y^ = 2r^RQ(x) + 2r^R^(x) + CpP(x) + Q(x) [14] 



where R (x) , Rj^(x), P(x), and Q(x) are polynomials of the sixth degree In x. 

 Corresponding to Equations [k], [5], [6], [10], [12] and [13] the polynomial 

 y^(x) satisfies the following conditions identically in r^, r^ , and C : 



(a) y^(0) = 



(b) i3^y^(0) = 2r^ 



(c) y^(l) = 



(d) |3^y^(i) = -2r^ 



(e) y^(m) = -^ 



(f) |3r/U) = o 

 and 



(g) Jy2(x)dx=J-C 



^' 



Since conditions (a) through (g) are satisfied identically in r , r^^ , and C , 



their application to [14] has the following consequences: 



Since y^(0) = 0, regardless of the values of r , r^^, and C , we must have 



RqCO) = R,(0) = P(0) = Q(0) = [15] 



Similarly we obtain the following equations: 

 From Condition (b), 



Rq'(O) = 1, R^'(O) = P'(0) = Q'(0) = [16] 



where the prime denotes differentiation with respect to x. Prom Condition (c) 



Rq(1) = R^d) = P(1) = Q(1) = [17] 



from Condition (d) , 



R,'(1) = -1. Ro'(l) = PJ(1) = Q'(l) = [18] 



