Panel Discussion 



77 



_ '"i , .>i.i u (14) 



z J - e z ^ ' 



and then substituting Zj for z in the above formulas. ■ ■ ■ "• -'-^^--^ - 



JOUKOWSKY TRANSFORMATION 



We shall now transform the profile in the z" plane to one in which the ratio 

 of the principal dimensions is unity. Put 



z" = c^ + I (15) 



and select the coefficients c and d so that the principal dimensions in the C 

 plane are each unity. Then we have, from (15), 



c+d=a",c-d=l, (16) 



where a" is given in (7), and hence 



c = ^ (a" + 1), d = Y (''"■ - ^) ■ ^^'^^ 



Set - ^ • 



z" = fi + iV. I = \ei* . (18) 



Then from (15) we obtain 



(19) 



/i. = (cA. + T-1 COS l/l , 



V = (ck - j] sin0 . (20) 



Eliminating ^ between (13) and (14) yields 



1^^ sec - V^ CSC i// = 4cd , 



or 



4cd sin"* i/' + (Ax2 + V^ - 4cd) sin^ - l^ ^ _ q , (21) 



a quadratic equation in sin^ . From its solutions one can obtain the corre- 

 sponding values of \ from (19) or (20). 



GERSHGORIN INTEGRAL EQUATION 



We now wish to transform the profile in the C plane into a unit circle 

 about the origin in the w plane, 



w = e^^ . (22) 



1623 



