Lioathkm — On 'two-Dimensional Fluid Motion. 13 



The obstacle is of finite dimensions, and is assumed generally to have a 

 pointed prow or vertex, and smoothly curved sides. The type of the 

 configuration is represented in figure 1. A stream-line divides at the vertex, 



and each branch follows the contour of the obstacle for a certain distance and 

 then breaks away as a free stream-line, which tends to ultimate parallelism 

 with the axis of x. Between the free stream-lines, and partially bounded by 

 the hinder surface of the obstacle, there is a semi-infinite region occupied by 

 liquid at rest, the " wake." 



The angles which the tangents to the two sides of the obstacle, just at the 

 vertex, make with the axis of x may be denoted by (p + s) it and - (p -s) n- 

 respectively. 



The relevant region in the z plane is the region of flow, whose boundary 

 is made up of the two free stream-lines and part of the surface of the obstacle, 

 including the vertex. 



The field of How is formally specified by the complex variable w s <f> + ixf/, 

 such that, if (u, v) be the velocity, 



d(p = u dx + v dy, d\p = u dy - v dx. 



A knowledge of the functional relation between w and z is what is required 

 for a definite specification of the motion. Such a relation must give a 

 conformal representation of the relevant region in the ; plane upon a 

 relevant region in the w plane, consisting, namely, of the whole of the w plane 

 bounded only by the two sides of a cut made along the axis of (f> from <j> = 





Fiq. 2. 



to (j> = + co, as indicated in figure 2. It is assumed thai w = at the 

 vertex. 



It is found convenient to introduce an intermediate variable £ ■ ? + hi 

 which is so related to z that the relevant region in the : plane is conformally 



[S*] 



