Let the initial circular cylinder be so placed on the s-plane that neither of the points 

 2 = i c lies outside of it. If one of them lies on the cylinder, an infinite velocity will occur 

 at the corresponding point in the transformed flow unless the original point on the initial 

 cylinder was a stagnation point. For dw/dz'= {dm/dz)/{dz'/dz), so that, at 3 = i c, 

 dm/dz '-»oo unless dm/dz -> 0. The singular point 3 = lies inside the cylinder and can be 

 disregarded. 



To study the general character of the transformation, let s'and z both be represented 

 for the moment on the same plane, with coincident axes. At infinity z' = z, so that the flow 

 is unaltered. The transformation gives to every finite point represented by 3 = re the dis- 

 placement cVs = (c'^/r)e~ ; the magnitude of this displacement is inversely proportional to 

 r and its direction lies at the same angle below the a;-axis as does the vector representing 

 2 above it; see Figure 114. Thus all points not on the ai-axis are moved toward this axis, and 

 all points not on the y-axis are moved away from this axis, provided r ^ c. Points on the 

 a;-axis are merely shifted along it, and similarly for the y-axis. 



Figure 114 — The Joukowski transformation, 



z' = z + c^ / z. 



Points lying on the circle |3| = c are brought on to the segment of the a;-axis between 

 3'= - 2c. Other circles transform into curves whose shapes vary widely. A circle centered 

 on the a;-axis is transformed into a curve that is symmetric with respect to the a;'-axis. If the 

 center of the circle is not on either axis, the transformed curve is asymmetric. 



The part of the 3-plane that lies outside the circle \z\ = c is thus mapped onto the 

 entire 2 '-plane, conformally except at 3 = i c. The transformation can be visualized by 

 imagining the circle |3| = c to be both flattened vertically and drawn out horizontally until 

 it becomes a segment of the real axis of length 4c, accompanied by a corresponding distortion 

 of all parts of the plane. Circles centered at the origin become converted into confocal 

 ellipses, while the radial lines outside the c-circle become joined at the ends to form hyper- 

 bolas confocal with the ellipses; see Section 81. The interior of the c-circle is likewise 

 mapped onto the entire 3 '-plane, as if it were turned inside out and also reflected in the real 

 axis while the origin recedes to infinity. 



174 



