13 



a = 4 + a 2 -yV^TV if a 3 > [40] 



a = 4 + a , if a < [4l ] 



2 3 



and, to the same order of approximation, 



m(x) =|(i +y +y l ln Tr)(-r + y 2 ) ^ 



and 



m(a) =l(l +|i +I 2 - ln^)a 2 iAa-a7, if a 3 ^ [43] 



m(a) =0, if a 3 < [44] 



It is seen that Kaplan's assumption that a = 4 gives the principal 

 term of the solution in [40] or [41 ]. The form [42] immediately suggests a 

 modification and refinement of the Munk-Weinblum approximation, Equation [23], 

 which will be considered in the next section. 



A graphical procedure for finding the roots a of Equation [35] is 

 also given in the Appendix. For this purpose the functions A(a) > B(et),... H(ce) 

 are tabulated in Table 10. 



AN IMPROVED FIRST APPROXIMATION 



According to its derivation the Munk approximation could be expected 

 to be useful only at a distance from the end points of a distribution. It was 

 seen, however, Equation [42], that under certain circumstances a distribution 

 which was a suitable approximation for the nose and tail of a body also ap- 

 peared as a generalization of the Munk-Weinblum approximation, [23]. This 

 suggests a procedure for obtaining an improved approximate distribution. 



It is desired to obtain a distribution m(x) which satisfies the fol- 

 lowing conditions: 



(a) m(x) assumes known values m and m, at the distribution limits a and 



b, i.e., 



m(a) = m a , m(b) = m b [45; 



(b) m(x) is nearly equivalent to the Munk-Weinblum approximation [23] at 

 a distance from the distribution limits, i.e., 



m(a) = Cy 2 for a « x « b 



