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 the fluid. The fluid round the vortex is always in motion, and one produces a definite movement in another. Lord Kelvin has proposed the hypothesis that portions of a fluid segregated in vortices account for the origin of matter. The properties of the ether in respect of its capacity of propagating disturbances can be explained by the assumption of vortices in it instead of by a property of rigidity. It is difficult to conceive, however, of any arrangement of the vortex rings and endless vortex filaments in the ether. Now, the further consideration of four-dimensional rotations shows the existence of a kind of vortex which would make an ether filled with a homogeneous vortex motion easily thinkable. To understand the nature of this vortex, we must go on and take a step by which we accept the full significance of the four-dimensional hypothesis. Granted four-dimensional axes, we have seen that a rotation of one into another leaves two unaltered, and these two form the axial plane about which the rotation takes place. But what about these two? Do they necessarily remain motionless? There is nothing to prevent a rotation of these two, one into the other, taking place concurrently with the first rotation. This possibility of a double rotation deserves the most careful attention, for it is the kind of movement which is distinctly typical of four dimensions. Rotation round a plane is analogous to rotation round an axis. But in three-dimensional space there is no motion analogous to the double rotation, in which, while axis 1 changes into axis 2, axis 3 changes into axis 4. Consider a four-dimensional body, with four independent axes, _x_, _y_, _z_, _w_. A point in it can move in only one direction at a given moment. If the body has a velocity of rotation by which the _x_ axis changes into the _y_ axis and all parallel sections move in a similar manner, then the point will describe a circle. If, now, in addition to the rotation by which the _x_ axis changes into the _y_ axis the body has a rotation by which the _z_ axis turns into the _w_ axis, the point in question will have a double motion in consequence of the two turnings. The motions will compound, and the point will describe a circle, but not the same circle which it would describe in virtue of either rotation separately.