1913] FINE SLITS IN THIN OPAQUE SCREENS 179
theorems relating to the flow of incompressible fluids lead to the desired conclusion. It appeared that (74), (75) give
showing that when b is small the transmission falls off greatly, much more than in case (i), see (20). The realised solution from (78) is
. , , , , /h_-.
s(nt~kr~M' ............... (79)
corresponding to ^m = 2 sin nt sin kx ............................ (80)
The former method arrived at a result by assuming certain hydrodynamical theorems. For the present purpose we have to go further, and it will be appropriate actually to verify the constancy of dtyfdfa over the aperture as resulting from the assumed form of ty, when kb is small. In this case we may take D = logr, where r2 = #a + (y - 77 )2. From (71), the suffix p being omitted,
... d-'D . t .
and herein -_=___._ = _.^_(?7 const.).
Thus, on integration by parts,
^ j /£m
- \V-j- \+ -^--j-dy ................... (81)
dy\ J _fl dy dy J
Tr, ..
dy ~ drdy~ (y -
and so long as r) is not equal to ± b, it does not become infinite at the limits (y=± b), even though so — 0. Thus, if "$f vanish at the limits, the integrated terms in (81) disappear. We now assume for trial
¥ = </(&•-?•), .............................. (82)
which satisfies the last-mentioned condition. Writing
y = b cos 6, rj — l} cos a, of •=• x/b,
d^r /"""(cos 6 — cos a)2 + cos a (cos B — cos a) , „ /ori.
we have - -r- = - -- ~( - ^ ---- --• — -,. --- -7 dd .......... (83)
dx J o (cos 6 - cos a)2 + so 2 v J
Of the two parts of the integral on the right in (83) the first yields TT when x = 0. For the second we have to consider
cos. 6 — cos a
----o (cos 6 — cos a)2 + cc*
12—2vations with the microscope, we restrict ourselves to