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 ty = 0, and let us further suppose that at the point of 

 divergence we have = 0. The forms of the boundaries in the 

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



z f 



w 



A' 



A 



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 ^r = 0, </> < 0. 



As in Art. 76, the transformations 



*' =log ,f } ax 



z. 2 cosh Zi ) 



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

 points # 2 = + 1. The further assumption 



* * (2), 



converts these into segments of the negative portion of the axis 

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

 the plane w, except that to w = corresponds 5 3 = oo , and con- 

 versely. The transformation is therefore completed by putting 



w=*r l (3). 



Hence, finally, t = ( - -Y + ( - - - ^ 



\ wJ \ w 



