  The mathematical study of vortices began with Herman von Helmholtz's
pioneering study in 1858. It was pursued vigorously over the next two decades,
largely by British physicists and mathematicians, in two contexts: Maxwell's
vortex analogy for the electromagnetic field and William Thomson's (Lord
Kelvin) theory that atoms were vortex rings in an all-pervading ether. By the
time of Maxwell's death in 1879, the basic laws of vortices in a perfect fluid
in three-dimensional Euclidean space had been established, as had their
importance to physics. Early vortex studies were embedded in a web of issues
spanning the fields we now know as "mathematics" and "physics" - fields which
had not yet become institutionally distinct disciplines but overlapped. This
paper investigates the conceptual issues with ideas of force, matter, and
space, that underlay mechanics and led to vortex models being an attractive
proposition for British physicists, and how these issues played out in the
mathematics of vortices, paying particular attention to problems around
continuity. It concludes that while they made valuable contributions to
hydrodynamics and the nascent field of topology, the British ultimately failed
in their more physical objectives.
