470 DE. J. G. LEATHEM ON SOME APPLICATIONS OF 



It need hardly be said that a reentrant angle of the polygon would, through the 

 corresponding Schwarzian factor, lead to a zero of dzfdw, and so to an infinite velocity. 

 In such a case some different transformation would apply, namely one which would 

 include a free stream-line starting from the reentrant corner, at least one of the 

 subsequent corners being in the region of still water. 



38. Free, Stream-lines when the Fixed Boundary includes Curves. The method 

 of the previous article suggests a step towards generalisation which consists in taking 

 part of the polygonal fixed boundary to have an indefinitely large number of corners 

 each of indefinitely small external angle, introducing a suitable curve-factor and 

 Schwarzian factor for each corner, and taking the limit of the product. 



The parameters b, c, ... are replaced by a real variable 9 which ranges between the 

 values assigned to w at the extremities of the curved part of the fixed boundary, and 

 the indices q, r, ... of the corresponding Schwarzian factors are represented by C^X/TT, 

 where x is the angle between the tangent, drawn in the sense of w increasing, and a 

 fixed direction. 



Thus there is obtained the resultant curve-factor 



* - Liin II {(iv 



winch is generally equivalent to 



the integration being extended over a suitable range of real values of 6. 



-V 



Fig. 14. 



For example if a liquid flow be interrupted by a symmetrical and symmetrically 

 placed obstacle consisting of finite curves CA, AB meeting at A at an angle 2pir, and 

 the values of w at B and A be taken as zero and a respectively, the transformation 

 would be of the form 



(95) 



