The lift and moment on a flat plate 189 
resultant velocity is constant; we shall take the usual approximate form of 
this condition which amounts to assuming the deformation of the free 
surface to be small and making the tangential component of the fluid 
velocity constant. 
Thus, instead of (2), we have the boundary conditions 
dw 
R= 6 Z2=x2+ia; z=2—1b. (57) 
Following out the same process as in § 2, the appropriate form is now 
dw matt Nn ! A, ies) F*(k) e—tke—2Kxa ay P*( => k) etke—2Kb d : 
dz Ore ntl 0 it en 2kd K 
20 F( + k) e—tke—2xd _ F(k) eike—2Kd 
_ c 58 
I a dk, (58) 
and it may be verified directly that this form satisfies the boundary con- 
ditions (57). ‘ 
It follows that the expressions for the lift and moment are now 
Kk) P* (x) e2*¢— F(—x) F*(—k) e- 2X? 
l — e—2Kd 
Y =pcl'— 2mp| a dk, (59) 
0 
oO A (pe * (jc) p—2Ka — Fl" ( — pe o3// — —2kb 
iq = 2npRi| c4,— | F'(k) F*(k)e F'(—k) F*(—k)e d 
= e—2Kd KK 
Ih {P'(k) F( = re F(k)} re ar | : (60) 
It is clear that we may write down the expansions from those in the 
previous sections by replacing each coefficient 7,, by —7,, and leaving the 
coefficients r/ unaltered in sign. 
Hence, instead of (45) we have, 
L/L) = 1+6, p+b. p? +b, p? +b, p?+.... (61) 
b, = —2r)sin8, 
by = 3r2 sin? @—r, — 27} cos 26, 
bs = —4r8 sin? 6 + 2797,(2 sin 6 — sin’ 0) 
+r, sin 7 cos 26 + 8797; sin 0 cos 26, (62) 
b, = Srésint 6 — 3r27,(3 sin? 0 — 2 sin @) + 37? + 373 cos 20 
— dror2(7 sin? 6 — 12 sin* @) — 187g7r; sin? 6 cos 20 
+ 37,7} cos 20 + 3777 cos? 26 + 3r5 cos 40. 
450 
