in such a way that, as s varies freely but without touching this curve, w varies continuously 

 along one branch of its values without the occurrence of ambiguities as to the value of dw/dz. 

 Examples of this procedure occur in the next chapter; Sections 40 and 61 may be mentioned. 



Many transformations are most conveniently defined in the inverse form as 2 = F (w). 

 Upon separating real and imaginary parts, equations of the form 3; = F. (c^i/f), y = F^ ^^^ '/') ^^^ 

 obtained. From these equations the equipotential curves and streatnlines, defined by constant 

 values of (p and i/r, may be traced. 



Finally, the physical significance of certain constants that may be introduced into a 

 transformation should be noted. 



Consider, in the first place, the effect of replacing 



w = f{2) [25j] 



by 



w = f(A2 + B),OT w = f\kiz-h) e I [25k,l] 



where k = \A\, a = -amp A, so that A = ke~'", and h = -Be^^/k. The value of w that is 



associated with any given value 2 ^ of 2 by the equation w = f{z) is assigned by (25^ to a 



value 2, such that 



-ia ia 



k(z^ -h) e =2,,or2„=— e + h 



V 2 ' 1' 2 ;;. 



Thus the vector representing z is obtained from that for 2, by changing its magnitude in the 

 ratio 1/k and also rotating it through the angle a, and then adding the vector representing h. 

 The resulting change in the plot of w on the 2-plane.can thus be described by supposing the 

 plot to be changed in scale in the ratio 1/k and also rotated counterclockwise through an 

 angle a , without moving the origin, and then to be given the translation represented by the 

 real or complex number h. The entire flow is thus rotated and displaced on the 2-plane in the 

 manner described. This constitutes an important means by which the solutions of hydrodynamical 

 problems can be modified to suit new conditions. 



The changes produced in and ip regarded as functions of x and y by the rotation and dis- 

 placement of the plot are the same as would result from an opposite rotation and displacement of 

 the X and y axes and thus possess in themselves no novelty. The change of scale, however, 

 is less familiar. It leads to the useful rule that all functions or expressions resulting from 

 a transformation w = f(z) may be generalized by replacing everywhere 2 by kz, or x, y by kx, 

 ky, where k is any real number. 



In the second place, consider the effect of replacing w - f(z) by 



Cw + D = f(z) [25m] 



49 



