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



where &amp;lt; a denotes the value of &amp;lt; for = a, the value of cUf&amp;gt;/dn 

 being zero over the circular part of the boundary. 



The case of the semicircle a = JTT will be of use to us later. 

 We then have 



l 2 



2n-I 

 and therefore 



191 



-j USIA/ ^ JU _L ^ X 



) a T(lT = Z 7; ~fT ^rt T c* T^T &quot;T ^ 



_ 7T 2 



1 &quot; 



7T 



Hence* 



2 x iwpw a 2 ...... (13). 



This is less than if the fluid were solidified, in the ratio of &quot;6212 

 to 1. See Art. 45. 



4. With the same notation of elliptic coordinates as in 

 Art. 71, 3, let us assume 



(j&amp;gt;+i^ = Cie-^ +ir ..................... (14). 



Since # 2 + y z c 2 (cosh 2f + cos 2?;), 



the equation (1) becomes 



Ce~^ cos 2?; Jwc 2 (cosh 2f + cos 2??) = const. 



This system of curves includes the ellipse whose parameter 

 is , provided 



or, using the values of a, b already given, 



a = Jo,(a + 6) 2 , 



so that -^ = J&&amp;gt; (a + b)~ e~^ cos 2?;, &quot;| 



^ = J B (a + 6) a e- 2 *sin2i7. J 



At a great distance from the origin the velocity is of the 

 order 1/Y&quot;. 



* Greenhill, I c, 



