CONFORM AL TRANSFORMATION TO PROBLEMS IN HYDRODYNAMICS. 441 



3. One object of the present paper is to try to extend the range of transformations 

 available for the conformal representation upon a half-plane of a region bounded by 

 a polygon some of whose sides are straight and some curvilinear. Aspects of the 

 subject will be discussed which have the theoretical advantage of being to some 

 extent free from the empirical character of Mr. PAGE'S method, but the application 

 of these to particular cases presents great practical difficulty. It is therefore proposed 

 to begin by studying those functions of w which, when introduced as factors into 

 formulce of transformation which are otherwise of the Schwarzian type, lead to 

 conformal representations of the required general character. Every increase in the 

 range of such functions made available, and every advance in knowledge of the theory 

 of such functions, diminishes the empirical element in their employment for the 

 solution of particular problems. 



The main object of the paper is the extension of the range of solvable problems 

 in two-dimensional liquid motion, and cases of such motion will be used as illustrations 

 of the pure mathematical theory. 



CONFOKMAL CURVE-FAOTORS. 



4. Definition and Characteristics. -Consider a transformation 



(w-a r }-^div, ......... (3) 



wherein A, a 1; a 2 , ... are real constants, ,, a,, ... real constants in ascending order of 

 magnitude, and ^ a function of u\ If ^ were absent this would be a Schwarzian 

 transformation giving a conformal representation upon the half-plane of w of the 

 region in the z plane bounded by a rectilineal polygon of external angles a 1( a,, a 3 ... . 

 Suppose ^ to be such that the transformation as it stands gives a conformal 

 representation upon the half-plane of w of the region in the z plane bounded by 

 a polygon having the external angles a ]; 2 , a 3 , ... as before, all its sides but one 

 straight, and that one side (say the side corresponding to a r+l > w> a r ) curvilinear. 

 A function ^ of w having this property may be called a " conformal curve-factor," or, 

 for brevity, a "curve-factor." In view of the possibility of our having to consider 

 functions which, when thus introduced into a Schwarzian transformation without 

 spoiling its conformal character, replace the rectilineal polygon by a polygon having 

 the same angles and not one only but several curved sides, it may be well to 

 distinguish between "simple," "double," "triple," &c. conformal curve-factors, 

 according to the number of curved sides which they introduce into the polygon. 



If a function ^ of w is a simple conformal curve-factor it must satisfy the following 

 requirements : 



(i.) The vector angle of the complex <$ must have a constant value for all real 

 values of w greater than a r+1 , and a constant value for all real values of w less than 



3 N 2 



