The lift and moment on a flat plate 187 
With these values we obtain 
1/2 1 /2p\2 
ifilg, = 145 ( 7) sino+7,(-”) (1+8sin?d) 
2 
1 /2p\*) ae 1 /2p\* Be Ble 
+352] (sin + 3sin 0) 52] (3 — 13 sin? 6 — 22 sin*9)+.... 
(48) 
This agrees with the expansion which may be found by the same method 
applied directly to this case. If we make a and d infinite, the stream is 
bounded by a lower plane wall. In this case the coefficients have the same 
numerical values, but ry and r, are now negative, and we see that the result 
is the same as (48) but with the terms in the odd powers of 2p/a negative. 
Another special case is when the mid-point of the plate is midway be- 
tween the walls, or a = b = 3d. In this case 
7 =0; 7, =7°/4d2; 1r2=0; 1s = 74/8d!; 
ry = 77/24d?; rr = 74/240d?, (49) 
and we obtain 
Ms 7? (2p\? Rig m* (2p\4 Sis ais eta 
L/L, = 145 (72) (1+sin 0)— sas) (11—53 sin? 6—22sin*@)+.... 
(50) 
This agrees with the expansion given by Tomotika (1934) for this parti- 
cular case. 
In general, calculations may be made from (45) and (46), and the variation 
in lift examined as the plate is moved across the channel. The following 
values illustrate this for one particular case: 
0 = 10°; 2p/d = 0-2 
ald 0-3 0-4 0-5 0-6 0-7 
Iii MOT 1-037 1-017 1-002 0-989 
8. We now obtain a similar expansion for the moment of the forces 
about the origin. If M) is the moment in an infinite stream, 
M, = 7pc*p? sin 0 cos 0. (51) 
Using (35) in (14), we obtain, after some reduction, 
M/M, = B,+kpry B, sin @ + 2kp*r, By sin? 0 
— 4p*r\ (kB, —4B}) sin? 0—} pr, {3k B,(sin 6 — 4 sin? 6) — 2B, B, sin J} 
—tp'r,(4kB, cos 20—2B, B,) sin? 0 
+ pri(kB,—2B, B,+2B3) sin? 6 cos 20+.... (52) 
448 
