Thomson — Stahility and Oscillation of a Perfect Liquid. 341 
the cube of the distance from the circular axis of the tore. The equa- 
tion of the curve is easily written down. It is calculated for the case 
in which the velocity at the centre of the tore (were there no vacuity) 
due to circulation through the tore, is equal to ^8/3 of the velocity 
at the boundary of the vacuous column at great enough distances on 
either side to be undisturbed by the circulation through the tore. This 
makes the maximum diameter of the vacuous core three times the 
undisturbed diameter. If the velocity- component, due to the dis- 
turbance, is small in comparison with the surface-velocity of the 
vortex column, the swelling will, of course, be but a small fraction of 
the radius of the undisturbed column. Try to get a corresponding 
problem of steady motion with rotational core, and you will see why I 
now abjure rotational motion, and definitively adopt vacuum for all 
cores. 
JS'ow, consider a uniform distribution throughout space, of vacuous 
vortex columns ; represented by section perpendicular to the length in 
rig. 2, PI. xix. ; red and blue, each representing vacuum, but with oppo- 
site circulations around them. But, instead of the proportions of the dia- 
gram, let the distance from each column to its three nearest neighbours 
be enormously great in comparison with the diameters of the columns, 
I think I can now prove that this arrangement of vortices is stable ; 
but its quasi-rigidity relatively to two-dimensional motion, without 
change of volume of the cores, is exceedingly small, and the corre- 
sponding laminar wave-motion exceedingly sluggish in comparison 
with the tensile quasi-rigidity, and corresponding wave-velocity 
which we should find by considering laminar motion in planes parallel 
to the plane of the diagram. 
Now, imagine a very great number of planes in all directions — as 
many within an angle of 1° of any one plane, as within 1° degree of 
any other. And let there be a distribution of straight vortex vacuous 
cores (as represented in Pig. 2), perpendicular to every one of these 
planes. The cores being thin enough, they may be placed along 
straight lines, no one of which intersects any other. The mutual 
influence of the vortices will produce disturbances from the straight 
lines in which we supposed them given, and slight swellings and 
deviations from exactly circular figure, in their cross sections ; and 
there will be sluggish motions of the cores, unless they are all placed 
so as to fulfil a definite condition giving steady motion. Even if this 
definite condition is not exactly fulfilled, the tensile quasi-rigidity, 
and corresponding velocity of laminar waves of the medium, thus 
kinetically constituted, will certainly differ but little from what they 
