This transformation may be obtained by superposing a uniform velocity upon the flow 

 out of a spout having parallel walls in such manner as to reduce the fluid to rest within the 

 depths of the spout. It is convenient first to reverse the flow through the spout by substitut- 

 — w, for w in Equation [62a], which gives 



—IV. 



2 = -w^ + e ^ [64b] 



For uniform flow at unit velocity toward negative x, the complex potential is w = z, 

 with <^2 = ^- Combining the two flows, the complex potential isw=w.+W2='W.+2. Sub- 

 stitution of w;, = w - 2 in Equation [64b] gives 2 = -w + z + e""*"*" ^ or m) = e~ "' "*" ^, which is 

 equivalent to Equation [64a]. 



Then, from z = x + iy, 'w = (f) + uf,, 



X = <p + %ln (<f>^ + xlf^ ), y = ip +tan-^A, [64c,d] 



^dz \~^ w (ji + i\p 



-U + IV - ' 



[64e,f] 





(1 +<?!>)2 +^/r2 (1 + 0)2 + ^2 



Since y changes sign with 0, symmetry exists with respect to the a;-axis. Furthermore, 

 the expression for y is many-valued, with a period of In. To make y single-valued, let 

 tan~^ ("A/^) have always the sign of and be numerically less than tt. 



Assume that > 0. Then, if i/» is large, so is y. Furthermore, along any streamline or 

 curve for constant 0, as </> ranges from + oo to - «, y continually increases, with a total in- 

 crease of ff, while X decreases on the whole from + » to - «>. If i/( > i/^, 



dx (h 



— = 1 + > 0, 



^ 02 + ^2 



since the equation, c^^ + <p + ip'^ = has no real roots for 0; hence x varies always in the 

 same direction along the streamline. If < y> < *^, however, x retrogrades for a time as 

 decreases, giving an /S-shape to the streamline, as illustrated in Figure 97. 



To locate the streamline for i/' = Oi keep constant and let i/r -» 0. Then, if > 0, 

 from [64c, d] y-»tan~^ (0/0) ->0, x -* <^ + In 0. Hence, with = 0, as varies from + ■» to 0, 

 X traces the entire a!-axis and -» as a; -♦ - «=. If, however, < 0, y -* tan~^ ("A/0) -* ~ t 

 and a; -> + ln(-0). Here (ln02)/2 is written as ln(-0) rather than as In because < 

 and the real logarithm is intended. Hence, as decreases from to - oo, a; first increases 

 from - oo to a maximum value of -1 at = -1, where dx/d<p = 0, and then decreases again to 



144 



