Untformly Valtd Soluttons of Laplace Equatton 
a(t) 
@ (x,y) is an arbitrary function and A me stands for the operator 
a1) aoe 
oxe gy2 
The same procedure is used to know the behaviour of the 
; (2) 
function ® Se 2 mil?) 
a— +74 = 
3 xy 
~ (2) ae et) (1) 
o® ak I ar = 95 ‘eis. el tian 
————-= 7 ! ——— + ff! ——— A ce) = 716) 
Oz - Ox mt Oy i xy : ae) 
The 7 -expansion in the vicinity of a point (x,y) on the planform (S) is 
(1) (1) 
~(2) (2) i 5 ONaDE- Se" OBI (1) L iden Ae 
$= (sy) +9 We +f 5 Frifdig Pe “zz AP pole 
If we consider a point (x,y) on the vortex sheet (£), the Taylor be- 
haviour of any function @) is: 
ql ae gu 
2 
= Po : ne 2, ) 
+ 0( 7 
The pressure continuity across the sheet gives additive 
conditions such as 
eS |) eal =} z 
__0¢k _ Ok | 
Ox Ox 
Outside SUX, p= . Consequently, rae is a discontinuous function 
on the curves (A), (L,) and (L,). At these points the flow velocity has 
no meaning and the outer expansion (2) is not valid. Elsewhere, this 
expansion is uniformly valid, as it can be shown easily in the follow- 
ing analysis corresponding to a point (x,y) located inside the plan- 
form (S). 
If the approximation (2) is not valid in the neighbourhood of 
(S), it must be within a region of thickness 0(€), because, in sucha 
layer, the boundary conditions must be written on the real body, and 
cannot be approximated by a condition on the plane z=0. Replacing 
n by €, the preceding results give the behaviour of the inner solution 
at infinity rewritten with the inner variable Z , as indicated by the 
matching conditions, 
1743 
