12 



EVALUATION OP P(x) 



Since P(0) = P'(0) = P(l) = P'(1) = P(m) = P'(ra) = and since P(x) 

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



P(x) = yx2(x - l)2(x - ra)^ [30] 



To evaluate the coefficient y, we have, from [21], j P(x)dx = — . rience, 

 writing ° ^ 



P(x) = y[x^ - 2x^(1 + m) + x*(l + Um + m^ ) - 2x^(m + m^) + m^x''] 



and integrating, we obtain 



y [y - j(l + m) + -^(1 + 4m + m^) - \{^ + m^) + -^m^] = \ 



or 



105 r,T 1 



y= 2(2 - 7m + 7m^) ^31] 



EVALUATION OP Qlx] 



Since Q(0) = Q'(0) = Q(1 ) = Q' (i ) = and Q(x) is a polynomial of the 

 sixth degree , it may be written in the form 



Q(x) = (5^ + «,x + i^x^lx^lx - 1)2 [32] 



The undetermined coefficients are evaluated as follows: Prom [19]. Q(ni) = "ir» 

 and hence 



d^ + ma^ + m2«„ = —-1 [33] 



Prom [20], Q'(m) = 0, and hence 



{b^ + i^m + b^x!?) 2m(m - l)(2m - 1) + (a^ + 2m52) m^(m - 1 j^ = 



or, dividing by m(m - 1) and simplifying, we obtain 



a^C^m - 2) + (J^m(5m - 3) + b^va^k^^va - 4) = [3^] 



