Brard 



-^ F*(t') j'(Vp,V', t'-T')dT' = G(T)^) = j'(Vp'+^) 



J dr 



where 



B , , , '^ , , , , 



— T J'C^P- t' -r') = - — ^ J (7]p, t -T ) 



[ ^ J(-^'p.-^'. t'-T')d7]', with J'(t7^,0) = 0. 



Using again the Laplace transform, we obtain: 



sf*(s) j'(77^,S) = jG(7];) = -J'(7]^+C0) 



Consequently, we may introduce a function h(s) and write: 



j'(^p,s) = G(77^) h(s) . 



Expression 



-^ H(t'-r')dT', 0(7)^) 



is the contribution in Giv^), when the motion is steady, of the area dS between 

 the two arcs deduced from d^ ^ by the translations -i^L(t' - t') and 

 -i^L(t ' - t' - dr') and the two streamlines of the relative motion coming from P 

 and P' . 



Function F*(t') is the solution of a singular Volterra's equation of the first 

 kind 



t ' 

 r F*(t') H(t' - T')dT' = 1 . 



Putting t' =kt', t'-r' = (1 - \)t', this equation becomes: 



1 



t' r F*[\t'] H[(l - \)t']d\ = 1 , 







what implies 



F*[kt'] n[(i-k)t'] = o{t'-i} (21) 



for t ' very small. 



840 



