CONFOEMAL TRANSFORMATION TO PROBLEMS IN HYDRODYNAMICS. 



449 



character of fig. 1 , wherein the arrows indicate the direction of w increasing, and the 

 relevant region is that on the left of the arrows. The angle ny must be positive 

 since that is TT times the order at infinity, and a negative order at infinity would 

 imply infinite velocity at infinity ;* the angle ny must not exceed TT as otherwise the 

 boundary would intersect itself. 



The form and dimensions of the curved part of the boundary depend on the form of 

 ^ and on the constants C and n. By equating 

 dx and idy respectively to the real and the 

 imaginary parts of C ^ n dw, when w is real and 

 between +c and c, the problem of expressing 

 x and y in terms of a real parameter w is reduced 

 to one of quadrature. The degree of arbitrariness 

 in the form of the curve corresponds to the extent 

 of the range of known forms of *& and the 

 adjustable parameters which these contain. 



A simple case is got by taking n = 1, and giving to 

 ^ 5 and ^ 7 ; this leads to 



dz \A(tc k) + ~B('t(> 2 c 2 )' l \ 



where 



A > 0, B > 0, and c > k > -c. 



Fig. 2. 



a form 



representative of 

 (25) 



The boundary in the z plane consists of two parallel lines extending to infinity on 

 the right, and smoothly joined by a curve on the left, as in fig. 2, the relevant region 

 being to the left of the arrows. Here z can be integrated, and if the origin of z be 

 taken at the point on the boundary where w k, the result is 



z = 



-<? log {w+(u? -< 



<? lo 



], . . . (26) 



whence, for points on the curved part of the boundary, 



x = 



_w (C 8 - 



) 1 '' -c" cos- 1 (tc/c) +c 2 cos- 1 (A/c)]. 



The co-ordinates of the extremities of the curve are readily deduced, and it is seen 

 that the distance between the straight parts of the boundary is JBcV. 



If v be the velocity of the liquid at the point z = 0, V,,- 1 is the value of dz/dw | 

 for w = k. Hence v^ 1 = B(c 2 -F) 1/2 , and the distance between the straight parts of 

 the boundary may be expressed in the form Jirc ! V 1 (c 8 -*')- 1/1 . In checking the 

 dimensions of this expression it should be remembered that c and k are both of the 

 dimensions of a length multiplied by a velocity. 



* See, however, 42, infra. 

 3 o 2 



