24 

 e o = m a - Cf = -0.00207; from [49], e^^ = 0. Finally we obtain from [47] 



m^x) = 0.328Y 2 - 0.00207 [83] 



We can now apply Equation [51] and the iteration formula [58] to ob- 

 tain the sequence of functions ^(x). Let us suppose that it is desired to 

 obtain a distribution m. (x) whose exact stream surface deviates from the given 

 surface by less than one percent of the maximum radius, i.e., An < 0.002. 

 Then, by [60], the sequence ip, (x) should be continued until i/>,(x) < 0.002 Vf (x) 

 for a^ x < b, unless the error, as represented by ^(x), begins to grow before 

 the desired degree of approximation is attained. In the latter case the best 

 approximation attainable would fall short of the specified accuracy. 



The Integrations in [50] and [51] may be carried out in the form 

 [7"1] in terms of 9 defined in [70]. For a fixed (x, y) on the given profile, 

 a and p are first computed from [73]. Then, to apply Gauss' quadrature formu- 

 la [76], the interval is subdivided at the points 9. given by [77] and the 

 integrands evaluated at these points. The corresponding values of t at which 

 m^t) in [51] or ^ 1 _,(t) in [58] is to be read are, from [70], 



t. = x - y cot 9. [70a] 



Since the values t. and sin 9. are used repeatedly in the successive itera- 



J J 

 tions at a given (x, y), these should be stored in a form convenient for 



application. 



The calculations for obtaining the integration limits a and for 



several values of x are given in Table 3- The values of 0. from [77], and the 



corresponding values of R, sin 9, for application of the Gauss 11 ordinate 



formula, and the values of t. from [70a] for each x are entered as the first 



three columns in Tables 5 a through 5h> in which are given the calculations for 



In order to compute ^ (x), m 1 (t) is computed from [83], then 

 m 1 R sin 9 is obtained. These are tabulated in Table 5. Gauss' formula then 

 gives Jm i sinfldtf. ^(x) is then obtained from [51]; its graph is given in 

 Figure 3- It is important to note that m (t) is obtained by calculation, 

 rather than graphically, in this operation. This procedure is recommended 

 since it gives greater accuracy in a critical step. In the subsequent opera- 

 tions on the ^'s considerably less percentage accuracy is required, since the 

 ^»'s are of the nature of first differences between the m's, so that graphical 

 operations are sufficiently accurate. 



