Milgram 



y 



^cos^ (7) 



with the simple Jacobian 



If = I Sin * . (8) 



Then the circulation can be expanded as 



. ' " " 00 



r[y(</')] = 2 Uq b V A^ sin n^ , (9) 



"' > n = 1 



where the A^'s are dimensionless. The usual relation for the lift distribution 

 holds; 



? [y(^)] = /OU(y) r(y) , (10) 



which gives 



00 



?(</.) = [2 /OUg^ b - pUg Kb^ cos 0] J^ A^ sin n^ . (H) 



n= 1 



Since the vortex field is solenoidal and Kelvin's circulation theorem is valid, 

 there must be a trailing vortex sheet of strength equal to the negative of the 

 spanwise derivative of the circulation: 



y[y(0)] -- -~ (12) 



^y 



and in terms of , 



-4Uq 2_, nA^cosni// 



yW ^^^— ^ (13) 



sin i// 



The solution of this problem is carried out by use of the method of images 

 as shown in Fig. 2, where the sign of the image of any vortex element is oppo- 

 site to the sign of the element in order to satisfy the boundary condition on the 

 image plane. The perturbation velocity is taken as the velocity induced by the 

 system of vorticity compriSBd of the lifting line and its trailing vortex sheet as 

 well as the image system. There is another source of velocity alteration. This 

 alteration occurs whenever the system of vorticity induces velocity parallel to 

 the direction in which the free stream speed varies. Because of the vertical 

 variation of free stream velocity, such induced flow convects fluid of a given 

 stream velocity to a region where the undisturbed stream velocity has a possibly 

 different value. For wind gradients commonly encountered in normal sailing 



1400 



