To establish this result, let the potential for the actual motion be written as the 

 sum of the potential for the approximate motion thus constructed and a fourth component 4>\ 

 or 



V r 



where 0r/ is the potential for the uniform flow; see Section 40. The partial motion represented 

 by (^' will then be one in which the net outflow or inflow across & vanishes, and in which the 

 circulation likewise vanishes around every closed curve that does not cut S. Hence, in par- 

 ticular, (;6'is singled valued, and so is the corresponding stream function i//'. In such a mo- 

 tion, it can be shown that the velocity vanishes at infinity at least as fast as lA^, as it does, 

 for example, in the symmetrical flow caused by a moving circular cylinder. At large distances 

 from S the motion represented by cyS'may accordingly be disregarded in comparison with the 

 other components, and only motions (a), (b), and (c) remain. 



The corresponding theorem for the complex potential is that, if dw/dz is differentiable 

 and single valued outside a curve S, and also finite at infinity, then, at sufficiently great 

 distances from S, dw/dz can be expanded in a Laurent series of the form, 



dv: ^' ^2 ^' ^ ^ 



— =.... — + — + — + a ' [72c] 



dz ^3 32 3 



as stated in Section 27. Hence, by integration, 



6. h 



- + — + a„ + cln 2 + a, 2 [72d] 



2 2 







where ....5,6,, a , c are constants, real or complex. The last two or three terms of this 

 series represent the approximation just described; the real part of c gives a term representing 

 the sources, the imaginary part a term representing the circulatory motion. The term in In 2 

 is many valued. 



If the surface S represents a rigid cylinder, there can be no source and c must be purely 

 imaginary. Furthermore, the largest term that depends upon the shape and motion of S is the 

 term b^/z, which, as in Section 37, represents a dipole. Thus at large distances the effect of 

 a moving cylinder with no circulation around it is that of a line dipole located in its neighbor- 

 hood. 



It is of interest, finally, to consider the effect of a conformal transformation upon the 

 distant motion. In an important class of single-valued transformations the plane is left un- 

 altered at infinity. Such a transformation from 2 to 2 'can be written, toward infinity, in the 

 form 



161 



