CONFOKMAL TRANSFOEMATION TO PROBLEMS IN HYDRODYNAMICS. 483 



the ends of the curve, any number of real parameters a 2> a x , ... may be taken, 

 subject only to the condition c > a, > a 2 > 3 > ...> c, and another set of para- 

 meters K!, K 2 , K S , ... associated with them, subject to the condition 



... =p + q ......... (129) 



Then the transformation 



(130) 



makes the axis of z real correspond to an open polygon in the plane such as that in 

 fig. 18. The external angles K^TT, K^TT, ..., so that the angle between the directions of 

 the arrows on the first and last sides is (K I +K.,+ ...) -w. The straight line in the 

 plane joining the points c must correspond to a convex curve in the z plane. 



The representation of the f configuration upon the half plane of w is of the 

 Schwarzian type, taking account of the corners in the boundary of the relevant 



Fig. 18. 



region. Thus if a, /3 be the values of w corresponding to the corners, the full (z, w) 



transformation is 



dz dw M3l) 



(-aj)* (*-,)*(-)*... ~ (w-*) p (w-W 



subject to the condition (129). 



Still further generalisation can be secured by making some of the screened-off sides 

 of the polygon in the plane curved. This can be done by introducing curve-factors 



