Weissinger and Maass 



, Reference 

 X cylinder 



Fig. 4 - The biplane in two- 

 dimensional theory and the 

 cross section of a ring airfoil 



can be split into the two equations 



q = 2 _[Vt(^)] , 



Gy + Ky = -Va(^') 



(1.19) 



(1.20) 



Since the right-hand side is known after evaluating the integral for v^ by means 

 of (1.19), an integral equation of the form 



Gg + Kg 



:i.2i) 



remains to be solved for g, where 

 tor, and K a regular operator. 



f is a known function, G the Glauert opera- 



The integral equations to be solved in the theory of ring airfoils and ducted 

 propellers are of the same type. Essentially, the following four methods have 

 been applied for solving (1.21). The first two were used at Karlsruhe, the third 

 at THERM, and the fourth at NSRDC. 



(1) If the kernel of K is developed into a double cosine Fourier series with 

 respect to the variable 6, and if the Birnbaum series (1.6) for g is introduced, 

 the left-hand side of (1.21) can be expressed as a cosine series with coefficients 

 that are linear combinations of the Birnbaum coefficients c^,. Equating these 

 coefficients with the Fourier coefficients of f, one obtains an infinite system of 

 linear equations exactly equivalent to (1.21). This is solved approximately by 

 truncation to a finite system and Gauss elimination. 



(2) This method (114) is a generalization of the above collocation method 

 for solving the Glauert equation Gg = a by (1.8). Since Kg is a regular integral 

 the same rectangular rule applied above to Gg can be used for the approximate 



1218 



