A different type of flow between such walls, in which there is no source on the axis 

 and hence no net outflow of fluid, was derived by Lbwy from the transformation tc = z'""'" *" 

 where m and n are real constants. 



Geometrical Properties of the Transformation 2' = In 2 



If 2 = re'*^ and 2'= a;'+ iy\ then a;'= In r, y'= ^. If is kept in the range - 77 < < tt, 

 the entire 2-plane is mapped onto a horizontal strip of the 2 '-plane extending frcm but not in- 

 cluding y ' = - n^ up to and including y' = n. The negative half of the real axis of 2 is mapped 

 onto the upper edge of the strip at y' = 77; the positive half becomes the parallel line y' = 0. 

 AH radii from the origin of 2, in fact, become lines parallel to the real axis of 2', each defined 

 by a certain value of or y'. Circles about the origin of 2, on the other hand, being defined by 

 fixed values of r or of a?' become lines parallel to the imaginary axis of 2'; the annulus between 

 two such circles becomes a strip in the same direction. AH other straight lines on the 2 '-plane 

 correspond on the 3-plane to logarithmic spirals with focus at the origin. 



The transformation can be visualized by imagining the 2-plane to be cut just below the 

 negative a;-axis and partially shut up like a fan, while the origin is spread out over a width 2tt 

 and simultaneously displaced to minus infinity. 



By selecting for 6 a different range of magnitude 2n^, the strip may be displaced verti- 

 cally into any other position. Or, if d is restricted to a range of width « , where < a < 2n, 

 the corresponding sector of apical angle a is transformed into a horizontal strip of width « on 

 the 2 '-plane. Finally, if Q is allowed to range without limit, the 2-plane is m.apped once on 

 every successive strip of width 2n. The complete transformation is thus infinitely many-valued. 



Sometimes it is convenient in such cases to'include both boundaries of the transformed 

 area. Thus, if - tt < 64, n, the negative a;-axis is used twice; with = - 77 it transforms into 

 the lower boundary of the strip at y'= - n, with d = n, into the upper boundary at y'= n. 



A simple closed curve not surrounding the origin on the 2-plane becomes a simple closed 

 curve on the 2'-plane, but one that surrounds the origin becomes an endless curve that is per- 

 iodic in the y' direction, with a period 2n, provided amp 3 is allowed to increase indefinitely 

 as the 2 curve is traced repeatedly in the same direction. If amp 2 is restricted to a limited 

 range, a closed curve about the 2' origin becomes an open one on the 2-plane, traced once for 

 each traversal of the 2' curve. 



The more general transformation, 2' = -4 In (az) = A\nz + A\n a, includes also rotation 

 about the origin through an angle equal to amp A, a uniform change of scale in the ratio |.4|, 

 and the displacement represented by A In a. 



(For notation and method; see Section 34; Reference 1, Article 64; Reference 2, Section 

 8.11, 13,10, 13.20, 13.21, 13.33.) 



85 



