57 



A(ee) = a{a - 4) 2 (5« 4 - 83a 3 + 288a 2 - 368a + 128) 

 B(a) = 72(a - 4) 2 (5« 3 - 25a 2 + 40a - l6) 

 C(a) = Ua(a - 4)(53a 2 - 148a + 128) 



D(a) = -288(a - 4)(5« 2 - l6a + l6) 



[138] 

 E(a) = -96a(3a - 4) 



F(a) = 11 52 (2a - 3) 

 0(a) = 48a(3a - 8) 

 H(a) = -1152(a - 3) 



have been tabulated in Table 10. In terms of these functions, Equation [125] 

 becomes 



A + a x B + a 2 C + a^D + a 2 E + a^F + a 1 a 3 G + a*a 3 H =0 [139] 



It is of interest to compare the approximate value for a from Equa- 

 tion [134] with the exact value for the prolate spheroid y 2 = ti(x - x 2 ) . In 

 this case we have 



a, = -a„ - — =-, a =0 



1 2 X 2 3 



and the exact value of a is 



a = 2 + 2i/l -\ = 4 - \ - -L 

 V X 2 A 4X 4 



But when the length-diameter ratio X is large, Equation [134] gives the ap- 

 proximate value a = 4 ~~Ts.» which is seen to consist of the first two terras of 

 the series expansion of the exact value of a. Table 11 shows that the approx- 

 imate formula gives excellent agreement with the exact values even for very 

 thick sections. Both the exact and the approximate formulas give m(a) = 0. 

 Thus the present approximate methods work very well for the prolate spheroid. 



