Figure 56 — Streamlines due to a Karman vortex train or "street." 

 (Copied from Reference 28.) 



Further Geometrical Notes 



Any given (f> circle has a radius R^ - c csch \{(ji/A)\, and its center is at ar =c coth 



{(}}/A). Thus the points {- c,0) are inverse points with respect to each of these circles; for, 



the distances of the points from the center of any circle are li = |c - c coth (cf>/A) | and c?, 



= I - c - c coth ((f)/A)\, and d d = c^ [coth-^ (4>/'^) - U = ^^^ • The equation of the circle 



might be written, from [41i], 



In— 1 = .^ = ±sinh-i— [41m] 



r2 A R^ 



the sign depending upon whether r or r is the greater. 



The ijj arcs, on the other hand, have a radius R,=c esc |(iAA4)|and are centered at 

 y = - c cot (il//A). The region between any two of these arcs is mapped onto a strip of the 

 w-plane lying between the corresponding values of tA. The entire 3-plane is thus mapped onto 

 a strip of width 2nA, between i// = - tt A and if/ = n A, and it is mapped again on each successive 

 strip of the same width. 



The arcs for lA = - n A/2 and i'/ = - Stt A/2 are semicircles which together form a circle 

 of radius c centered at the origin and passing through the singular points at (- c,0). Thus the 

 transformation can also be used to transform the interior of a circle into an infinite strip. The 

 strip is n'l,4| wide and parallel to the real axis of i//; it can be shifted so as to lie between the 

 lines lb = - TT A/2 by adding z ^/l to the value of w and hence n- 4 to that of t'/, and, since i n 

 = In(-l), tlie transformation can then be written 



w = AWn (c + 2) - In {c-z^ [41n] 



Here amp (c-s) has been chosen so that c - z = r^ e and In (c-s) = In (z-c) - m, as 



shown in Figure 57; thus now ili = A ($2-0, + n). The quantity rr - 6^ also equals the internal 

 angle 9, between cO and cz measured positively clockwise; in terms of this angle. 



90 



