We shall assume that f, 6 and y a^£ small of first order, and neglect 

 terms of fourth and higher order in (37). Then, since r 



2 3u^ 8(rv^) 



'^^O 3 



= 2r - — is small of order f , (37) yields 

 dx 



Mn(x) =.T^^ [fV(x,f)] 



9r 



0^ ' ■ 4 dx ' I 



or, since u (x,f) = U (s,0) cos y '-■ U (s,6) cos y 



Mq(x) = H^ [f\(s,6) cos y] (38) 



Similarly, the source distribution M (x) for the irrotational flow about 

 the surface displaced by the displacement thickness 5* is given by 



M, (x) = T 3- [(f+6* sec y)^ U^*(s,6) cos y] 

 i 4 dx i 



where U * denotes the s-component of the velocity. Then, assuming that 

 U (s,6) = U *(s,6), we obtain 



M^(x) - Mq(x) = "I^ [(2f6*+6*^ sec y) U^(s,6)] (39) 



Similarly, when the surface of revolution is generated by r = g(x) and 

 the axial distribution is M(x), the Gauss flux theorem gives 



M(x) = 2TTg(x) V(s,6)(l+k6) dS + 27Tru(x,r) dr 



)r, by (27), and integration-by-parts of the last integral. 



14 



