Untformly Valtd Soluttons of Laplace Equatton 
first approximation, we must choose: r(s)-=s°andu(s)=s 
Then 
ee 
ne ro (a, Wee zag F ] +0 ( 5% ( 10) 
Om i: 
This expansion is shown to be uniformly valid in a neighbourhood of 
the point I, after verifying that : 
the limit Y —-cogives expression ( 8 
the limit YW-.o0 gives expression (9 
— — 
III. 2 The vicinity of a round leading edge 
This paragraph is devoted to the study of the singular beha- 
viour of the outer solutions "and gi), in the vicinity of a point A 
belonging to a round leading edge. Let us start our investigation with 
the function @) solution of the equation (11 ). 
- fi! ( sheeylewe-n SS (es) 
which is rewritten using ae curvilinear Sopramnates a) 
Mc ( —2 94 yc 
Ae iy ot 8) 8. OEE pis See) ae. G(s oe = 
1=YC(s) de® OS. 1-¥C)a5.0 8 
= 
Sia 1 = Yc(s) (zA(s) +H" )- sina( s ) Hy" ae) 
(12) 
There are many solutions to this problem, due to the fact that we cannot 
take into account the condition at infinity, when looking for a beha - 
viour. Any solution of equation ( 12 ) is obtained from a particular 
one, after addition of a solution corresponding to the homogeneous 
problem oe F Z—»0O 
The homogeneous problem 
In order to investigate the behaviour of the homogeneous 
solutions in the vicinity of A , we define the new variables 
~ 
Mie in/ fn Z = (2 -¢2,(#)) / 
with 7 <1 
We look for an 7 - expansion of the function 
@!") (« n¥, cay 2) +92] = © (ms, *Z8) 
1747 
