78. CIRCULAR ARCS BY THE JOUKOWSK! TRANSFORMATION 



Let the initial circle representing a circular cylinder be centered now on the y-axis, at 

 (0, h), and let it have such a radius a as to pass through the points 2 = i c, as in Figure 116. 

 Then it transforms into a circular arc with ends at the points s'= i 2c, which may represent 

 a lamina of arcuate cross section. 



Figure 116 — The s-circle goes into an ate on the 3 'plane. See Section 78. 



For, each of the following two equations is equivalent to Equation [77a] 



(2'- 2c) z = {z - cf, (2'+ 2c) z = (z + cf. 

 Division of these equations gives 



s'-2c z - c 



a'+ 2c \ z -v c 



[78a] 



Write 



iQ, idy , id' id' 



z-c = T^e \z + c^T^e ,z'-2c = T^e '■,3'+2c = T^e ^. 



The angles thus introduced are illustrated in Figure 116. Then, equating complex amplitudes 

 on both sides of Equation [78a], 



?i'-0;= 2(^1-02). 



[78b] 



177 



