436 Lieut.-Col. A. R. Richardson on Stream-line 



II. £ has an essential singularity for some value of iv. 



It can have no zeros in the relevant part o£ the w-plane 

 otherwise g = co at the corresponding points in the fluid. 



Hence it" f contains an algebraic factor a zero, w G , of this 

 factor must lie in the non-relevant part of the w-plane. 



Let i|r=-vjr be the stream-line through w , and w 1 a point 

 on -v/r in the relevant part of the plane. 



The path in the £- plane corresponding to t/t must pass to 

 infinity as w approaches w along -yjr = ty from w lf 



Hence there must be either a pole or essential singularity 

 of z at some point on i/r between w 1 and iv . 



Again consider the case of any disturbed motion of a free 

 stream-line such that takes the same value an infinite 

 number of times ; e.g. a disturbance over the surface of 

 a jet. 



Hence f takes the same value an infinite number of times 

 and has an essential singularity for some value of w. 



In such cases £= — f- = f(iv) has an essential singularity 



for some value of w, say w = d. 



Hence z will also have an essential singularity at this 

 point. 



It is therefore necessary to examine the solution with 

 a view to seeing if any physical meaning can be attached 

 to the integral. 



o 



III. Examination of the integral of -^- = f(w). 



J J '' dw v ' 



To simplify the discussion suppose w = d is the sole 

 essential singularity of f{w) which is holomorphic elsewhere. 

 Suppose further that the whole of the ^(;-plane corresponds 

 to the whole of the ^-plane. 



To one value of w corresponds one value of z. The 

 converse is not true, for in the neighbourhood of w = d 

 z takes any value (not an exceptional value) an infinite 

 number of times. 



Hence corresponding to any value z of z there will be an 

 infinite number of values of w. 



The stream-lines, correspond ; ng to the lines -v/r = constant 

 which pass through such values of w, will all pass through z . 



The following results show that this infinite branching 

 arises solely from points w in the vicinity of d, the essential 

 singularity. 



