For the case when the field point r (1 is given in cylindrical 

 polar coordinates. 



r o/*o \ *« R « 



^l — , x p . = — - i + — — e t (0) 

 D\L) R » / D - : ' 



ind 



s W b xi V . x R > x R . . 



— 0?.x R > = ~ i + — £ r ^(t?,x R ) + e b J 



■(106) 



X (??. x R ) x, e 



= + 7} sin P 



D D p 



(rj, x R ) = d le + 2n cos p /x R 



i= — - — U R -x R cos(0+0 b -0)| i 



cos P sin (0 + ft, - 0) 



(107) 



D D 



sin 



+ — — (x R - x R cos (9 + d b - 0)j 



/ x o x\ 



( — -— )cos0 p cos<0 +6 b - 0) 



+ sin 0p — sin (8 + d b - 0) 



e r (0) 



COS 0p r 



— — U R - x Ro Cos(0 te +fl b -0 



- r; sin p ) cos p sin I 8, + fl b - 

 cos0 p \ sin p / 



+2T? ^ R -j + -r-lx 



COS 0p\' 



-x R cos(e te + 6 b -0 + 2r,^— 



n sin p l cos p cost fl te 



cos p \ sin 0p 

 + d h - + 2tj ) + — ^— x R siniu t 



+ 8 b - + 2tj 



e r (0) (108) 



A computer code has been developed for a subroutine to 

 compute the approximate value of W: 



f' 1 "D" D f 



e, X^ dr, + 



r o S *b 1 



dij 



(109) 



Options are included in the subroutine call statement 

 to permit the first integral to be evaluated either by the 

 trapezoidal rule with equal increments in rc or by Simpson's 

 rule with equal increments in \A/ . This second integration 

 procedure insures more dense spacing of the integrand near 

 the blade and a more accurate computation. To insure an 

 even number of intervals, the number of points specified in 

 the call statement is doubled when this more accurate pro- 

 cedure is employed. The second integral is evaluated by the 

 trapezoidal rule but is not employed when r/., < rjj 



For a given value of r)j and specified number of double 

 intervals, N, the constant increment in y/rj is 



\Aq" 



for which the increment in r; between successive points is 

 Arjj = Tjj- I7j_ j 



(2i- !)• 



(110) 



and the increment in angular variable 9 between successive 

 points is 



A0, = 2 Arjj cos 0p/x R 



= (2i- 1) 2cos0 p t),/(x r 4N 2 ) 



(111) 



77 , 2 cos ( 

 4N 2 X R 



4N - 1 2 cos *p 



A0 2N = — T7, = (4N-DA0, 



4N 2 x R 



Generally n, = n-, and the equal increments of y/rj are used 

 for integration with N = 2 ■ rj-, double intervals. A value of 

 n 2 = 10 has been satisfactory to date. When the distance 

 between points becomes small, special fine point spacing in n 

 is employed to insure convergence. Accuracy of calculations 

 was determined by comparison with analytical results for the 

 tangential velocity component due to a circular arc vortex 

 filament, the axial velocity component at the origin for a 

 general helical filament, all velocity components for a straight 

 line vortex, and induction factors ( 1 7) for general helical fila- 

 ments. At individual points of this comparison for filaments, 

 accuracy to the third decimal point was found with the 

 selected parameters and an overall accuracy of the non- 

 dimensional induced velocity component of the sheet to one 

 or two units in the fourth decimal point was found. 



12 



