22 



SEVENTH-DEGREE POLYNOMIAL FORMS 



After the work in connection with the sixth-degree polynomial had 

 been completed, it was of interest to determine how well a form was defined 

 by the parameters m, r , r^^ , and C . For this purpose it was decided to intro- 

 duce an additional degree of freedom and to develop the equations for the 

 class of forms represented by polynomials of the seventh degree. For then, 

 by holding the original parameters constant , allowing the new parameter to 

 vary, and comparing the form determined by the polynomial of the sixth degree 

 with those of the seventh, an answer to the above question could be obtained. 



DERIVATION OP BASIC POLYNOMIALS 



The new parameter was chosen to be a^ , the coefficient of x^ in 

 Equation [2]. Then y^ is a linear function of r , r^^, a^ , and C and may be 

 written in the form 



t = 2r^S^(x) + 2riSjx) + a2U(x) + CpV(x) + W(x) [53] 



where S (x), Sj^(x), U(x), V(x), and Q(x) are polynomials of the seventh degree 

 in X. To determine y^(x) we have the same conditions for these polynomials 

 as (a) through (g), tabulated on page 9 and, in addition, the equation 



(h) dfiy^^O) = 2a^ 



On the basis that these conditions must be satisfied Identically in 

 r , r , a^ , and C , we obtain, as for the polynomials of the sixth degree, the 



1 2 p r ^ ^ 



relations 



1 

 S^{0) = Sq"(0) = S^d) = Z^'{^) = S^(m) = S^'(m) = j S^{x)dx = 







S^'(O) = 1 



1 

 SJO) = S^'(O) = S/'(0) = Sjl) = S^(m) = S^'(m) = js^(x)dx = 

 • 



S,'(l) = -1 



1 

 U(0) = U'(0) = U(1) = U'(1) = U(m) = U'(m) = J U(x)dx = 







U"(0) = 2 



