104 MOTION OF A LIQUID IN TWO DIMENSIONS. [CHAP. IV. 



whence we find 



dw 



Along the stream-lines ty = 0, i/r = TT, f is real so long as $ lies 

 between and oo . To find the form of the free boundaries, let 

 us consider the portion of the stream-line -^ = for which &amp;lt;/ 0. 

 The first term of (59) is then real, the second imaginary, so that if 

 we write 



=i(cos04 ~ 

 q (C 



we have, along the line in question, 



e-* ........................... (60). 



If s denote the arc of this line, measured from the edge of the 

 aperture, we have 



whence s= &amp;lt;f&amp;gt;. The equation (60) then gives 



dx 

 T - 

 ds 



and therefore 



x=l+e- ........................... (61), 



if the origin of z (hitherto arbitrary) be taken at the edge of the 

 aperture. The final width of the stream is given by the difference 

 of the extreme values of ^ (the velocity being unity), i.e. it is 

 equal to TT; and since when s = oo , x = 1, the abscissa of the 

 centre of the stream is 1 + J TT. The width of the aperture is there 



fore 2 + TT, and the coefficient of contraction is - ~ , or 611. 



7T + 2t 



Again, from (58) or (60) we find 



whence 



/I * 11 l + N/ 1 -* 2 * 



^^1-^-1^== 



