The lift and moment on a flat plate 183 
with G, = Xing” J, (vpe—**) J, (Kpe*”), 
Gy = E(— 1)" nJ,(vpe!) J, (xepe), 
G, = EnJ,(vpe!) J, pe), 
Gy = 2(-1)" ng" J, (vpe—”) J, (Kpe”). 
(28) 
FLAT PLATE BETWEEN PARALLEL WALLS 
5. We consider the limiting case obtained by making &, zero, that is, 
by putting q = 1 in the previous results. The cylinder reduces to a flat plate 
of width 2, at an angle 6 to the direction of the stream, and with its mid- 
point at distances a, b from the upper and lower boundaries, respectively. 
We write T= 2nkepsin6; F(x) = cpsin @f(k). (29) 
The equation for f(x) is 
S(k) = keJo(Kpe"*) — a, (pe) 
+ {f(v) Gy + £*(v) Ga} oP + {f*(— v) Gy + f= v) Gah er?” dv 
0 
| —e-2vd v 
vi I @{f*(=0) @ +f(=2) Ga + flr) Ga+ f(r) Gert" dv (a) 
) 
0 tes e2vd 
Gy, Go, Gs, G, being given by (28) with g = 1. 
We approximate to f(k) by successive substitution of approximations 
for f(«) in the integrals of (30), repeating the process as far as may be desired. 
Our object is to obtain the various quantities ultimately in power series in 
p/d, or alternatively in p/a or p/b, assuming these ratios to be less than unity. 
The expansion for f(x) is most readily obtained by replacing the Bessel- 
functions in (30) by their power series as far as necessary so as to give all 
terms up to a required order in the final results. We shall develop these 
expansions up to terms of order (p/d)4; except for the length of the expres- 
sions, the expansions could readily be taken to a higher order. Itis sufficient, 
for the present purpose, to take as the first approximation 
f(v) = k— hivpe® — dhv2pe? + Livi pes? + dkvtptet?, (31) 
Further, to this order, it is sufficient to replace G, by 
G, = txpe™ (Supe !? — gev'pre*”) 
fu LK2p?e?"9(Ly2pe—28) = qigk3 p33? (Lupe), (32) 
and G,, Gs, G, by similar expressions. 
444 
