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



where and i/>- are the velocity-potential and stream-function of the liquid motion, 

 so that the velocity components u, v are (when z = x + iy) 



v = -fy/dy = ty/dx ..... (22) 

 With this interpretation it is known that 



dz/dvv = q~ l exp(ix), ......... (23) 



where q is the resultant velocity, and x the angle which its direction makes with the 

 axis of x. 



When a transformation is such that for w real and tending to + there is a 

 definite limit for dz/dw, it will be convenient to denote this limit by V" 1 . This 

 means that the limiting velocity at points indefinitely distant in the direction of the 

 axis of x is a velocity V parallel to the negative direction of that axis. 



The interpretation of other constants in the transformation formulae might be 

 emphasised by explicitly indicating a factor V in every constant which is associated 

 with w by addition or subtraction ; for example (wcj 1 ' might be replaced by 

 (/c c'V)''-', where c' has the dimensions of a length and depends only on the 

 geometrical configuration in the z plane. But, in order to avoid an unnecessary 

 appearance of heaviness in the formulae, this device for emphasis will not be employed ; 

 it will be easy and sufficient to remember that the velocity of flow corresponding to 

 a given transformation can be altered everywhere in the ratio of V to V, without 

 changing the geometrical configuration, by replacing V where it occurs explicitly 

 by V, and c, in an expression such as (w c) >! '\ by cV/V. 



14. Flow round a Semi-infinite Barrier in the Form of a Wedge with Smoothly 

 Rounded Apex. A transformation in which a curve-factor is not accompanied by 

 any Schwarzian factor is typified by 



.......... (24) 



where C is a constant and ^ a curve-factor of definite linear range (say c to.c) and 



definite angular range y. For example *& may be 

 any one of the curve-factors already enumerated. 

 The boundary in the z plane corresponding to 

 w real is free from corners. As the point w moves 

 along the negative direction of its real axis the 

 point z moves backwards parallel to the axis of x 

 until w reaches the value c ; then z describes a 



_. curve whose tangent (with the direction corre- 



sponding to w increasing) makes a continually 



increasing angle with the axis of x until that angle attains the value ny ; after that, 

 for w still decreasing, the boundary is straight. The configuration is of the general 



