178 Mr Love, On the Theory of Discontinuous [May 4, 



Peop. II. It is possible to find a transformation by means of 

 a relation between two complex variables Z and Z' by which any 

 given polygon in the Z' plane can be transformed into the real 

 axis in the Z plane, and points within the polygon in the Z' plane 

 correspond singly to points in the Z plane whose imaginary part is 

 positive. 



This transformation and the conformable representation of the 

 polygon upon the half-plane, which it involves, have been in- 

 vestigated by Schwarz and Christoffel, and it can be shown that 

 the relation between Z and Z' is given by the equation 



%=AIi{Z-X r f^- 1 (B), 



where A is a constant and X r is the point on the real axis in the 

 Z plane which corresponds to the internal angle a r of the polygon 

 in the Z' plane. 



To verify this observe, 



(i) That dZ'/dZ is never zero or infinite except at points on 

 the real axis in the Z plane : 



(ii) That if Z be real, and lie between two consecutive zeros or 

 infinities of the function dZ'/dZ, say X r and X r+l , the argument of 

 dZ'jdZ or dZ'jdX remains the same for all the values of X, so that 

 the argument of dZ' remains the same, and all the points Z' which 

 correspond to points on the real axis between X r and X r+1 lie in 

 one straight line in the Z' plane. 



By combining (i) and (ii) it appears that the points on one side 

 of the real axis in the Z plane correspond to points within a 

 polygon in the Z' plane, and the points X x , X 2 , ... correspond to 

 the corners. 



We must further observe (iii) that if Z be very near to X r all 

 the other factors on the right of (B) may be considered constant 



except (Z-X r ) a ^ ir ~ 1 , so that in the neighbourhood of X r the 

 change of the argument of dZ'jdZ is the same as that of this 

 factor — and, in passing through X r in the positive direction, this 

 change is an increase by it — a r , which is the same as the increase 

 of argument of dZ' in going round a corner of the polygon where 

 the internal angle is a r . 



We note that it is in general possible in a given problem to 

 choose arbitrarily three of the points X v X 2 , ... "and then the rest 

 will be determined. 



2. The problems to which the theory is applicable are such 

 as are concerned with the motion of fluid in space bounded 

 partly by fixed plane rigid walls and partly by one or more 

 free stream-lines along which the velocity is constant. These 

 include the escape of a jet from a polygonal vessel, flow into 



