (0 - a) 4- sinn (<£ + a)} log (2 sin 0)
Jo
f Jr+Ja
+ 2 |         rf<^> sinn </> sinn (0 - a) log (2 sin <£)
. JT /•*"•
- 2           d0 smn ^ sm^ (0 + a) log (2 sin <£).
J Jir - Ja fin"
= 2      c?</> sinw </> (sinn (<f> - a) + sinn (</> + «)} log (2 sin 0), .... (35 )
Jorent 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