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



gives the motion of a liquid contained within a hollow elliptic 

 cylinder whose semi-axes are a, b, produced by the rotation of the 

 cylinder about its axis with angular velocity o&amp;gt;. The arrangement 

 of the stream-lines ty = const, is given in the figure on p. 99. 



The corresponding formula for (f&amp;gt; is 



The kinetic energy of the fluid, per unit length of the cylinder, 

 is given by 



( 



dy) j -dxdy = 1 -^- x ^6 . . .(5). 



This is less than if the fluid were to rotate with the boundary, as 

 one rigid mass, in the ratio of 



/a 2 -fry 



U+&v 



to unity. We have here an illustration of Lord Kelvin s minimum 

 theorem, proved in Art. 45. 



2. Let us assume 



i/r = Ar 3 cos 30 = A (x 3 3xy-\ 

 The equation (1) of the boundary then becomes 



A (x 3 3#2/ 2 ) \w (a? + 7/ 2 ) = C (6). 



We may choose the constants so that the straight line x = a shall 

 form part of the boundary. The conditions for this are 



Aa 3 - i&amp;lt;wft 2 = C, 3Aa + \G&amp;gt; = 0. 

 Substituting the values of A, C hence derived in (6), we have 



x 3 a 3 Sxy~ -\- 3a (a? a~ + 2/ 2 ) = 0. 

 Dividing out by x a, we get 



or x + 2a = + \/3 . y. 



The rest of the boundary consists therefore of two straight lines 

 passing through the point ( 2a, 0), and inclined at angles of 30 

 to the axis of x. 



