Panel Discussion 



One sees then that the Bieberbach method, which necessarily operates with 

 the inverse transformation (2), cannot be assured of success. Furthermore, a 

 corollary of the foregoing discussion is that the probability of success is much 

 higher for a nearly circular section than for an elongated one. This indicates 

 the desirability of a preliminary transformation of the Joukowsky type, such as 

 that used in the Theodorsen method of conformal mapping, which first maps the 

 given profile into a near circle. 



BRANCH- POINT TRANSFORMATIONS 



Consider a ship section which intersects the free surface at an angle a at 

 A and the vertical centerplane at an angle /3 at b . The double ship section will 

 then have corners of angle 2a at A and 2/3 at B . We wish to transform the con- 

 tour of this double section into one without corners. 



A transformation which eliminates the corners at A and its image in the y 

 axis is [5] 



'^'. P = 2(1 -^), 1 < p < 2 . (3) 



This transforms the point B to a point B in the z plane with coordinates 

 (0,b'), where 



b' = cot — , — = cot y ■ \^) 



P a 



Since the point A is transformed into a point A' with coordinates (1,0), we see 

 that 



y 



(5) 



is the angle o'b'a', where 0' denotes the origin in the z plane. Then we have 



y' < y. 



Next, we wish to eliminate the corner at B' in the z' plane. Put 



'' - ^^"'"^^' - ^ 2(1 -4), 1 < q< 2 . (6) 



q 



In the z" plane, the points a and b are now transformed into A" and b" with 

 coordinates (a" ,0) and (0,1), where 



1 1 pq 2 \ ^] 



Here, 7" is the angle o"b"A" in the t plane. 



1621 



