192 
HINTON. 
enough for the particles at E and C to be separated as they 
are at F and D, they can rotate round to the position D and 
F, and a similar motion is pos¬ 
sible for all other particles. 
There is no matter or obsta¬ 
cle to prevent them from 
moving out in the w direc¬ 
tion, and then on round the 
circumference as an axis. 
Now, what will hold for one 
section will hold for all, as 
the fourth dimension is at 
right angles to all the sec¬ 
tions which can be made of 
the sphere. 
We have supposed the matter of which the sphere is com¬ 
posed to be three dimensional. If the matter had a small 
thickness in the fourth dimension, there would be a slight 
thickness in Fig. 12 above the plane of the paper—a thick¬ 
ness equal to the thickness of the matter in the fourth di¬ 
mension. The rods would have to be replaced by thin slabs. 
But this would make no difference as to the possibility of 
the rotation. This motion is discussed by Newcomb in the 
first volume of the American Journal of Mathematics. 
Let us now consider, not a merely extensible body, but a 
liquid one. A mass of rotating liquid, a whirl, eddy, or vor¬ 
tex, has many remarkable properties. On first consideration 
we should expect a rotating mass of liquid immediately to 
spread off arid lose itself in the surrounding liquid. The 
water flies off a wheel whirled round, and we should expect 
the rotating liquid to be dispersed. But we see the eddies 
in a river strangely persistent. The rings that occur in puffs 
of smoke and last so long are whirls or vortices curved round 
so that their opposite ends join together. A cyclone will 
travel over great distances. 
Helmholtz was the first to investigate the properties of 
vortices. He studied them as they would occur in a perfect 
