10 



From Condition (e), 



R (m) = R^(m) = P(m) = 0, Q(m) = -J- [19] 



Prom Condition (f ), 



R^'(m) = R^'(m) = P'(ra) = Q' (m) = [20] 



and from Condition (g) , 



1 I 1 1 1 



rp(x)dx =-, jR^(x)dx = JR^(x)dx = JQ(x)dx = [21] 



4 



The values of R (x), R (x), P(x), and Q(x) will now be derived on the basis of 

 the relations in Equations [15] through [21]. 



EVALUATION OP R^Cx) 



Since R^lO) = R^(1) = Rq'(1) = Rjra) = Ro'(ni) = 0, and since R^{x) 

 is a polynomial of the sixth degree, it may be written factorlally in the form 



R^(x) = (a^ + a^x) X (x-l)2(x-m)2 [22] 



The coefficients a and a may be evaluated as follows: Prom Equation [l6] 



1 



R '(0) = 1; whence, from [22], 



«o=^ [23] 



^ m 



Equation [22] may be rewritten as 



R^(x) = «o[x^ - 2x^(1 + m) + x^(1 + 4m + m^) -2mx^(l + m) + ra^x] 

 + ajx® - 2x^(1 + ra) + x^'d + 4m + m^) -2mx='(1 + m) + ra^x^] 



Hence, from [21 ], 



JR^Cxjdx = a\^ - |(1 + m) +-J(1 + 4ra + m^) - |m(l + m) + -^m^J 







+ a J^ - ^( 1 + m) + -^( 1 + 4m + m^ ) - ■^m( 1 + m) + ^m^j = 



