:■:'■ Weissinger and Maass 



The kernels u^(t7) are antisymmetric continuous functions of the argument 



V = H^-^') -- ^^^ • (2,18) 



So, all these integral equations are of the type (1.21) and can be solved by the 

 methods described in Sec. 1.3. 



The most important of these operators is T^, defined as 



(2.19) 



Ko(^,^') = '^ Gj(k2) = ^^ [(l + k'2)E(k) - 2k'2K(k)] 



(2.20) 



2 



k^=V-7' k'2=i-k2, a;=i + :L, (2.21) 



where 



r 

 .ml COS 2m!3 ,„ ,„. 1 r i. • x 



G (k^) = (-1) I di9 , (Riegels function) 



(l-k2sin20)^/2 



7r/2 



f COS 2rrn? ^ 



Caj") = dy , (Legendre function) 



■'^ ^-./2 [2(co- 1) + 4sin20]i/2 



^77/2 



(2.22) 

 (2.23) 



K(k)=r ^^ , E(k) = r (l-k2sin25)i/2d0 . 



-i (l-k2sin2f?)l/2 J (2.24) 



Formulas and numerical tables for the kernels of the operators 7^ have been 

 presented by means of Riegels functions at Karlsruhe (102, 103) and by means of 

 Legendre functions at THERM (135, 137, 138, 139). 



The basic ideas of lifting-line theory and of generalized lifting-line theory 

 ("three -quarter -point method") can also be applied to the ring airfoil (102), thus 

 obtaining very simple formulas, e.g., for the lift. For the latter theory the 

 agreement with the exact results is very good over the whole range of X, 

 < \ < oo; lifting-line theory is valid only for small values of k. 



In Karlsruhe, ring airfoils with central bodies have also been investigated 

 (110). The axisymmetric body is assumed to be slender and to have a small 

 maximal radius r^, such that it can be represented by an axial distribution of 

 sources and doublets. To satisfy the boundary conditions at the body and the 

 ring airfoil, only the leading terms in the Taylor expansion with respect to 

 Pm - ^max/^ ^^^ retained. For geometry and notation see Fig. 5. 



1222 



