23 



The initial behavior of the distributions corresponding to each of 

 the three roots, as determined from Equations [33] through [36], and [39], is 

 shown in Table 2. It is seen from the table that the distribution for a = 12.1 

 begins with practically a zero value for m(a), with a small negative slope and 

 with upward curvature. Since the distribution curve cannot continue very far 

 with upward curvature, there must be an inflection point nearby. In contrast, 

 the distribution corresponding to the other two roots begin with positive 

 slopes and downward curvatures and hence must be considered simpler. Further- 

 more, the distribution for the first root is considered simpler than for the 

 second since the distribution curves are practically identical except that, 

 for the second root, the curve is extended a distance 4a = 0.0011, in the 

 course of which m(a) changes from a positive to almost a numerically equal 

 negative value. If we take the point of view that the positive and negative 

 values of this extension counterbalance each other, the curve without the 

 extension, corresponding to the first root, must be considered the simplest. 



TABLE 2 

 Characteristics of Initial Distribution 



a 



a 



m(a) 



C x 



C 2 



3-65 

 3.85 

 12.1 



0.0219 



0.0208 

 0.0066 



+0.0000216 

 -0.0000191 

 +0.0000008 



+0.0375 

 +0.0376 

 -0.0064 



-0.103 



-0.109 

 +0.35 



Hence, for the purpose of obtaining a first approximation, we will 

 assume a = 3.65 and, correspondingly, a = 0.022, m(a) = 0.000022. Often, as 

 in this case, the labor of obtaining a and m(a) can be considerably reduced 

 by using the less exact equations [40] through [44] instead of [37] through 

 [39]- Since, as will be seen, the iteration formulas rapidly improve upon the 

 first approximation, great effort should not be expended to determine an 

 initial value for m(a). 



The values a = 0.022 and m(a) = 0.000022 have been derived for the 

 profile in the £, 77 -plane. The corresponding values in the x, y-plane are 

 a = -O.956 and m = 0.000088. By symmetry we also have b = -a, m, = m . 



A first approximation can now be obtained from [47], [48], [49], and 

 [50]. Since A = 5.0, we have k 1 = 0.059- Also, from [78]: f = O.OO659, 

 jfVdx = 0.0640, J" /dx = O.O637. Hence from [50], C = O.328. Then, from [48], 



