33 



ERROR IN DETERMINATION OF p/q 



Let 4(p/q), An, Av , and Am denote errors in p/q, u, v, and m. Then, 

 from [69] , we have 



A 2- = -2(u Au + v Av) 



from [68], 



Av = y 1 4u 

 and from [67] and [72], except near the stagnation points, 



4u » 34« r- sin 3 , w = i+4H 

 y 2 Jo y 2 



Hence 



. _p _ 8uA m 



(1 + y' 2 ) 



If now we assume u s 1, y 1 s 0, y 2 s 4m (Munk's approximation), we obtain 



. p _ 24m 



Thus an error of one percent in the determination of m would introduce an 

 error of 0.02 in p/q. 



In the foregoing example the minimum value of p/q was about -0.20. 

 Hence an error of one percent in m would have produced an error of ten percent 

 in the minimum value of p/q. It was found, in fact, that the results with 

 Gauss' 7-° rdinate rule deviated from the values of p/q given by the 11 -point 

 rule by less than 0.003 for the entire body. The 7-point rule would have 

 sufficed if an accuracy of only 0.003 in p/q were required; see Figure 5- 



If greater accuracy is desired the integrals can be evaluated in the 

 forms [71a] and [72a]. If the latter forms are used in conjunction with the 

 Gauss quadrature formula the values of x should be chosen identical with the 

 t's required by the Gauss formula. This enables the entire calculations, in- 

 cluding the iterations and the velocity determinations, to be made arithmeti- 

 cally, without resort to graphical operations, so that the method becomes suit- 

 able for processing on an automatic-sequence computing machine. In order to 

 obtain sufficient accuracy in the integrations and to obtain the velocities 

 and pressures at a sufficient number of points along the body a Gauss formula 



