Darrozes 
5 ge” + 1 
¥ = a AA 
a (2) + 5 (1) + (1) + 2 ) 
ee SE SE OS Vix) € (5) 
We are going to consider now, the validity of the so-called ''outer 
expansion" (2 ), 
It could be plausible that some difficulties arise when looking for the 
behaviour of outer solutions @ '*) in the vicinity of the part (S) _ of 
the plane z = o, because the liquid cannot go through the body, and 
the first approximation seems to be a non uniform flow. 
II, 2 - The region of uniform validity in the plane z =o 
It is known that, any point in the xoy - plane is regular for 
the outer expansion (2 ), except the points located on the lines ( A ) 
( L,) and ( Lg). This result is easily proved by introducing a new ex- 
pansion in the following way. 
Any function A (x,y, z) is written A (x,y, 7z) = A (x,y,z7). 
The formal expansion of the function A , when z goes to zero, is iden- 
tical to the expansion of the function A , when 7 goes to zero witha 
fixed value of z . The 7 -expansion must depend upon the cluster 
ne \ since A does not depend upon 7 
Inserting this formalism into the problem governing the 
e - term @!'1) of the potential velocity, the behaviour of the solution 
for vanishing values of z is given by the corresponding 7 - expansion, 
(ih) 
a Oxy. 2, 0) <8" my) + 1O["( say, Z) + 
@ (x,y,7z) =® 
ae n2A ® hava 
922 xy 
3p) ae ee 
ts = a Ze O , (aye ee. 
Zz 
The following result is obtained without any difficulty 
1 ~ 
® Al) 11,363 
y ee 
af 1zf-=1 DW AyyP -=7Z 
(1) | re eS “11 5 
a xyix+55" 2 xy Day B'+0(7 )-1a) 
1742 
