304 -58- 



The proof is similar to the proof of the Gauss 

 quadrature formula. This theorem indicates an exact 

 evaluation of the integral involving 2n parameters 

 (a polynomial of degree 2n-l) in terms of n specially 

 selected points. It is exactly analogous to the more well 

 known G-auss quadrature formula, but it is simpler in two 

 ways: the zeros x-,..o,x of T (x) are easy to obtain, 

 since they are merely the zeros of cos n 9 where x = cos 9; 

 and the weight factors imiltiplying f(xi) are all equal to 

 1/n, 



If f(x) is not exactly equal to a polynomial of 

 degree 2n-l but can be approximated by one, vie can write 

 the approximate quadrature fonmila 



f(x) 



V^ (^ 



^-kX ^(-1^ 



2 11 f : I 



The accuracy of this approximate formula depends on the 

 closeness with which f (x) can be approximated by polynomials 

 of degree 2n-l» 



The integrals In question in part III are of the form 



.TT- 



da 



3 k 



where v is some exponent, and a , a. are the two zeros of 

 the expression under the square root sign. This can be 



written as 



/v _V^**l - -^v- - «o^ da 



V ^(a^ - a)(a - a^) 



