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



(a) If we assume 



i/r = Ar* cos 20 = A (x 2 - y 2 ), 

 the equation (46) then becomes 



which, for any given value of A represents a system of similar and 

 coaxial conic sections. That this system may include the ellipse 



we must have 







Hence ^ = - Jo, . -^ (x* - y 2 ) 



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 corresponding formula for &amp;lt; is 



The angular momentum of the fluid, per unit length of the 

 cylinder, about the axis of rotation, is 



Hence the cylinder rotates under the action of any external forces 

 exactly as if the fluid were replaced by a solid whose moment of 

 inertia about the axis of rotation is 



per unit length, M being the mass per unit length of the fluid. 

 (6) Let us assume 



v/r = Ar* cos 30 = A(x*- 3xy*). 

 The equation of the boundary (46) then becomes 



A (x* - Z.r/) + (r 2 + y*) = C ............. (47). 



