Brard 
29 _(M') 
2 1 1D Se 
div curl |— a ee ee ED on | aD (M)<29o ; 
ip s/f, MM : 
It follows, as in the case of (3.9), that the solutions of (5.7) are 
T Ber + \n, 
(M) = TY (M) +3, a 
ith — é 
wane n T) = constant ys 
and therefore that there exists one, and only one, vortex os which 
is tangent to > and satisfies (5.7). E 
One has 
7 ahacin. Vy Ake =n AV" : y 
T' (M) = -n,, af (M_) -V a) ny, AVM) on »2 (5.10) 
At each point M on oe é (n}, : Ou ‘ tna) are three unit vec- 
tors making a right-handed system ; n is normal to in the inward 
direction, and 7 is in the direction of T!'. The lines @' tangent to 
@' and the lines’ tangent to 7' determine on 3 two systems of 
orthogonal curvilinear coordinates, | the arcs o' on €@' and s' on 
being oriented in the direction of @ and r' respectively. 
! 
Let us consider (Figure 5.1) two lines € (o ) and 
Sa (0',) close to each othey and two lines Co (sh) Nees (s', + ds'). The 
flux dT” of the vortex 1 through the area Mg, Mg, Mj, Mi, is 
equal to V'(M) 8 a4 do' (where do! =o,' -¢,'). Through the area 
Me Me, — Le , it is equal © V'(M}) 6,41 do! = dT -wn de ds' 
1 ape > 4 
: VecACte eo Fit a a. ak - 
with ds' = s', - s';. Hence Sar r (s,@) o.n 29. .n,) This 
results from the fact that the vortex ribbon whose edges are ‘(o') 
and & '(c',) loses, between € (s')) and © (s',) , vortex filaments 
entering D;. 
The . are closed rings, since the ends of any segment of 
L2iz2 
