On Two- Dimensional Fields oj Flow, 191 



what is wanted is a con formal representation of the relevant 

 region in the z plane on the half-plane of w for which yjr is 

 positive ; but when there are singularities the {z, w) relation 

 is not conformal, and an intermediate variable £=£ + iv is 

 introduced, and relations are established between z and f, 

 and between w and f. The (z, f) relation is a conformal 

 representation of the relevant region in the z plane upon 

 that half-plane of f for which 77 is positive, and is generally 

 a differential relation formulated in terms of Schwarzian 

 factors and curve-factors by the methods, and subject to the 

 limitations, explained in the previous paper. The (10, t) 

 relation is an explicit formula for win terms of f, constructed 

 by taking account of the specified singularities, with such 

 images as are required to make ^ = when r] = 0. The 

 difficulty of integration, inevitable in problems of this kind, 

 arises only in connexion with the (z, f) relation. 



3. The geometrical relation. — In constructing the (z, £) 

 relation it is to be noted that a gap in the boundary at 

 infinity, corresponding to the parallelism or divergence of 

 two parts of the boundary, is to be regarded as a corner and 

 dealt with by means of a suitable Schwarzian factor. Curves 

 in the fixed part of the boundary are represented by curve- 

 factors selected from known types of suitable angular range, 

 and these curve-factors (unfortunately not usually the curves 

 themselves) are among the data of the problem. Curves 

 which are free stream-lines are represented by curve-factors 

 which are among the quaesita of the problem. 



4. The field relation. — In constructing the (w, f) relation 

 a distinction must be drawn between singularities in the 

 boundary and singularities at other points of the field. The 

 only singularities contemplated are logarithmic, namely, in 

 cases of liquid flow, sources and vortices, doublets being got 

 by differentiation if desired. A source in the boundary at 

 the point f=c, whose rate of output into the field is m, is 

 represented in w by a term —(m/ir) log (f— c). A vortex 

 in the boundary is inadmissible since it involves an infinity 

 of f. 



If the point f = c corresponds to a point at infinity in the 

 z plane, the singularity represented by the above term is a 

 source at infinity, that is an influx of liquid at the rate m 

 through an infinitely distant gap in the boundary. 



A source whose rate of output is m, situated in the field 

 at a point f = a. + i{3, must be counterbalanced by an image 

 source at the point £==a — ifi. Hence it is represented in w 

 by a term 



-(in/2,r)log{(r-«-i/8)(r-« + ;/8)}. . . (1) 



