Flow from a Disturbed Area. 437 



(i.) About d describe a small domain B in the 2#-plane. 



Of all the stream-lines through z in the z-plane only a finite 

 number lie on lines ty = constant in the iv-plane which do not 

 pass through B ; and all except a finite number of the 

 points iv which correspond to z lie inside B . 



.For i£ not there will be a point of condensation 



IV] , W 2 , • • . .W n > W w 



outside B c . 



Hence io w is the limit of a set of values w n at each of 

 which z = z 0) i.e. w M is an essential singularity . contrary 

 to hypothesis. 



(ii.) A stream-line can have only a finite number of double 

 points over a length AB for which the corresponding points w 

 do not lie inside B . 



For if an infinite number exist there will be a value iu M such 

 that near the corresponding z M there is infinite oscillation < f 

 the stream-line. This is contrary to the assumption that f(w) 

 is holomorphic near w w . 



(iii.) The domain B may be extended so that for values of w 

 outside it the corresponding portions of the stream-lines. have no 

 singularities. 



Describe a curve W in the w-plane not a locus of double 

 points to cut every line i/r = constant but not passing 

 through B . 



Let C be the corresponding curve in the ^-plane. 



To the intersection of G with any stream-line yjr will 

 correspond a finite number of points, in the w-plane, not 

 lying on W but which are outside B . 



From these points blacken out that part of every line 

 <\Jr = constant remote from W. 



Hence on any line ty = constant there will be a finite 

 interval between W and the blackened part of the wr-plane. 



Since the transformation is holomorphic, to a small deter- 

 mination of will correspond a small deformation W of W 



(except where ~ = or go) and of the blackened part of 



v r div ' l 



the w-plane. This deformation on any stream-line can be 

 taken so small that W does not enter the blackened area 

 unless it passes through a double point on the stream-line. 



Hence a finite domain can be determined a hour W such 

 that inside the corresponding domain about 0, in the 

 ^-plane, every stream-line is free from singular points 

 except such as arise from places w inside P> supposed 

 extended to coincide with the blackened area. 



