Multiplying through by -c'*/[r (a^ - h^)]: 



— (cos 6 cos T] + sin 6 sin -q) - = U. 



This can be written 



-A2 t a^_h^ 



2 



— cos d + A' cos 7/1 + sin - A' sin r/ I = a'^ , [79b] 



c A C' a r^„ ,, 



A'= , a'= . [79c, d] 



a2 _ ^2 a^ - A^ 



Now c^ cos 0A, -c^ sin Q/r are the coordinates of the point c^ / z. Hence, Equation [79b] 

 shows that c^ / z lies on a circle of radius a' drawn about the point C" or (-A'cos 77, A 'sin 77) 

 as center. Clearly OC" and OC are equally inclined to the y-axis but on opposite sides of it. 



If, in particular, the initial circle passes through the point 3 =- c or S, C" lies on 

 the radius EC. For, the slopes of BC and BC'are, respectively. 



A sin 77 A' sin 77 c h sin q 



c+hcosr, c- A'cos 77 c(a2 - A^)- c^Acos 77 



from Equation [79c]. But, from the triangle BOC, 



a'^ - h^ = c'^ + 2c h cos rj. 



Hence the second slope equals the first. Since the point g = - c or 6 is itself on the locus 

 circle, the two circles touch at B. 



According to the results of Section 78, a circle centered at Cj, the intersection of the 

 radius BC with the y-axis, would transform into a circular arc of total angular length 4^, 

 where /B is the angle between the radius BC and the avaxis. This arc, with ends at 

 2'= - 2c, lies inside the transformed contour as a sort of skeleton. 



The construction of an airfoil contour in this manner is shown in Figure 124 for 

 A = 0.87c, R = 34° 40'. The skeleton arc is also drawn. The graphical procedure is 

 discussed further by Ruden. 



185 



