Writing C = C^ + i C^ where Cj and C^ are real, Cw = C(<p + ii/j) = C^4,-C^4j + {(C^^ + C^iP). 

 Thus C changes the potential and the stream function from <p and i/f to 



Here, in the terms containing C,, i/i may be regarded as a second possible potential and - 

 as the corresponding stream function. It is already known, however, that a new potential and 

 the associated stream function can be constructed by making a linear combination of other 

 potentials and the same combination of the associated stream functions. 



The addition of Z? to Cw then merely adds constants to 0' and i// ', which, as hydrodynami- 

 cal quantities, contain arbitrary constants in any case. 



In view of all these results, it is often convenient to study a transformation in skeleton 

 form, with the omission of constants such as A, B, C, D. The equations thus obtained may not 

 be dimensionally balanced, from the physical standpoint. The results can then easily be gen- 

 eralized as desired by adding constants to and tA, or by multiplying both of them in all 

 equations by the same real constant, or by making suitable combinations of these functions, 

 or by changing axes on the z-plane, or, finally, by multiplying z, x, and y in all equations by 

 the same real number. In this way, also the dimensional balance can be restored if desired. 



In practical problems a boundary condition is usually specified. If the fluid is confined 

 by fixed bounding surfaces, the streamlines must be tangential to these surfaces, and over 

 each of them tlr must have a fixed value. The mathematical problem is then to find a trans- 

 formation w = f(2) such tiiat the curves representing these surfaces on the 2-plane transform 

 on the w-plane into straight lines parallel to the axis, along each of which lA has a constant 

 value. 



No practical general method of discovering the necessary transformation is known. It 

 can sometimes be found by means of the Schwarz-Christoffel transformation, which will be de- 

 scribed presently. Many types of flow have been discovered by assuming some transformation 

 and then investigating the flow that it represents. 



26. THE TRANSFORMATION OF IRROTATIONAL MOTIONS 



The solution of a new problem can sometimes be obtained by transforming the known 

 solution of an old one. Thus, let w = + z t/f = /(g) be the complex potential for a known 

 problem; and let z be connected with a new variable Z by the transformation 



z = X + iy = F{Z), Z = X + lY • ^- 



The result is equivalent to a single transformation from Z to w: 



w=f[z(Z)] =g{Z) 



50 



