16 



^(x) = m i _ 1 (x) + j f(x) 



H 



6m 1 _ 1 (t) 



dt 



Hence, deducting [56a] from [56] and making use of [57]. we get 



•6 ^.jU) 



•'a 



Also, from [57] we obtain 



-I 1 

 j=1 



dt 



[56a; 



[58] 



[59: 



Thus, in order to obtain m. -(x) , we first assume an m (x), then determine 

 ip, (x) from [51]- i> (x) , \p (x), ... can then be successively obtained from 

 [58], and finally m i+1 (x) from [59]- 



It has been stated that the magnitude of ip, (x) is a measure of the 

 proximity of m. (x) . This property of ^, (x) can be given a geometrical in- 

 terpretation. Corresponding to the distribution m. (x) there is an exact 

 stream surface on which the stream function ip, (x, y) = 0. Let An, be the 

 distance from a point (x, y) on the given body to this exact stream surface, 

 measured along the normal to the given body, positive outwards. Let u be 

 the tangential component of the flow along the body. Then we have 



u s= "y 



1 e^U.y) 



en 



1 ^(x.y) 



y An. 



But Ail> = -<P ( x )» since <p, (x, y) = on the exact stream surface. Hence 



An, 



^(x) 



yu< 



[60] 



Since, for an elongated body, u = 1 , except in the neighborhood of the stag- 

 nation points, it is seen that ip, (x) enables a rapid estimate to be made of 

 the variation from the desired profile of the exact stream surface correspond- 

 ing to m. (x). This is an important property because it can be used to monitor 

 the successive approximations. Thus, the sequence ifrAx) can be terminated 

 when An, becomes uniformly less than some specified tolerance; or, since there 

 is no assurance that the infinite sequence ^.(x) converges, the sequence can 

 conceivably give useful results even without convergence if it is continued as 



