1 64 Sir William Thomson on the 



liquid's revolution, is 



-n=i©mVAog-^+-1159\ . . . (43) 



This is very small in comparison with 



2o>+^v(log~+-1159), . . . (44) 



the angular velocity of the direct wave ; and therefore clearly, 

 if the initial normal velocity of the surface when left free after 

 being displaced from its cylindrical figure of equilibrium be 

 zero or any thing small, the amplitude of the quicker direct 

 wave will be very small in proportion to that of the reverse 

 slow one. 



Case III. 



A slightly disturbed vortex column in liquid extending 

 through all space between two parallel planes ; the undisturbed 

 column consisting of a core of uniform vorticity (that is to 

 say, rotating like a solid), surrounded by irrotationally revol- 

 ving liquid with no slip at the cylindric interface. Denoting 

 by a the radius of this cylinder, we have 



T = cor when r < a, -\ 

 and n , (..... (45) 



v > a. j 



T=a>- „ r 



T 



Hence (13), (14) hold for r<a, and (23), (24) for r>a. 



Going back to the form of assumption (6), we see that it 

 suits the condition of rigid boundary planes if Oz be perpen- 

 dicular to them, in one of them, and the distance between 

 them 7r/m. 



The conditions to be fulfilled at the interface between core 

 and surrounding liquid are that p and w must have the same 

 values on the two sides of it : it is easily proved that this 

 implies also equal values of t on the two sides. The equality 

 of p on the two sides of the interface gives, by (13) and (23), 



I 4cw 2 — (ico— n) 2 I \dr/ r=a v f 



** -* r=a 



and from this and the equality of w on the two sides we have 

 ,. vfV ./rZw\ internal , 2z'ft)-| 



V J l y \wdr/ r=a _aj /dw\«ternal 



W-(ico-nf - \^dr) w ' (47 J 



