10G MOTION OF A LIQUID IN TWO DIMENSIONS. [CHAP. IV. 



Along the free boundary ty = 0, &amp;lt; &amp;gt; 0, we have in the same 

 way as before 



?-r*-li 



ds 

 or x= 1 s e~ 2 *, 



if the origin of x and of s be at the extremity of the fixed wall. 

 Also 



ds 

 whence 



?/ = - J TT 4- e- l - e 2 * -f arc sin e~*, 



the constant of integration being so chosen as to make y for 

 s = 0. When s = oo , ^ = JTT, so that, the final breadth of the 

 stream being as before equal to TT, the breadth of the canal is 2?r. 

 The coefficient of contraction is therefore \. 



This example, the first of its class which was solved, is due to 

 Helmholtz (lc. Art. 92). 



IT 



If in (58) we write f a for f we obtain the solution of the case 

 where the inclination a of the walls of the canal has any value 

 whatever. 



98. Example 12. A steady stream impinges directly on a 

 fixed plane lamina. The region of dead water behind the lamina 

 is bounded on each side by a surface of discontinuity at which q is 

 (for the moving fluid) constant (say =1). 



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 for which -v/r = 0, and let us further assume that at the 

 point of divergence we have &amp;lt;/&amp;gt; = 0. The forms of the boundaries 

 in the planes of z, f, w are shewn in Fig. 9. The region occupied 

 by moving fluid corresponds to the whole of the plane of w; but 

 the two sides of the straight line -fy = 0, &amp;lt;f&amp;gt; &amp;gt; are internal bound 

 aries. The assumption w = *Jw transforms this double line into 



