Brard 



It is possible also to consider a free vortex of the f^ -family as made of 

 three vortices of intensity dy^ : vortex (i) on the arc P'm'P and the segment PP'; 

 vortex (ii) on the arc PKP' of (Jg j (k in the (z,x) -plane), and on the segment 

 P'P; and vortex (iii) on the arc P'KP of Q^^, and on the stream lines of the rela- 

 tive motion starting from P and P ' and leading to the infinite downstream. 



Vortex (i) is equivalent to a distribution of normal doublets on the part 

 S'(P) of S' behind the arc P'm'P and on the part S^CP) of the surface I/P) 

 generated by the segment pp' when P and P' describe Q.^^; 



Vortex (ii) is equivalent to a distribution of normal doublets on the surface 



Vortex (iii) is equivalent to a distribution of doublets on the part 2(P) of the 

 wake which edges are the arc P'KP, and the streamlines starting from P and 

 from P ' . 



Because the distributions of doublets on ^^(P) are equal and opposite, the 

 contribution in ^^ of the vortices (i), (ii), (iii) is due only to the doublets dis- 

 tributed on S'(P) and on I.(P). 



A similar reasoning may be repeated for a vortex of the f^ -family. 



Finally, the contribution in Wg of the vortices just considered is 



d\ = dYo + dy^ , (19) 



with 



^^0 = -if/ d^ ii^^(^)'^^'^o(P). 



dy' =^ -T^ ^ dS'(m') xdy' + ;j^ If -T^ ^dS"(m")xd7" 



s'(P) '"' '"'M ^0 477 JJ^_^ dn^„ ^"^j V ; /o 



V (19') 



In these formulae, the unit vector n is normal to 2, and therefore, approx- 

 imately identical to i^; the unit vectors n^/.n^j,,, are positive outwards. 



The total potential y^ is therefore given by 



^o(M) = - 4^ J r„(P)dyp ^ ^ d^(M) , (20) 



^0 1 



where P describes the arc P^'K • Pj, P^ and P^ being the extremities of d^^, and 

 x(yp) the abscissa of P on 0.^^. The coordinates of /^ on 2 are ^, yp. 



On the other hand, the total potential YqCM) , generated by doublets on s, may 

 be written 



830 



