14 Proceedings of the Royal Irish Academy. 



represented upon the half-plane t; > 0. This implies a relation between 

 to and £ such that the relevant region in the w plane is also conformally 

 represented upon the half-plane ij > 0. Thus there are two functional 

 relations, the "geometrical relation" between ; and £, and the "field relation" 

 between w and £. Figure 3 represents the £ diagram, and it is convenient 



c 



K.g. .5. 



to specify i »■ «i ) 1 1 -^ of interest in the other diagrams by i In- values assigned 

 to fat these points, namely c at i he prow, -a and " at the points of 

 departure of the free stream-lines, and - oc and + x for infinite remoteness 

 along these stream-lines. 



A suitable form for the field relation is 



w=£K(£-i (I) 



the main problem is to Hud a geometrical relation which shall satisfy 

 nol only the geometrical data but also the special requirements of free 

 stream-lines. When tins problem is dealt with by the method of curve- 

 factors no variables are required but those above specified. 



'■>. If the obstacle i- symmetrical about a line parallel to the ultimate 

 Btream, the distribution of flow is symmetrical about the same line. Such a 

 configuration is typified in figure 4. In this case it is convenient to adopt a 



Fio. 4. 



different formulation and to regard half the field of flow as constituting the 

 relevant region in the ; plane, the boundary being completed by the straight 

 stream-line in the line of symmetry. The corresponding region in the 

 "■ plane is the half-plane on the positive side of the real axis, so that the 

 intermediate variable £ is not required, £ and w being equal. The value 

 assigned to £ or w at the prow is now zero, and the value at the point of 

 departure of the free stream-line is a. 



