The lift and moment on a flat plate 179 
function within the contour C, and the last term is to be determined so as 
to satisfy the boundary conditions on the walls. These conditions are 
I dw 
Game Z2=x+1a,2=2—ib, (2) 
where J denotes the imaginary part. 
To satisfy these conditions we replace the series in (1) by 
|, Fe e@dk, forz=x+ia, 
0 
|| F(—k)e“*dx, forz=2—ib, 
0 
where E(k) — Ac (3) 
0 
We may build up an expression for dw,/dz by successive images. Taking the 
expressions in (3), a single reflexion at a plane wall changes F(«) into the 
conjugate complex F*(«); if the reflexion is at the upper wall (y = a) the 
contribution to dw,/dz valid in the liquid is 
| F*(k) e—tkz—2Ka dk, (4) 
0 
while if the reflexion is at the lower wall (y = — 6), the corresponding form is 
=| F*(—k) et? dk. (5) 
0 
Taking successive reflexions at the two walls, the contributions of the 
infinite sets of image systems may be summed, and we obtain finally 
dw imintA, 
dz rags 0 gntl 
(ve) *(4-) p—ikz—2ka __ F* —K eike—2Kb 
+] Pints) “{ ) dk 
0 l—e-* 
dk, (6) 
i F( a k) ea ikz—2Kd =: F(k) etkz—2kd 
0 l —e—2kd 
where d = a+b, and F(x) = LA, kK”. 
It may easily be verified directly, by using (4) and (5), that (6) satisfies 
the boundary conditions (2). 
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