Now, a very remarkable fact about such a vortex is that the ends of the vortex cannot remain suspended and isolated in the fluid. They must always run to the boundary of the fluid. An eddy in water that remains half way down without coming to the top is impossible.
The ends of a vortex must reach the boundary of a fluid—the boundary may be external or internal—a vortex may exist between two objects in the fluid, terminating one end on each object, the objects being internal boundaries of the fluid. Again, a vortex may have its ends linked together, so that it forms a ring. Circular vortex rings of this description are often seen in puffs of smoke, and that the smoke travels on in the ring is a proof that the vortex always consists of the same particles of air.
Let us now enquire what a vortex would be in a four-dimensional fluid.
We must replace the line axis by a plane axis. We should have therefore a portion of fluid rotating round a plane.
We have seen that the contour of this plane corresponds with the ends of the axis line. Hence such a four-dimensional vortex must have its rim on a boundary of the fluid. There would be a region of vorticity with a contour. If such a rotation were started at one part of a circular boundary, its edges would run round the boundary in both directions till the whole interior region was filled with the vortex sheet.
A vortex in a three-dimensional liquid may consist of a number of vortex filaments lying together producing a tube, or rod of vorticity.
In the same way we can have in four dimensions a number of vortex sheets alongside each other, each of which can be thought of as a bowl-shaped portion of a spherical shell turning inside out. The rotation takes place at any point not in the space occupied by the shell, but from that space to the fourth dimension and round back again.
Is there anything analogous to this within the range of our observation?
An electric current answers this description in every respect. Electricity does not flow through a wire. Its effect travels both ways from the starting point along the wire. The spark which shows its passing midway in its circuit is later than that which occurs at points near its starting point on either side of it.
Moreover, it is known that the action of the current is not in the wire. It is in the region enclosed by the wire, this is the field of force, the locus of the exhibition of the effects of the current.