This paper was the beginning of, and the impetus for, the great deal of development in hydrodynamics that Kelvin referred to as "formidable" while Maxwell, in the review of the vortex atom for the 1875 Encyclopedia Britannica called the mathematical difficulties "enormous" though also following that "the glory of surmounting them would be unique."
On the apparent necessity of this work Hicks pointed out "The simple incompressible fluid necessary on the vortex atom theory is quite incapable of transmitting vibrations similar to those of light." (Hicks 1885)
(FitzGerald 1880) What is unclear is why it took roughly a decade for Kelvin to use this same solution of MacCullagh's aether to resolve his own problems with elasticity--shared by Maxwell's theory--which were the major barrier for the vortex atom theory.
The interesting crossover point of note is that George FitzGerald whose work was strongly focussed upon development of Kelvin's vortex atom theory, is well known for collaborating with Hendrik Lorentz on the hypothesis of physical contraction as an explanation for the Michelson-Morley experiment.
Hicks (1885) "On the Constitution of the Luminiferous Ether on the Vortex Atom Theory", p.
(2002), The Vortex Atom: A Victorian Theory of Everything.
In 1867 Lord Kelvin published "On Vortex Atoms" in which he begins by stating that when he first discovered Helmholtz's laws of vortex motion in inviscid fluid, it occurred to him that the ring vortices Helmholtz described must be the only true form of atoms.
The Cosserat work, however, is a mechanical and physical theory of electromagnetism which is compatible with Maxwell's electromagnetism, Kelvin's ring vortex atoms, and Poincare's relativistic electromagnetism.
The concept of the vortex atom was based on the presumed existence of the ether and implied the chimerical claim that matter and ether consist principally of the same substance and that their only difference is the substance's state of motion.
Indeed the vortex atom became much more significant in popular understanding than in the sciences themselves, since the model didn't solve any of the problems with which physicists were struggling at the time; instead it supported a hypothesis--of the existence of the ether--that, frustratingly, could never be proved.
(Horz II, 31) Kelvin's "On Vortex Atoms" appeared first in volume VI (1867) of the Proceedings of the Royal Society of Edinburgh and was reprinted the same year in the Philosophical Magazine (volume XXXIV).
Even here, Stewart and Tait take their prompt from Kelvin, who had mused in his first publication on vortex atoms: "To generate or to destroy 'Wirbelbewegung' in a perfect fluid can only be an act of creative power" (Thomson 1).