where n is a real number not less than 1/2. In terms of <^ = re'", 



2=rl/"e'^/" [60c] 



Thus the real axis of ^, corresponding to = or ff, is bent at the origin into two radii on the 

 2-plane enclosing an angle n-/n, and the space above the real axis of ^ is transformed into the 

 space in this angle. The angle is concave if n > 1, convex if 1/2 ^ n < 1. Under the trans- 

 formation, the upper half of the ^-plane may be imagined to expand or contract as the negative 

 half of the real axis rotates into the proper position. 



An infinite wall lying along the real axis of ^ thus becomes an angle formed by two 

 semi-infinite planes joined at their edges. Suppose that there is also a singularity in the flow 

 on the (^-plane, such as a source, vortex or dipole, located at a point which lies above the 

 ar-axis and is represented by ^ = A^e'^" or (A" cos /Sn, A" sin /3n). Assume that = /3 ^ n/n. 

 This singularity will have a line image in the plane wall, located as if behind a mirror or at 

 ^ = A" e~'^". The effect of the transformation [60a, b] will then be to transform this singu- 

 larity into a similar one on the 2-plane, located at 2 = A e'" or (A cos /3, A sin /3) inside or 

 on the angle. 



Equations [43a], [42a], and [50a] are readily adapted to the geometry of the present case by 

 changing 2 to ^ and making the proper substitution for c or A. Let all amplitudes d, except that of 

 e~'P", be taken in the range £ d < 2 tt. The complex potentials w on the 2-plane thus obtained 

 are, when written in terms of 2, 



Source: w = -A [In ('2"-A"e'^"; + In (2"-h"e-'^")] . 

 Vortex: w = iA [111(2" -h'^e'^'*) - in (z-'-h^e'^")] . 



Dipole: w = ^ [e'« ('2''-A"e'^T^ - e^'^z" -h^e-'^T^I . 



According to the results of the last section, the source on the 2-plane emits a volume 

 of fluid equal to 2TrA per unit length, and the circulation around the vortex is still F = 27tA, 

 as on the ^-plane. The transformed dipole moment, however, is ij./(nh"~^), and its axis is 

 directed at an angle ^-{?i-l)/3 to the positive a;-axis on the 2-plane. For, as 2 -» Ae'F*, 



2"_A"e^" 2-Ae^ V 3-he^' z - heW \^^ ' 3 = he^^ ""^"'^ z -he'ft 



If n = Y^, the "angle" becomes a semi-infinite plane, as in Section 39. A few of the 

 streamlines due to a symmetrically placed vortex near such a plane are illustrated in Figure 

 91. In this figure /3 = ff, so that on the ^-plane the vortex lies on the y-axis and its stream- 

 lines are circles, like those shown in Figure 61 and suggested briefly in Figure 92. Refer- 

 ences: Greenhill^^, Hamel'*'*, Paul'*^''*^, and for the vortex, Miyadzu"*^; with flow past the 

 corner and perhaps finite q at the edge, Uslenghi^*; with a line source on the edge, 

 Kucharski"*^. 



(For notation and method; see Section 34: Reference 2, Section 8.51.) 



135 



