95, 96.] 



VENA CONTRACTA. 



103 



96. Example 10. This is an illustration of the theory of the 

 vena contracta. Fluid escapes from a large vessel through an 

 aperture in a plane wall. The forms of the boundaries in the 

 planes of z, f, w are shewn in Fig. 7; fixed boundaries being de 

 noted by heavy, free surfaces by fine lines. The figure in the 

 plane of f is a lune of angle J TT having its angular points at f= + 1. 

 The figure in the plane of w has the limiting form just noticed. 



Fig. 7. 



We assume for simplicity that the parameters of the limiting 

 stream-lines are -$&amp;gt; = 0, ^r = TT. Applying then the rule developed 

 in the last article, we assume 



,(57). 



The arbitrary constants must satisfy the conditions 



(a) when &amp;lt; = cc , = x , 



(b) when $ = -t oo , ?= i; 



and if we further take the equipotential surface passing through 

 the edges of the aperture as that for which &amp;lt;/&amp;gt; = 0, we must have 



(c) w = when f= 1. 



Of these conditions (a) gives B = D, (U) gives A = - C, and 

 (c) gives A = - B, so that (57) becomes 



.(58), 



