14 



(c) ra(x) satisfies the virtual mass relation [24] which may be written 

 in the convenient form 



fm(x)dx = jj-(l + k ) fVdx [46] 



J a Jo 



where 



and 



It is readily verified that Condition (a) is satisfied by the 

 distribution 



m(x) = Cy* + e + e^ [47] 



e o =FaK -n + C ( af b - bf a>] [48] 



e x = b4iK - m a + C ( f a " V] ™ 



If the linear term e + e x in [47] is small in comparison with m(x) at a dis- 

 tance from the ends, then Condition (b) is also satisfied. Finally, Condition 

 (c) can be satisfied by a proper choice of C in [47]. This is accomplished by 

 writing m(x) in the form 



mlv \ p/,,2 b-x - x-a „ \ b-x m , x-a 

 substituting it into Equation [46], and solving for C. We obtain 



T 1+k i I'y 2 ^ " T(t>-a)(m +m ) 

 C —2 -f^ 2 a_b_ 5Q 



JV<* --i(b-a)(f +f ) 



SOLUTION OP INTEGRAL EQUATION BY ITERATION 



Now that we have derived a good first approximation to the doublet 

 distribution function in the integral equation [19 ]» it would be very desir- 

 able to apply it to obtain a second, closer approximation. This can be accom- 

 plished by means of the iteration formula which we will now derive. 



Let m^x) be a known first approximation and \}> (x) the corresponding 

 values of the stream function if/ on the given profile y 2 = f(x). Then, from 

 Equation [l8], 



