The Vibrations in the Lucretius atom

Two ring atoms linked together or one knotted in any manner with its ends meeting, constitute a system which, however it may be altered in shape, can never deviate from its own peculiarity of multiple continuity.

It makes 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 vortex atom has a very important property based on the spectrum-analysis established by Kirchhoff and Bunsen.

Professor Stokes taught this dynamical theory to me before September 1852. Since then, he has taught it in his lectures in the University of Glasgow.

It required the ultimate constitution of simple bodies to have one or more fundamental periods of vibration. This is similar to a stringed instrument, or an elastic solid consisting of tuning-forks rigidly connected.

This is an assumed property in the Lucretius atom.

It gives it that very flexibility and elasticity for atomic constitution in aggregate bodies.

Lucretius and his followers created a hypothesis of atoms and vacuum that accounts for the flexibility and compressibility of tangible solids and fluids.

If this were correct, then a sodium molecule, for instance, should not be an atom. Instead, it would a group of atoms with void space between them.

Such a molecule would not be strong and durable. It would lose 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 discovery 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 formidable character, but certainly far from insuperable in the present state of mathematical science.

I had not attempted, hitherto, to work it out except for an infinitely long, straight, cylindrical vortex.

For this case, I was working out solutions corresponding to every possible description of infinitesimal vibration, and intended to include them in a mathematical paper which he hoped soon to be able to communicate to the Royal Society.

One very simple result is the following.

Let such a vortex be given with its section differing from exact circular figure by an infinitesimal harmonic deviation of order i.

This form will travel as waves round the axis of the cylinder in the same direction as the vortex rotation, with an angular velocity equal to (i-1)/i of the angular velocity of this rotation. Hence, as the number of crests in a whole circumference is equal to i, for an harmonic deviation of order i there are i-1 periods of vibration in the period of revolution of the vortex.

For the case i=1 there is no vibration, and the solution expresses merely an infinitesimally displaced vortex with its circular form unchanged.

The case i=2 corresponds to elliptic deformation of the circular section; and for it the period of vibration is, therefore, simply the period of revolution.

These results are, of course, applicable to the Helmholtz ring when the diameter of the approximately circular section is small in comparison with the diameter of the ring, as it is in the smoke-rings exhibited to the Society.

The lowest fundamental modes of the two kinds of transverse vibrations of a ring, such as the vibrations that were seen in the experiments, must be much graver than the elliptic vibration of the section.

It is probable that the vibrations which constitute the incandescence of sodium-vapour are analogous to those which the smoke-rings had exhibited.

It is therefore probable that the period of each vortex rotation of the atoms of sodium-vapour is much less than 1/525 of the millionth of the millionth of a second, this being approximately the period of vibration of the yellow sodium light.

This light consists of 2 sets of vibrations coexistent in slightly different periods. These are equal approximately to the time just stated, and of as nearly as can be perceived equal intensities.

Thus, the sodium atom must have 2 fundamental modes of vibration for their respective periods. These vibrations are equally excitable by such forces as the atom experiences in the incandescent vapour.

This last condition renders it probable that the 2 fundamental modes are approximately similar. They are not merely different orders of different series chancing to concur very nearly in their periods of vibration.

In an approximately circular and uniform disk of elastic solid, the fundamental modes of transverse vibration, with nodal division into quadrants, fulfil both the conditions.

In an approximately circular and uniform ring of elastic solid these conditions are fulfilled for the flexural vibrations in its plane, and also in its transverse vibrations perpendicular to its own plane.

But the circular vortex ring, if created with one part somewhat thicker than another, would not remain so. Instead, it would experience longitudinal vibrations around its own circumference. It cannot have 2 fundamental modes of vibration similar in character and approximately equal in period.

This assertion [1] can be practically extended to any atom consisting of a single vortex ring, however involved.

This is illustrated by those of the models shown to the Society which consisted of only a single wire knotted in various ways.

It is therefore probable that the sodium atom may not consist of a single vortex line. Instead, it might consist of 2 approximately equal vortex rings passing through one another like two links in a chain.

It is, however, quite certain that a vapour consisting of such atoms, with proper volumes and angular velocities in the to rings of each atom, would act precisely as incandescent sodium-vapour acts—that is to say, would fulfil the “spectrum test” for sodium.

The possible effect of change of temperature on the fundamental modes cannot be pronounced upon without mathematical investigation not hitherto executed.

Therefore, we cannot say that the dynamical explanation now suggested is mathematically demonstrated so far as to include the very approximate identity of the periods of the vibrating particles of the incandescent vapour with those of their corresponding fundamental modes at the lower temperature at which the vapour exhibits its remarkable absorbing-power for the sodium light.

Cross section illustration of vortex around a ring