194. T. H. Havelock 
ELLIPTIC CYLINDER 
11. The expressions for a flat plate have been obtained as limiting cases 
of those for an elliptic cylinder. We shall consider now the general case 
when the cylinder is in a stream between plane parallel walls, and we shall 
examine the moment of the forces; further, in order to simplify the caleu- 
lations, we assume in this case that there is no circulation. 
Referring to § 4, we have to determine F(x) from (27) and (28) with 
ISO, 
The process of approximation is carried out as before, and we record the 
result up to terms necessary to give the moment to the required approxi- 
mation. We obtain 
P(k) = — $ep(dxpB, e + x2p?B, €29 — 1 x33 B. e310 
~ ork p Bye + sag p? Bee" + ...), (79) 
B, = e —ge- + I p?r, (2qe — q2e-19 — e-i#) 
— h pri (e3! — get? — ge-Bi8 + ge~i?) 
— 5 p'r,{2q(e + e308) = (1 ae q’) (e? + e8t8) 
+ TEP ri{(1 + 39?) e” —9(3 +97) e-} 
a 3 pry ry {4qe3? ata (1 ab 3q7) (e? ale GW) ae 2q(1 dt q°) Ck (80) 
ab tptrsqen = Creme + q(l as q’) e—t0 — 2q7er? 4k ger? ie eb 
+4 piri(qe9 + Ge 89 — ge3i0 4 e5i8) 4 
IB Ne Gem tee 
B, = e—qe" +... 
B, and B, are of order p? and do not contribute to the value of the moment 
up to terms in p?. 
Using (79) and (80) in (14), we obtain, after some reduction, 
M |mpe*p? sin 0 cos 6 = 1+ p*{4qr,+73(q— 2 cos 20)} 
+76P {ri(1 + 3g?) + 7r3(1 + 9? —4 cos 20) + 4r,7/(1 + 3q2— 8 cos 26) 
+ 4r7°7(3 + 3q° — 8q cos 26 + 6 cos 40) + 275(3 + 2 — 8¢ cos 20 + 6 cos 40)}+.... 
(81) 
In this expression, @ is the angle the major axis makes with the direction 
of the stream, a, b are the distances of the centre of the ellipse from the two 
walls, and the coefficients r are given in (46); further, if a’, b’ are the semi- 
axes of the ellipse, we have p? = a’?—b’? and q = (a’ +b’)/(a’ —b’). 
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