Comparison of these equations with Equations 

 [76d] and [76h] shows that here b^ = a'^ f/e"'", 

 6j'= - c^; also, here y = -a . Eence from 

 Equation [76e,i], in which S'= 0, the kinetic 

 energy of the fluid per unit length of the 

 lamina is 



fj = —pS{U^ = 7TpU^{a^-c^ cos 2a), [78q] 



since (R)e '" = cos 2 a. A more useful form 

 is obtained by writing 6 for 2c, the half-chord 

 of the lamina, and introducing its central 

 height d above the chord. Using Equations 

 [78g] and [78d], d^ = C^ {4:cf = b^ tan^ (3 

 = b^ (aVc^-1), whence a^ = (b^ + d^)/4:. 

 Thus 



pnU' 



Figure 122 — Pressure differences above and 

 below the lamina shown in Figure 121. 



b^ s\r\'^ a + — 



[78r] 



(See Reference 1, Article 70.) 



79. THE JOUKOWSKI AIRFOILS 



By displacing the initial circle so that it passes through only one of the points 

 s = - c and surrounds the other, the Joukowski transformation can be made to yield a contour 

 that is pointed at one end and rounded at the other. If the circulation is then chosen so as to 

 make the velocity finite at the pointed end, it is finite all round. According to a hypothesis 

 proposed by Joukowski, a properly designed airfoil automatically develops in the fluid around 

 it, by means of friction, a circulation of such magnitude as to remove the tendency for the 

 velocity to become infinite at the trailing edge, which is usually made comparatively sharp, 

 and measurements have shown this hypothesis to be close to the truth. 



The general Joukowski transformation is most easily handled by a graphical method. 

 It is desired to construct points representing 2 'where 



183 



