55 



The equations can be solved explicitly in the case of a very elon- 

 gated body for which a 1 , a £ , a , ... in [115] are all very small. First let 

 us suppose that they are so small that all the terms in [125] containing them 

 are negligible, so that the first product term alone may be equated to zero, 

 i.e., 



a(a - 4) 2 (5a 4 - 83a 3 + 288a 2 - 368a + 128) = [131] 



whose real roots are a = 0, 0.547, 4.0, 4.0, and 12.429. 



a. 

 Let us consider the solution a = 4; i.e., a = x— . Since the radius 



of curvature at x = is a.^2, this solution is seen to be in accord with 

 Kaplan's assumption for the end points of the distribution. Furthermore, sub- 

 stituting a = 4 into Equations [129] and [130], we obtain, to the same order 

 of approximation, 



2 



a a 

 D = 64, c = -^, c x - j±, c 2 = 



whenc e 



2 



a a 

 m(x) = -yg-+ Tp x, m(a) = [132] 



In order to obtain a second approximation it will be assumed that 

 not only a,, a„, a . ... but also (a - 4) are small to the first order. Then, 



" 1 2 3 



neglecting terms of third and higher order, Equation [125] becomes 



-3072(a - 4) 2 + 6l44a £ (a - 4) - 3072a* + 768a 1 a 3 = [133] 



whenc e 



a = 4 + a ± 4-l/aTa - [134] 



2 2 r 1 3 L ' 



provided 



a Z 



3 



Corresponding to this value of a we obtain from Equations [126] through [129], 

 to the same order of approximation, 



*The smaller of these two roots has given the preferred solution in all cases tried thus far. 



