SMALL VIBRATIONS OF A HOLLOW VORTEX. 
189 
The effect of the correction to the form of the hollow is to make the section slightly 
elliptic with the major axis perpendicular to the plane of the ring, and with the inner 
side slightly flatter than the other. 
The value of x obtained above is, when k is infinitely small, larger than the critical 
value given in (18). The fluid carried forward will therefore be ring-shaped. If for a 
rough approximation we take the two first terms of the expressions, the value of k for 
a hollow vortex in steady motion, and carrying forward a simply connected mass of 
fluid, will be found from 
T 47T+ 1 
2 
or 
4 TT+ 1 
k= 4e“' — 4e -6 ' 78 
= '00522 very nearly 
this would make R/r=10 2 , about. Since k is so small our approximations are very 
close, and it follows that for even extremely small cores of hollow vortices, the fluid 
carried forward is a mass without aperture. For infinitely small ones this is not the 
case. 
The form of the hollow has been determined above by the value of k. If the normal 
variation from the circle be denoted by Sn 
bn =-—— S k= — —(12L— 15)F cos 2v 
du dk — c' ' 
= -tttr ^ 12 lo ^ =- 15 j cos 2v 
The figure below represents the form given by this expression to a circle in which 
r[R= '2. Though this value is large, it shows the nature of the change of form better 
than a smaller value. 
