] 06 MOTION OF A LIQUID IN TWO DIMENSIONS. [CHAP. IV 



are sufficiently illustrated by the figures. We thus get 



or ? = e w + (e aw -l)* (2). 



For the free stream-line starting from the edge A of the aperture 

 we have t|r 0, &amp;lt;/&amp;gt; &amp;lt; 0, whence 



dxjds = e~-\ dyfds = (1 e~* M ) 

 or x=l-e~ s , y = (l-e- M )*-i 1 gi ?r 



the origin being taken at the point A. If we put dx/ds = cos 6, 

 these may be written 



x = 2 sin- ^0, y = sin 6 log tan ( J TT -f i 0) (o). 



Line of Symmetry. 



When 5 = x , we have & = 1 ; and therefore, since on our scale 

 the final breadth of the stream is TT, the total width of the aperture 

 is represented by TT+ 2; i.e. the coefficient of contraction is 



7T/(7T + 2), = 611. 



* This example was given by Kirchhoff (I.e. ], and discussed more fully by Lord 

 Rayleigh, &quot; Notes on Hydrodynamics,&quot; Phil. May., December 187G. 



