25 

 The border curve for Condition [51], f(x) = 0, can now be expressed 



_ g(x) rc7i 



and graphed with a as ordinate and x as abscissa, for ^ x g 1 . The curve 

 is shown in Figure 5, where the nonpermlssible side of the curve is cross- 

 hatched. The values of a^ for which the lines a^ = constant intersect the 

 curve [56] must be excluded since, for these values. Condition [51] is not sat- 

 isfied for all values of x . In this way it is seen from Figure 5 that only 

 values of a in the range -26 g a_ ^ 11 satisfy Condition [51]- 

 Similarly Condition [52] defines the boundary curve 



2[a^U(x) + g(x)] [a^U'-Cx) + g"(x)] - [a^U'lx) + g'(x)]^ = 



a2^(2UU" - U'2) + 2a^(gU" - g'U' + g"U) + 2gg" - g'^ = [58] 



from which a^ may be graphed as ordinate against x as abscissa, for ^ x g 1 . 

 This curve is also shown in Figure 5. and its nonpermlssible side is also in- 

 dicated by shading. The curve shows that the limits of a to satisfy Condi- 

 tion [52] are - 00 < a^ g 158, and -2 g a^ g b. Hence Condition [51] and [52] 

 are satisfied simultaneously only by values of a in the range -2 ^ a^ ^ 6. 



COMPARISON OF SIXTH- AND SEVENTH -DEGREE POLYNOMIAL FORMS 



The class of bodies represented by polynomials of the seventh degree, 

 y^ = a^x + a x^ + ... + a^x'', includes as a subclass, when a^ = 0, those of 

 the sixth degree. As a check on the formulas for the coefficients occurring 

 in the basic polynomials [5'+]. the example shown in Figure 3 to Illustrate the 

 polynomial of the sixth degree was also applied to the seventh degree. The 

 form was assumed to have the same primary geometrical characteristics as be- 

 fore, viz, m = 0.40, r = O.SO, r = 0.10, C = O.65, and a was chosen to 

 '0 '1 p 2 



have the value in the resulting sixth-degree polynomial, a^ = 2.1497, as given 

 in Figure 3- Now if these values of the parameters and the values of the co- 

 efficients listed in the preceding section are substituted into Equation [53], 

 the resulting polynomial obtained is found to be identical with the original 

 sixth-degree polynomial, with a reducing to zero. 



