Brard 



There is no velocity potential in n. since q^O, and curl v^ = 2qi • But we may 

 write: 



-grad O. 



.(M) = ^ curl ||[ ^^ dS(m) . f |[[ ^ dO,(Mi)| , M in fi, . (10) 



We have to define t^jCm) (Fig. 3). 



In order to do that, let us consider on s the rings c ^ and c '^ located in the 

 planes of abscissae x and x + dx, (dx<0). Letm'.mJ be two points on 03, m' 

 being on the starboard side, m^ on the portside, with z(mj) = z(m'). A, origin of 

 the arc a^ on Cj, is chosen on the upper arc of the contour C along which the 

 planes tangent to s are parallel to i . A' being on c^ and on the lower arc of 

 e, we consider a point m on the arc m'A'mJ. Let dCT2(m'), do-jCm) be the distances 

 measured on s, at m' and at m, respectively, between c^ and Cj. These dis- 

 tances are considered as positive. Moreover, i^Cm) is the unit vector tangent 

 to c^; ijCm) is positive with respect to the x-axis. The arc da^ > has a di- 

 rection identical to this of ij. 



Now we define at m an element of bound vortex dt^jCm) = i2(m)dt'(m) by the 

 condition that 



o , 2U . 



dt (m)(Sda ) = + 1 (ndo- da ) . 



■^ m L ^ ^ ^ m 



Consequently, the filament vortex which intensity is equal to 



- — i„dx dz( m ) 

 L y 



on the segment m^m' and to dtQ2(m)(Bda'^)^ on the arc m'A'mj, is closed and this 

 intensity is constant along the filament vortex. It is the flux of the vortex 

 (2U/L)iy through the small area (da^do-'^)^, on s. 



The total vortex at m has an intensity given by 



m _ 



(11) 



This vortex is equal to zero at A. 



It constitutes with the vortices (2UA)i located in n^, a family of closed 

 filament vortices having a constant intensity along their length. Consequently, 

 the vector 



V.,«=ic..{|^.s.„,./£f./-^} (12, 



826 



