Simple methods for the treatment of thin airfoils were described by Jeffreys^^, 

 Munk^^, Glauert'*' ^^, and Millikan^^, who also considered biplanes. 



l^ater work has been concerned chiefly with practical methods of making calculations 

 for given profiles of any shape. See especially papers by Theodorsen and Garrick^°, and by 

 Gebelein^^, also Theodorsen^^, Schmieden^-^, Garrick^'*, Kaplan^^. Line sources on the 

 axis of the airfoil are used by Fistolesi^^ and by Goldstein^'', both sources and vortices by 

 Keune^^. Jones and Cohen^^ show how to use the Joukowski transformation itself in order 

 to effect small changes in a given profile. 



Approximate methods for double or biplane airfoils have been discussed by Millikan 

 and, with use of elliptic functions, by Garrick^°°. 



The theoretical literature on airfoils is naturally extensive, but most of it either makes 

 little use of potential theory or deals with systems of vortices and so lies outside of the 

 scope of the present discussion. 



VARIOUS CYLINDERS 

 81. CIRCLES INTO ELLIPSES 



The transformation of Section 77, 



2 = 2"+ [81a] 



z" 



can be used to convert a circular cylinder into one of elliptic cross section. 



For, consider a circle on the 2 "-plane centered at the origin, which can be described 

 in terms of polar coordinates r, Q as follows: 



2" = a;"+ iy" = re , x" = t cos 6, y'' ^ t sin 6, [81b, c,d] 



where t is constant. For the corresponding transformed curve on the 2-plane, from Equation 

 [81a], 



■ •■ c^ ■ 



z = x+iy=Te'^+ e~'^, ... - [8*Le] 



T + 1 COS 6, y = [ r I sin 6. [81f, g] 



190 



