11 



Then, simplifying, substituting a^ = — 5' ^^^ solving f or a , we obtain 



° rr •■■ 



^ ^ _ 7(1 - ^m + 3m^) [24] 



^ 2m^(2-7m + 7m^) 



EVALUATION OP R^(x) 



Since R^(0) = R^'(O) = R^(l) = R^M = R^'(m) = 0, and since R^(x) 

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



R^(x) = ()3q + ^, x) x^ (x - l)(x - m)2 [25] 



The coefficients B and B are evaluated as follows: 



1 



From [l6], R^'(1) = -1; whence, from [25], 



()S^ +^J(1 - m)2 = -1 [26] 



Equation [25] may be rewritten as: 



R^(x) = /8o[x^ - x^d + 2m) + x3(m2 + 2m) - m^x^] 



+ /Sjx^ - x=(l + 2m) + xMm" + 2ra) - ra^x^] 

 1 

 But, from [21], J R (x)dx = 0. Hence, integrating the above expression for 

 R^(x) and simplirying, gives 



7/3^(2 - 6m + 5m^) + yS^dO - 28m + 21m^) = [27] 



Solving for [26] and [27] simultaneously, we obtain 



B = 10 - 28m + 21m^ ^28] 



° 2(1 - m)2(2 - 7m + 7m^) 



7(2 - 6m + 5m^) 

 ^1 = "2(1 - m)^(2 - 7m + 7m^) 



[29] 



