The lift and moment on a flat plate 195 
The moment for a flat plate in a stream between plane walls, and without 
circulation, has been obtained, by conformal transformation, by Tomotika 
(1933), who also gives an expansion for the case a = b = 3d; it is 
2 2 
M/M, = 1+55(7) (14 2sin26) 
48\ d 
4 4 
~ 000) (11—106sin?0—66sin*@)+.... (82) 
If in the general result (81) we put q = 1 and use the values of the coefti- 
cients given in (46) and (49) we obtain again this particular result. 
We shall use (81) to illustrate one point, namely the change in the moment 
when a flat plate is replaced by an elliptic cylinder whose major axis is of 
length equal to the width of the plate; thus we examine the effect of rounding 
the edges of the plate and giving it a finite thickness. 
To simplify the calculation, we take the cylinder in the position given by 
a = 6 = 3d. Then (81) gives 
2) /a'\2 
M |7pec?a’* sin 6 cos 6 = aja +7) (2q — cos 20) 
ie mA? (a’ 
23040\d 
where A=1-b/a2, g = (a'+b')/(a’—0’). 
4 
{99 + 1689? — (300 + 44¢) cos 26 + 33 cos 46} + | , (83) 
We begin with a flat plate of width 2a’, and then keeping a’ constant we 
increase b’; to simplify the calculations we have taken the position given 
by 6 = 45° and the following table shows the result of the calculation for 
various values of the ratio a’/d. 
a’/d 0 0-1 0-2 0:3 0-4 
b’/a’ 
OR 1-0 1-0165 1-0673 1-1561 1-2885 
0-05 0:9977 1-0159 1-0714 1-1690 1:3149 
0-09 0-9917 1-0113 1-0717 1-1734 1-3361 
0-13 0-9830 1-0038 1-0679 1-1799 1-3484 
0-2 0-9600 0-9829 1-0535 1-1773 1-3640 
0-5 0-7500 0-:7780 0-8655 1-0227 1-2664 
For an infinite stream (a’/d = 0), this process of increasing the ratio 
b’/a’ with a’ constant gives a moment which steadily decreases to zero when 
b’ =a’. Aninteresting point which arises from these calculations is that in 
a stream of finite width, with plane walls, the moment rises to a maximum 
456 
