62 



g(x) = y^P^ + y,P,(0 + ... + yjj^) 

 or 



g(x) = g(x) - y^P^l^) [8] 



Now, from Equation [6], 



Jg(x)P^($)d§ = rl Jp/(l)d| = ^r, [9] 



and, from [3] and [7], 



g{x) =C^(^)% ... =|f|^+ ... [10] 



where only the coefficient of ^^ is shown since, by a well-known property of 

 Legendre polynomials, the terms in the powers of ^ less than the fifth give 

 zero when substituted into [9]- Also 



Psff) = ■§■(63?^ - loe + 15?) [Tl] 



Hence, substituting from [10] and [11 ] into [9], gives 



_ ^ 



Also, from [7] and [11 ], 



P^(x) =-^[63(2x - 1)^ - 70(2x - 1)3 + 15(2x - 1)] 



= 252(x=-|x^.^x3-fx^.^x-^) 



12] 



[13: 



Hence, substituting from [12] and [13] into [8], gives Equation [2] as we 

 wished to prove. 



