Untformly Valtd Solutions of Laplace Equation 
Z 
+ 
{ body trailing 
if edge fl ¥ 
Figure 6 
If theN-expansion of the velocity potential has aterm  o (1) 
gy 2 H 
= 
the function ae is a solution of the homogeneous problem 
A S41" = oz, %,)= Ao (Z) +Ao (Z,) +i (Bo (Z) - Bo (Z)) 
but A, (Zz) and Bg (Z) are holomorphic functions except on the real 
axis. ce the negative values of Y , Ao (Z) and Bo (Z) are not defi- 
ned when Z goes to zero, and the limit functions are denoted At (Y) 
and B#(¥). 
In the same way, we denote by AQF) and BO\z) the corresponding 
limits for <s o. As an exemple, we solve the problem involving 
Ag (2). 
Var |- 
Ag = Ajo Sulie 
k = 0, because the function A)(Z) cannot contain %. Then Ap (zZ)isa 
constant, but this constant may have different values in the upper pla- 
ne and in the lower plane (Z <0). 
Ag (z) = cot (s) Zope 
cl (27) 
Arp 28) “Seg es) Z Giile 
+ 2 
It is easily proved that Bg (Zz) =ieg(s) so a ds jt eo (s) 
a4). 
The following term is the Ri expansion of ® is not of order 7 %. In 
fact, if sucha term 746!" does exist, with 
®) 1. A,(zZ) +A, (z ane B, (Z) - B, (24) te pressure condition 
(26) shows that 
AO, alo +i pi®. = Als !@y 5 Ex Bl © (28) 
£Y53 
