The lift and moment on a flat plate 18] 
ELLIPTIC CYLINDER 
4. We take the contour C to be an ellipse, of semi-axes a’ and b’, with its 
major axis making an acute angle 0 with the positive direction of Ox, and 
we take the origin at the centre of the ellipse. 
In terms of a complex variable € (=&+7in), we take 
z = pe cosh €, (15) 
and the contour C is given by 
£=&); pcoshé =a’; psinhé, = b’. (16) 
(17) 
We now write 
dw 
de 
the second and third terms being in a suitable form in the elliptic co- 
ordinates; to obtain F(x) in terms of the new coefficients b,, we have to 
compare these two terms with the series in (6), noting that 
dz/d€ = pe sinh €. 
Hf) 0 oll” ‘ A du 
= epe sinh €+ 5— — > 17b,e-"5 + pe” sinh ¢ aa 
Ue 2 
For this purpose we put the series in (6) into the form in (3) valid for the 
upper surface; under the same condition it can be shown that 
i" f ae =D (fi F : . AK 
di"b,e-" = pe sinh € oe Jy(Kpe’) +iXb,, J, (Kpe®)) e dk. (18) 
0 ~ 
It 
Hence, by comparison, we obtain 
TAGS) = = Jo(Kpe’) +iXb,, J, (Kpe’), (19) 
d Ee Yess 
and a = cpe’ sinh €+ = —Di"b,e-é 
Bibs 0 ANE (pe e—tkz—2ka _ Pe => etks—2Kb 
+ pe sinh c(, its) = —_ és) dk 
ie, © J F — P\ pH PF -\ pikar ,— 2d 
—pesinhf| * (Se as) eS Te 
~Jo 1— e-2Kd 
We now express this in the form 
dw ecole 
de = X(C,, em ar Dp Gs) 
