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 &>. The arrangement 

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



The corresponding formula for <f> is 



a 2 -6 2 

 <b w . ,- . xy ........................ (4). 



a 2 + 6 2 9 



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

 is given by 



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

 one rigid mass, in the ratio of 



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

 theorem, proved in Art. 45. 



2. Let us assume 



^ = Ar* cos 30 = A(a?- 30y 2 ). 

 The equation (1) of the boundary then becomes 



^(^-3^ 2 )-ia>(tf 2 + 2/ 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 - |o>a 2 = C, 3Aa + \w = 0. 

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



a? -a 3 - 3#?/ 2 + 3a (x~ - a 2 + y*) = 0. 

 Dividing out by x a, we get 



x z + 4>ax + 4a 2 = 3y 8 , 

 or x + 2a = + V-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. 



