THE RECOGNITION OF THE FOURTH DIMENSION. 197 
If matter in its small particles is four dimensional we 
should expect this double rotation to be a universal charac¬ 
teristic of the atoms and molecules, for no portion of matter 
is at rest. The consequences of this corpuscular motion can 
be perceived, but only under the form of ordinary rotation 
or displacement. Thus if the theory of four dimensions is 
true we have in the corpuscles of matter a whole world of 
movement which we can never study directly, but only by 
means of inference. 
The rotation A, as I have defined it, consists of two equal 
rotations—one about the plane of z w, the other about the 
plane of x y. It is evident that these rotations are not nec¬ 
essarily equal. A body may be moving with a double rota¬ 
tion 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 rota¬ 
tion about a plane; for, the first being x to y and 2 to w, 
and the second being x to y and w to 2 , when they are 
put together the z to w and «i to 2 rotations neutralize 
each other, and we obtain an x to y rotation only, which 
is a rotation about the plane of z w. Similarly, if we 
take a B rotation, y to x and z to w, we get, on cofnbining 
this with the A rotation, a rotation of z to w about the x y 
plane. In this case the plane of rotation is in the three- 
dimensional space of x y z , 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 proven 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 sec¬ 
tion. 
