Details of the curve fitting procedure are contained in Ref- 

 erence 14. Once the trigonometric polynomials are specified, 

 they may be integrated to any point in the closed interval 

 to n. The resulting expressions may be arranged to produce 

 a set of coefficients which are multiples of each tabulated 

 functional value. Then a summation of the coefficients 

 times the tabulated value yields the integral value. A com- 

 puter code was developed for a subroutine to determine 

 these coefficients for an array of arbitrary values of the 

 independent angular variable between and tt when the 

 tabulated values were fit by either a half-range sine or 

 cosine series. The coefficients are stored in the subroutine 

 for repeated use. 



For example, the meanline offset is 



MVV .J"" 3 _D siivc 



5 3 x,. 2 



J 1_ <- 



(86) 



The slope of the meanline is assumed to be finite at 



3x c 



both the leading and trailing edges which results in a sine 

 term approximation for the term in square brackets: 



sin a v~" 



— = Z a " Sln 



and the integral evaluated analytically (15). 



v I r" F (a; a x Ru )da 

 V n T cos a - cos a„ 



N -n 



1 ^- £ cos na da 



n Z-> n I cos a - cos c» 



n = JO ° 



N 



" sin na Q 



L-, n sin a 



(92) 



(93) 



The numerical procedure consists of evaluating the coeffi- 

 cients a in terms of values of the integrand ( Equation 

 91 ) at equal angular intervals of a from to it. Finally, 

 Equation (93) is rearranged to be the sum of the product 

 of the integrand values at discrete points and coefficients 

 R „ which can be determined once and for all: 



(87) 



V- I [?^vw]„R m K) 



where (F) are a table of values of Equation (91 ) at 



(94) 



vhere a n are given explicitly ( 14) as functions of the values at 



E„(a„,x p ) 



5 = 1 a "~ 



which can be rearranged in the form: 



E„K,x B ) V/ a rf 



'o> *R> V I D sina\ 



(89) 



where Tj(a ) are a set of coefficients dependent on the 

 interval spacing and desired evaluation point a . They can. 

 be calculated once and for all and stored for repeated use. 

 Similar procedures are followed for all regular integrations. 



For the integral with a singular point, the Cauchy 

 Principal Value must be obtained (see Equation 28). First 

 the integration in the radial direction is performed, resulting 

 in an equation 



v(«o.*R > ,f- 

 V = -] F<a;a ,x Rc 



cos a - cos a 



(90) 



The regular part of the integrand, F (a. ; a , x Ro ) can be 

 expanded in a half-range cosine series in a: 



N 

 F (<*;a o , x Ro ) = \ a n (<* o , x R(J ) cos na 



(91) 



N 



< m < N , and R m (a ) are a set of coefficients 



dependent on the interval spacing as well as the desired 

 evaluation points, a . Details of this procedure as well as 

 FORTRAN statements for a computer subroutine are given 

 in Reference 16. 



To determine a value of the radial integrand at the 

 singular point x, = x c , a linear expansion of the position 

 vector s about the fixed pointy is derived. This linear 

 expansion results in a form of the integrand which can be 

 integrated analytically in the radial direction by holding the 

 other terms in the integral constant at x R . The expansion 

 of s about r produces the equation ° 



^R ) 



• + (x„- x R ) ae, + 



3x„ 



(95) 



Nt X e 



J>_c_ 

 2 D 



(96) 



The approximate distance from the reference point is 



I r - s | =» p(x R - x R , x -x )-D 

 o o 



where 



p(x R .x c ) = ^Ax R 2 +Bx R " c + (^y \- : 



(97) 



10 



