162 
MR. W. M. HICKS OK THE STEADY MOTION AND 
pamphlet published at Halle in 1864, with the title ‘ Theorie der Elektricitats- und 
Warme-Vertheilung in einem Hinge.’ 
The theory of the motion of vortices is interesting, not only from the mathematical 
difficulties encountered in its treatment, but also from its connexion with Sir W. 
Thomson’s theory of the vortex atom constitution of matter. In an abstract of the 
present paper intended for the Proceedings of this Society, I have given some physical 
speculations which induced me to take up the question of the motion of a hollow 
vortex—that is, where cyclic motion exists in a fluid without the presence of any 
actual rotational filaments—in which case there must be a ring-shaped hollow in 
the fluid, however great the pressure may be, so long as it is finite. The essential 
quality of all vortex motion is the cyclic motion existing in the fluid outside the 
filament, and not the rotational motion of the filament itself. Whether the filament 
be present or not, it is often possible to get some general idea of the motion that 
ensues in many cases without recourse to actual calculation. Thus, for instance, the 
treatment by Sir W. Thomson of the action of two vortices on one another,'" and of 
the form of the axis of a ring, along which waves of displacement are running,! may 
be cited. The same course of general reasoning, which was applied in a paper on the 
steady motion of two cylinders in a fluid,J will also apply to illustrate the mechanism, 
so to speak, which causes a single vortex ring to move with a motion of translation. 
Thus suppose a single vortex ring, which is for a moment at rest. It is clear that the 
velocity of the fluid just inside the aperture is greater than outside, and therefore 
the pressure less inside than outside, whilst the pressure is the same at corresponding- 
points in the front and hinder portions. The consequence of this is that the ring 
begins to contract without a general motion of translation. But the effect of this 
contraction of aperture itself produces velocities in the surrounding fluid, which, com¬ 
bined with the cyclic motion, increase the velocities in front of the ring, and decrease 
them behind. The consequence of this is a difference of pressures, which urges the 
ring in the direction of the cyclic motion through the ring, and it begins to move 
forward with increasing velocity. After a time this translatory motion would increase 
so much as to make the velocity within the aperture approach to that without; the 
state of motion will therefore be one in which the translatory velocity tends continually 
to a limit. 
The present communication is divided into three sections. In the first, new 
functions are introduced to give the stream lines. These functions are connected 
with, and have analogous properties to, the Toroidal Functions; are, in fact, given by 
B = Sc/P/dA and T= — S dQ/du. They have the property of being single-valued, even 
when they represent cyclic motion—a motion wdiich the single-valued Toroidal 
Functions cannot by themselves represent. At the end of the section the values of 
* “ Yortex Motion,” Trans. Roy. Soc. Edin., xxv. 
f “Yortex Statics,” Proc. Roy. Soc. Edin., ix. 
t Quart. Jour. Math., xvii. 
