156-158] STABILITY OF A CYLINDRICAL VORTEX. 251 



velocity, d\lr/rdd, must be continuous at the boundary of the vortex, for which 

 r = a, approximately. Assuming for the equation to this boundary 



r=a + acos(sd o-) .............................. (iv), 



we have still to express that the tangential component (d^/dr) of the velocity 

 is continuous. This gives 



r + s cos (s6 - <rt] = f -- s cos (s6 - a-t). 

 a T a 



Substituting from (iv), and neglecting the square of a, we find 



Ca=-sA/a .................................... (v). 



So far the work is purely kinematical; the dynamical theorem that the 

 vortex-lines move with the fluid shews that the normal velocity of a 

 particle on the boundary must be equal to that of the boundary itself. 

 This condition gives 



dr d\jr dty dr 



~dt~ ~ rdd ~ ~dr Vd6 ' 



where r has the value (iv), or 



A sa ... 



o-a = * - + ................................. (vi). 



Eliminating the ratio A /a between (v) and (vi) we find 



<r = (-l)f .................................... (vii). 



Hence the disturbance represented by the plane harmonic in (iii) consists 

 of a system of corrugations travelling round the circumference of the vortex 

 with an angular velocity 



a-/s=(s- !)/. ................................. (viii). 



This is the angular velocity in space; relative to the previously rotating 

 fluid the angular velocity is 



the direction being opposite to that of the rotation. 



When s = 2, the disturbed section is an ellipse which rotates about its 

 centre with angular velocity ^. 



The transverse and longitudinal oscillations of an isolated rectilinear 

 vortex- filament have also been discussed by Lord Kelvin in the paper cited. 



158. The particular case of an elliptic disturbance can be 

 solved without approximation as follows*. 



Let us suppose that the space within the ellipse 



* Kirchhoff, Mechanik, c. xx., p. 261; Basset, Hydrodynamics, Cambridge, 

 1888, t. ii., p. 41. 



