[79£ 



and c is real. 1( s = re 



Thus the point representing the complex number c /z lies on a line inclined at the angle 9 

 below the a;-axis, and at a distance c /r from the origin. The vectors representing c'^/z and 

 3 are easily constructed and can then be added vectorially to obtain z. 



For values of z representing points on the initial circle, the operation can be simpli- 

 fied by first constructing the locus on which c^/s must lie. Let the center C of the initial 

 circle, whose radius is a, be displaced a distance h from the origin in a direction making an 

 angle rj with the positive a;-axis, as illustrated in Figure 123. Then, when z lies on the circle, 



(r cos 9 - h cos rj)^ + (t sin 9 - h sin rj)'^ - a^, 

 T - 2 hr (cos 6 cos 7/ + sin 9 sin 77) + A^ - o^ = 0. 



Figure 123 — Illustrating relations for a general 

 Joukowski transformation. 



184 



