Hinton - The Fourth Dimension.pdf

218
THE FOURTH DIMENSION
all, as the fourth dimension is at right angles to all the
sections which can be made of the sphere.
We have supposed the matter of which the sphere is
composed to be three-dimensional. If the matter had a
small thickness in the fourth dimension, there would be
a slight thickness in fig. 12 above the plane of the paper
—a thickness equal to the thickness of the matter in the
fourth dimension. The rods would have to be replaced
by thin slabs. But this would make no difference as to
the possibility of the rotation. This motion is discussed
by Newcomb in the first volume of the American Journal
of Mathematics.
Let us now consider, not a merely extensible body, but
a liquid one. A mass of rotating liquid, a whirl, eddy, or
vortex, has many remarkable properties. On first
consideration we should expect the rotating mass of
liquid immediately to spread off and lose itself in the
surrounding liquid. The water flies off a wheel whirled
round, and we should expect the rotating liquid to be
dispersed. But see the eddies in a river strangely persistent.
The rings that occur in puffs of smoke and last
so long are whirls or vortices curved round so that their
opposite ends join together. A cyclone will travel over
great distances.
Helmholtz was the first to investigate the properties of
vortices. He studied them as they would occur in a perfect
fluid—that is, one without friction of one moving portion
or another. In such a medium vortices would be indestructible.
They would go on for ever, altering their
shape, but consisting always of the same portion of the
fluid. But a straight vortex could not exist surrounded
entirely by the fluid. The ends of a vortex must reach to
some boundary inside or outside the fluid.
A vortex which is bent round so that its opposite ends
join is capable of existing, but no vortex has a free end in