15 



i rb m At) 

 * x (x) = 4-f(x) + f(x) -i dt 



Ja r> 3 



[51 



Thus \j> (x) is a measure of the error when m (t) is tried as a solution of the 

 integral equation [19]- If m(t) is a solution of [19], Equation [51] may be 

 written in the form 



^(x) = f(x)| 



b m (t)-m(t) 



dt 



[52] 



But, on the same assumptions as were used to derive Munk's approximate distri- 

 bution, Equation [22], we obtain as an approximate solution of the integral 

 equation [52] 



m (x) - m(x) = y^iU) 



or, denoting the new approximation to m(x) by m (x), 



m 2 (x) = m i (x) - y i/>Ax) 



Hence, from [51 ] 



•bm (t) 



1 1 f m 1 

 m 2 (x) = m^x) +{f(x) j- -J -A. 



dt 



[53; 



[54] 



[55] 



Since the foregoing procedure can be repeated successively, we obtain the iter- 

 ation formula 



b m,(t) 



1 1 f m i 

 m i+1 (x) = m i (x) + j f (x) j -J — 



dt 



and 



m i+l' x ^ " m i^ x ^ = "tf^W 



[56] 



[57] 



It is seen that \]j. is the value of the stream function on the given 

 profile corresponding to the i" 1 approximation m (x) and hence serves as a 

 measure of the error when m.(t) is tried as a solution of the integral equa- 

 tion [19]. 



Although successive approximations to m(x) may be computed directly 

 from [56], an alternative form, which is both more convenient and more signif- 

 icant, will now be derived. From [56] we may write _ 



