The lift and moment on a flat plate 185 
Hence, writing A, = te w, 
A, = J,(Kpe’*) a iJ, (Kpe*’), ( 
this gives 
2) * —2ka _ Cl ees —2Kb 
+(-1)"C, = — Jepe”— pe A, F*(k)e Bell (—k)e as 
0 l—e-« 
2 — = a —2«d 
+ he] {4, F(-«) A, ED ike (60) 
i 0 1 Se-2 
Hence the equation for k, to give the required circulation 27kcp sin 0, is 
, ; 20 4 fi(k)e*t*-—A f*(=k) e—2Kb 
bes 1+ Ipe| s aero dk 
of NV —2xkd 
~ Ipe'#| (Af oa dx, (41) 
d 
with A,, A, given by (39). 
We substitute (35) in (41) and also use the power series for A, and A,; 
carrying out the integrations and using the same notation as before, we find 
k = 1+kprysind + 4p, (k cos 20+ B,) — p*ri(k + B,) cos 20 
— 41 p*r.(k sin 30 + B, sin @ — B, sin @) 
— 1, p1r,(k cos 40 + 2.B, cos 20 + 2B, + Bs, cos 20) 
+i ptr(k+2B,+2B,+B,)cos40+.... (42) 
Finally we substitute from (36) and solve the equation for /; we obtain 
k= 1+4, p+. p?+a,p?+a,p*+..., 
a, =7osind, 
dy = r2 sin? +r, cos? 0 — 27; cos 20, 
dz = resin? 6 + 2r97, sin cos? 0 — tr, sin 30 — 4797; sin 8 cos 20, (43) 
a, = résin?6 + 3r27, sin? 0 cos? 6 — drory sin @ sin 30 
+472(3 —6 sin? 6 +4 sin‘ 0) —+4r5(2 + 3 cos 20 + cos 40) 
, 
— 6r27} sin? @ cos 20 — 37,17; cos? 0 cos 20 + 371? cos? 20 + 3r3 cos 40. 
7. We may now obtain the lift from (11) and we express it in terms of 
the corresponding lift in an infinite stream; that is, we write 
VS lig Vn = iby = 2s 
446 
