76-77] 



VENA CONTRACTA. 



107 



77. The next example is of importance in the theory of the 

 resistance of fluids. We suppose that a steady stream impinges 

 directly on a fixed plane lamina, behind which is a region of dead 

 water bounded on each side by a surface of discontinuity. 



The middle stream-line, after meeting the lamina at right 

 angles, branches off into two parts, which follow the lamina to the 

 edges, and thence the surfaces of discontinuity. Let this be the 

 line -\/r = 0, and let us further suppose that at the point of 

 divergence we have &amp;lt; = 0. The forms of the boundaries in the 

 planes of z t f, w are shewn in the figures. The region occupied 



i 



by the moving fluid corresponds to the whole of the plane of 

 w, which must be regarded however as bounded internally by 

 the two sides of the line ^ = 0, (f&amp;gt; &amp;lt; 0. 



As in Art. 76, the transformations 

 z l = log f, 

 z = cosh # 



give us as boundaries the segments of the axis y. 2 = made by the 

 points a? 2 = + 1. The further assumption 



*-- (2), 



converts these into segments of the negative portion of the axis 

 2/3 = 0, taken twice. The boundaries now correspond to those of 

 the plane w, except that to w = Q corresponds 3 = oo, and con 

 versely. The transformation is therefore completed by putting 



w = zr l (3). 



Hence, finally, f= ( - I) + ( - I - 1)* (4). 



