and hence, substituting (10), (11), and (12) into (9) and solving for M(x), 

 we obtain 



TTM(x) =1^ [g(x) u(x,0)-U(6-6^)] (13) 



Since, as is seen from Figure 3, 



g(x) = f(x) + 6 sec Y (1^) 



and 



U ■==. u(x,g) sec Y ^ u(x,0) sec Y (15) 



we can write (13) in the form 



TTM(x) = -^ {u(x,0)[f (x)+6, sec y]} (16) 



dx 1 



Here we have assumed that f(x), 6, 6 , (u-U.) , v and Y are small quantities 

 of the first order and terms of third and higher order have been neglected. 



For the irrotational flow about the body without a boundary layer, (16) 

 yields the well-known, second-order approxim.ation for a centerplane 

 distribution. 



^M^(x) = X- [u(x,0) f(x)] (17) 



(J QX 



Hence, the additional s'ource strength due to the boundary layer is given by 



M(x) - Mq(x) =.^~ [u(x,0) 6^ sec y] (18) 



The .result in (18) resembles the Preston-Lighthill formula (3) , especially 

 when the body slope is small. The distribution strength is approximately 



