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



it is seen that their respective moduli (on the linear range) are 



{2ce~ Y (ccoshy w)Y k and {2ce Y (c cosh y w)} 1/a , 



and it is to be remembered that the angular range of ^ 23 is zero. Thus the 

 transformation 



j u ww asmnb^-m /m \* 



dz= , . , - ^. ^ , (91)* 



U (2c) p (w c cosh y) p 



representing a configuration of the kind indicated in fig. 11, gives \dz\dw\ = U" 1 along 

 the curved part of the boundary, so that the curve may be a free stream-line along 

 which the velocity is U. If m be taken equal to p the angular range of the whole 

 transformation is zero, so that the first and last straight stream-lines are parallel. 



The extension to boundaries with a greater number of corners is obvious. 



36. Another simple example, leading to well-known results, is afforded by the use 

 of^ 13 . As the modulus of a ! - + w ! ' ! for ?r negative is (a ;)'''-, the combination of 



fl K/\c ccshf 



Fig. 11. 



a power of ^ 13 with a Schwarzian factor consisting of a suitable power ofwa will 

 give a free stream-line. 



In fact, the transformation 



gives a liquid flow of the character indicated in fig. 12, there being a fixed obstacle 

 consisting of two planes meeting at an angle Zp-n-, and stream-lines extending to 

 infinity and tending to parallelism with the undisturbed stream. 



The case of p = ^ is that of an obstacle consisting of a single-plane wall at right 

 angles to the stream. 



37. The previous transformation suggests a method of building up a transformation 

 applicable to symmetrical flow past an obstacle in the form of an open polygon. The 

 method consists in employing for the range of w negative, (the range which is to 



* An equivalent form of this transformation is 



dz = 



