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



where < a denotes the value of < for 6 = a, the value of d<f>/dn 

 being zero over the circular part of the boundary. 



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

 We then have 



and therefore 



p _*>a 4 y 1 ( 1 2 1 ) 



Jo * a ~"^r Z 2n+ 3 (2^1 ~ 2rc + l + 2rc + 3) 



87* 

 Hence* 



x l7r/!)a) 2 a 2 ...... (13). 



This is less than if the fluid were solidified, in the ratio of '6212 

 to 1. See Art. 45. 



4. With the same notation of elliptic coordinates as in 

 Art. 71, 3, let us assume 



(f)+i^ = Cie- 2 ^ +i ^ ..................... (14). 



Since x 2 + y* = Jc 2 (cosh 2f + cos 2?;), 



the equation (1) becomes 



Ce~^ cos 2?7 |o)C 2 (cosh 2f + cos 2ij) = const. 



This system of curves includes the ellipse whose parameter 

 is > provided 



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



= Jo)(a + 6) 2 , 

 so that ^r Jo> (a + 6) 2 e~^ cos 2?;, 



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

 order 1/V 5 . 



Greenhill, I c. 



