If we suppose the spherical shell to be of four-dimensional matter, our representation will be a little different. Let us suppose there to be a small thickness to the matter in the fourth dimension. This would make no difference in fig. 44, for that merely shows the view in the _xyz_ space. But when the _x_ axis is let drop, and the _w_ axis comes in, then the rods CD and EF which represent the matter of the shell, will have a certain thickness perpendicular to the plane of the paper on which they are drawn. If they have a thickness in the fourth dimension they will show this thickness when looked at from the direction of the _w_ axis. Supposing these rods, then, to be small slabs strung on the circumference of the circle in fig. 45, we see that there will not be in this case either any obstacle to their turning round the circumference. We can have a shell of extensible material or of fluid material turning inside out in four dimensions. And we must remember that in four dimensions there is no such thing as rotation round an axis. If we want to investigate the motion of fluids in four dimensions we must take a movement about an axis in our space, and find the corresponding movement about a plane in four space. Now, of all the movements which take place in fluids, the most important from a physical point of view is vortex motion. A vortex is a whirl or eddy—it is shown in the gyrating wreaths of dust seen on a summer day; it is exhibited on a larger scale in the destructive march of a cyclone. A wheel whirling round will throw off the water on it. But when this circling motion takes place in a liquid itself it is strangely persistent. There is, of course, a certain cohesion between the particles of water by which they mutually impede their motions. But in a liquid devoid of friction, such that every particle is free from lateral cohesion on its path of motion, it can be shown that a vortex or eddy separates from the mass of the fluid a certain portion, which always remain in that vortex. The shape of the vortex may alter, but it always consists of the same particles of the fluid. 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.