22 



We will also need the slope of the profile which, from [78], is 



v , _ f'(x) _ - 0-^ 3 



2y (l . x 4)i/ 2 



The profile and f(x) are graphed in Figure 2. 



[82 



Figure 2 - Graphs of y(x) and y^x) for y 2 (x) = 0.04(1 - x 4 ) 



First let us find the end points of the distribution. We have, from 

 [81], a x = 0.08, a 2 = -0.24, a 3 = 0.32. The approximate formula [40] then 

 gives a = 3.68 or 3-84, whence a = J: = 0.0217 or 0.0208. An examination of 

 the complete polynomial [37] with the aid of Table 10 shows that its zeros oc- 

 cur at a = 3.65, 3.85, 12.1. In the application of Table 10 to determine these 

 roots the regions of possible zeros should be determined by inspection, the 

 values of the polynomial in these regions calculated from Equation [37] and 

 Table 10, and then graphed to obtain the zeros. It is seen that in the pres- 

 ent case the approximate formula [40] would have been sufficiently accurate 

 for the determination of the roots near a = 4. The solution of the complete 

 polynomial equation will always yield an additional large root, corresponding 

 to the large root of Equation [131] of the Appendix; in general, however, this 

 root should be rejected since as will be shown, the initial doublet distribu- 

 tion corresponding to it is less simple than for the roots near a = 4. 



