The flow on the g-plane is thus transformed into a flow on the <-plane in which the 

 real axis is a streamline. A possible complex potential for such flow is w = ct. With this 

 potential, as t -* <=<>, so that (t - 1) ' -* t and predominates over the logarithm, s -> ht/n and 

 w - ct -» cnz./h. But the assumed flow at infinity requires that on the 2-plane w -* Uz. Hence 



c = h V/n, and 



kU 

 w = (J3 +14) = ;;. [107c] 



77 



By substitution, z can easily be expressed in terms of w, but ^ and \Jj cannot be separated 

 in terms of ordinary functions. 



The relation between w and z is more conveniently studied in terms of real coordinates 

 ji, V on the 3-plane defined by writing 



i = cosh (fi + if). [107d] 



Using hyperbolic formulas listed in Section 32, and also separating real and imaginary parts in 

 z - X + iy and in w, it is found that 



h 

 2 = — [^ + iv + sinh {fi + i v)\, [107e] 



h . h . ^ ^ 



X = — (^ + sinh fi cos v), y = — {v + cosh /^ sin v), Ll07f, g] 



hU hV 



— cosh |U cos V, ijj = — sinh /x sin v). [107h,i] 



The coordinates /x, v are single valued within the region of interest provided ^ /i, 

 Q ^v ^ JT. At infinity, f/ -> °o, sinh fi -► cosh //, ^/sinh fi -> 0, and (j> -* Vx, as it must. The 

 coordinate curves on which /i has a constant, positive value begin on the positive a;-axis, 

 where v = 0, cross the y-axis, and end on the line AB, on which u = n, y - h. The curve for 

 fi = is the segment OB. 



Some streamlines are shown in Figure 177. The point on OB at which g = \U\, or 

 u = 77/2, y = (2 + n)h/2n = 0.818A, is marked by a shortline. 



The line COBA is the streamline for i/* = 0. As the streamlines proceed from right to 

 left, they all rise through a total distance h. 



From Equation [107a], and Equation [107c] and K = h/rr, 



ldw\ 2 Idw I dz\^ 2 ^ - ^ 

 \dzl ~ \ dtl dt / ~ t + 1 



264 



