Brard 

 -grad [OjjCm^) + O^ 3(111^)] nCm^) = [- grad ^02(11^) + V^2('"e)] "("^e) 



= ^ [iyAOm^] n(m^) . (14) 



Therefore $02 + $02 fulfils on s^ the same condition as the wanted potential 

 Oq 2 • Because these two potentials are regular in Q.^ and at the infinity, one has 



(I)o2(M) = $0 2(M) + %2(^) + constant in fi^ . (15) 



This equation defines t^^C"^)' ^^^ ^^^ 



to2("') = |{-grad a)o2(me) - iyAOmJ An(m^). (16) 



This solution does not depend upon the choice of the rings c^ since the potential 

 $02 is perfectly defined (with the exception of an additive constant), by the con- 

 dition on Sg. 



3.4. The Vortex Distribution When the Fluid 



is not Quite Perfect ' 



In this case a vortex wake exists. Let us assume firstly that 



u=0, q=0, -jjM- 



The total velocity potential may be written: 



(17) 



where 



W WW 



^00 = ^00+ 'Poo ' '^oi u = "^01 u + ^01 U • 



In these expressions Oqq, and % i(w/U) are the solutions obtained in par. 

 3.2. The potential ^^^ has to be added to ^qq when a wake already exists for 

 w/u = 0; the potential Wg ^Cw/U) has to be added to o^ i(w/U) when a wake exists 



for w/u M- 



Figure 4 suggests that the wake is made of free filament vortices shed 

 along a not necessarily closed line d^ ^. 0.^ ^ is approximately in the (x, y) - 

 plane . 



For reasons of generality, we consider a closed line d^ ^ which contains the 

 arc (Jqj. On the arc (Jgi - fig j no vortex is shed. 



It is possible to consider ^P = ^Pqo + "^o iC^/U) as generated by two families 

 fg, fg of free and bound vortices (Fig. 5). 



828 



