2M(x) =. ~ (Uf6^) - 



2 

 ^ (Uf) - ^^ (U cos Y) +Tfe [g^u(x,g)] 



dx 



2 dx 



2 dx 



The same analysis, applied to the source distribution M (x) , yields 



2^0 (-) '---'k (V) -^k^\ -^ ^) +i^ [g\(x.g)] 



and hence, assuming U(s,5) = U (s,6) and u(x,g) = u (x,g), we obtain 



M(x) -M^Cx) =|^(Uf63_) 



(40) 



Comparison of (39) and (40) now shows that M (x) = M(x) if 



1 2 



f6* + y 6* sec Y = fiS 



(41) 



This agrees with the definition of 6* in (25) when Y is small. Otherwise, 

 the displaced surface should be taken in accordance with (41) rather than 

 (25). 



A Vorticity Theorem 



Consider a steady mean flow of an incompressible fluid about a body at 

 rest in an otherwise unbounded fluid, with mean vorticity present in the 

 boundary layer and wake, BLW, bounded internally by the body surface S and 

 externally by the surface T; see Figure 6. The boundary conditions to be 



Figure 6 



15 



