be supposed to exist on the boundary. The general principles and theorems that hold for 

 irrotational motion will hold within the moving fluid taken by itself, but they may not be 

 applicable throughout a region that includes part of the surface of discontinuity. 



The theoretical flow of a frictionless fluid past an obstacle as obtained on such an 

 assumption shows more resemblance to the flow of an actual fluid than the theoretical flow in 

 which the motion is everywhere continuous. In particular, a force is exerted on the obstacle. 

 But in real cases a great deal of vortex motion is observed to exist in the wake. 



In the mathematical theory of two-dimensional motion, constancy of the velocity q 

 implies constancy of \dw/dz\ along a free streamline. It is found convenient to work with 

 the variable 



dz 



C= , [110a] 



dw 



or, in terms of the velocity components u and v, 



I dw\~^ 1 u + iv 



\ dz I -U + IV 2 



where the last member is obtained by rationalizing the denominator and using q = u + v . 

 Thus \C\ =1/?. 



Regarded as a function of s, ^(2) effects a transformation from the s-plane onto a 

 ^-plane. Let this plane be drawn parallel to the 2-plane and with the real axis of C parallel 

 to the aj-axis. Then the vector representing i^ has the same direction as the particle velocity 

 at the corresponding point on the 2-plane, since it makes with its real axis the angle 

 tan~' {v/u). Therein lies the special utility of the variable ^. 



Each streamline on the 2-plane transforms into a curve on the 4^-plane, and this curve 

 can be regarded as the corresponding streamline in a transformed motion. From the properties 

 of ^ it is clear that any straight portion of a 2 streamline will transform into a segment of a 

 parallel radius from the origin. As the 2 point traces a curved free streamline, on the other 

 hand, along which q is constant, the ^ point traces the arc of a circle of radius l/g, centered 

 at the origin. 



Further transformations may then be made in terms of other variables until the problem 

 is converted into a form in which the solution can be guessed. 



In the alternative "hodograph method" of Prandtl, dw/dz, or -1/^, is used as an 

 auxiliary variable instead of ^■, see, for example, Betz and Petersohn.^*^ 



(For notation; see Section 34.) 



271 



