The rotation A, as I have defined it, consists of two equal rotations—one about the plane of _zw_, the other about the plane of _xy_. It is evident that these rotations are not necessarily equal. A body may be moving with a double rotation, in which these two independent components are not equal; but in such a case we can consider the body to be moving with a composite rotation—a rotation of the A or B kind and, in addition, a rotation about a plane. If we combine an A and a B movement, we obtain a rotation about a plane; for, the first being _x_ to _y_ and _z_ to _w_, and the second being _x_ to _y_ and _w_ to _z_, when they are put together the _z_ to _w_ and _w_ to _z_ rotations neutralise each other, and we obtain an _x_ to _y_ rotation only, which is a rotation about the plane of _zw_. Similarly, if we take a B rotation, _y_ to _x_ and _z_ to _w_, we get, on combining this with the A rotation, a rotation of _z_ to _w_ about the _xy_ plane. In this case the plane of rotation is in the three-dimensional space of _xyz_, and we have—what has been described before—a twisting about a plane in our space. Consider now a portion of a perfect liquid having an A motion. It can be proved that it possesses the properties of a vortex. It forms a permanent individuality—a separated-out portion of the liquid—accompanied by a motion of the surrounding liquid. It has properties analogous to those of a vortex filament. But it is not necessary for its existence that its ends should reach the boundary of the liquid. It is self-contained and, unless disturbed, is circular in every section. [Illustration: Fig. 15 (143).] If we suppose the ether to have its properties of transmitting vibration given it by such vortices, we must inquire how they lie together in four-dimensional space. Placing a circular disk on a plane and surrounding it by six others, we find that if the central one is given a motion of rotation, it imparts to the others a rotation which is antagonistic in every two adjacent ones. If A goes round, as shown by the arrow, B and C will be moving in opposite ways, and each tends to destroy the motion of the other. Now, if we suppose spheres to be arranged in a corresponding manner in three-dimensional space, they will be grouped in figures which are for three-dimensional space what hexagons are for plane space. If a number of spheres of soft clay be pressed together, so as to fill up the interstices, each will assume the form of a fourteen-sided figure called a tetrakaidecagon.