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



are sufficiently illustrated by the figures. We thus get 



?+?- l = 2e w , 

 or f = e w + (e^-l)* ..................... (2). 



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

 we have ty = 0, (/> < 0, whence 



-(l-e-^ ............ (3), 



x = 1 - *-, y = (I - e~^ - i iogi- ...... (4)*, 



the origin being taken at the point A. If we put dx/ds = cos 0, 

 these may be written 



x = 2 sin 2 0, y = sin 6 - log tan ( J TT -f 1 0) ...... (5). 



Line of ' Symmetry. 



When s = oc , we have x 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 



7r(ir + 2), ='611. 



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

 Rayleigh, " Notes on Hydrodynamics," Phil. May., December 1876. 



