20 



and 



Hence 



Cll m (t)dt = r-^-fm(t)-m(x)]dt + ra(x) f-^- dt 

 Jo r 5 Jo r 5 Ja r 5 



j 3- m(t)dt = j -^-[m(t)-m(x)]dt + m(x)(cosa - cos/3) [71a] 



f'_y! m (t)dt= f-^fm(t)-m(x)ldt 



+ m(x) [cos a - cos - -=■ (cos 3 a - cos 3 £)j [72a] 



Gauss 1 quadrature formula is a convenient and accurate method of 

 evaluating these integrals. The formula may be expressed in the form 



£p U )d|= JjR nl FU ni ) [7^] 



where the £. are the zeros of Legendre's polynomial of degree n and the R . 

 are weighting factors. These have been tabulated 26 for values of n from 1 to 

 16. These numbers have the properties 



R ni - R n,n-i + l and *ni ' A.n-i+1 ™ 



The value of the integral given by Formula [7^] is the same as could be ob- 

 tained by fitting a polynomial of degree 2n-1 to F(x). The values of R . ar 

 £ . are tabulated in Table 1 for n = 7, 11, and l6. 



When the limits of integration are a and , as in Equations [~f\ ] 

 and [72], Gauss' formula becomes 



a n 



p-a 



where 



plfilw-.^^yiy [76; 



=£^£ +o±£ [771 



w i 2 ? ni 2 Lm 



