96 Proceedings of the Royal Society 
its own peculiarity of multiple continuity, it being impossible for 
the matter in any line of vortex motion to go through the line of 
any other matter in such motion or any other part of its own line. 
In fact, a closed line of vortex core is literally indivisible by any 
action resulting from vortex motion. 
The author called attention to a very important property of the 
vortex atom, with reference to the now celebrated spectrum analysis 
practically established by the discoveries and labours of Kirchof 
and Bunsen. The dynamical theory of this subject, which Pro- 
fessor Stokes had taught to the author of the present paper before 
September 1852, and which he has taught in his lectures in the 
University of Glasgow from that time forward, required that the 
ultimate constitution of simple bodies should have one or more 
fundamental periods of vibration, as has a stringed instrument of 
one or more strings, or an elastic solid consisting of one or more 
tuning forks rigidly connected. To assume such a property in the 
Lucretius atom, is at once to give it that very flexibility and elas- 
ticity, for the explanation of which, as exhibited in aggregate 
bodies, the atomic constitution was originally assumed. If, then, 
the hypothesis of atoms and vacuum imagined by Lucretius and 
his followers to be necessary to account for the flexibility and com- 
pressibility of tangible solids and fluids, were really necessary, it 
would be necessary that the molecule of sodium, for instance, 
should be not an atom, but a group of atoms with void space be- 
tween them. Such a molecule could not be strong and durable, 
and thus it loses the one recommendation, which has given it the 
degree of acceptance it has had among philosophers ; but, as the 
experiments shown to the Society illustrate, the vortex atom has 
perfectly definite fundamental modes of vibration, depending solely 
on that motion, the existence of which constitutes it. The dis- 
covery of these fundamental modes forms an intensely interesting 
problem of pure mathematics. Even for a simple Helmholtz ring, 
the analytical difficulties which it presents are of a very formid- 
able character, but certainly far from insuperable in the present 
state of mathematical science. The author of the present com- 
munication had not attempted, hitherto, to work it out except for 
an infinitely long, straight, cylindrical vortex. For this case he is 
working out solutions corresponding to every possible description of 
