80 MOTION OF A LIQUID IN TWO DIMENSIONS. [CHAP. IV. 



The branch-points of a function w are also branch-points of 

 -y- , and vice versa, as is easily seen from the meaning of the latter 

 function. 



Hence if an assumption of the form (26) or (27) be hydro- 

 dynamically intelligible, the portion of the plane xy occupied 

 by the fluid must not include any branch-points ; otherwise the 

 component velocities would be ambiguous at every point of the 

 fluid. Branch-points may however occur on the boundary of this 

 portion, in which case circulation round them is impossible. 



80. We can now proceed to some applications of the foregoing 

 theory. 



Example 1. Assume 



w = Az n , 



and suppose A to be real*. Introducing polar co-ordinates r, 0, 

 we have 



&amp;lt;/&amp;gt; = Ar n cos n&, 



\lr = Ar n sin n0. 



(a) If n = 1, the stream-lines are a system of straight lines 

 parallel to x, and the equipotential curves are a similar system 

 parallel to y. In this case any corresponding figures in the planes 

 of w and z are similar, whether they be finite or infinitesimal. 



(b) If n = 2, the curves &amp;lt;f&amp;gt; = const, are a system of rectangular 

 hyperbolas having the axes of co-ordinates as their principal axes, and 

 the curves ty = const, are a similar system having the co-ordinate 

 axes as asymptotes. The lines 6 = 0, 6 = ^TT are parts of the same 

 stream-line i/r = 0, so that we may take the positive parts of the 

 axes of x, y as fixed boundaries, and thus obtain the case of a fluid 

 in motion in the angle between two perpendicular walls. 



(c) If n = 1, we get two systems of circles touching the 

 axes of co-ordinates at the origin. Since now &amp;lt; = - , the 



* If A be complex, the curves $ = const., \p = const, are not altered in form but 

 only in position, being turned round the origin through an angle arc tan /?, if 



A = a. + ? /5. 



