34 
a number of monocyclic circuits for each of which the 
circulation in the circuit, or the cyclic constant, is equal to 
the strength of the enclosed vortex tube upon any surface 
drawn bounded by the circuit. We have therefore shown 
that cyclic motion requires the space in which it exists to 
be made multiply continuous either by solid boundaries or 
by vortex filaments. 
Cyclic motion is therefore a condition of space, and vortex 
motion is a result of it. Cyclic motion can be produced by 
suitable impulses applied as shown by Thomson (Yortex 
Motion, Trans. Edin., R.S.), and vortex motion in a perfect 
fluid can only appear as a consequence, either of cyclic or 
discontinuous motions. 
Since the cyclic constant can not be a function of the 
time, we also infer 
But from this we may not conclude that the vortex motion 
is steady in the sense that it is always the same at the same 
point in space. For if the circuit retained a fixed position 
in space, the portion of fluid moving rotationally might 
move out of it. The true conclusion is that while the core 
of vortex motion remains within any circuit the strength of 
the core on any shell experiences no variation with the 
time, and as the circuit may at any time be replaced by a 
reconcileable circuit, this theorem may be extended to all 
that a circuit which embraces a core of vortex motion 
moves so as to continue to embrace the same core the same 
number of times. As each circuit may be diminished by 
the substitution of reconcileable paths, we may apply this 
to the simplest circuits embracing a core. We thus arrive 
at the known properties of vortex motion accompanying 
cyclic motion. 
space. 
Again, since 
D,/( 
= 0, we may conclude 
