  We consider the linear stability of Hill's vortex with respect to
axisymmetric perturbations. Given that Hill's vortex is a solution of a
free-boundary problem, this stability analysis is performed by applying methods
of shape differentiation to the contour dynamics formulation of the problem in
a 3D axisymmetric geometry. This approach allows us to systematically account
for the effect of boundary deformations on the linearized evolution of the
vortex under the constraint of constant circulation. The resulting singular
integro-differential operator defined on the vortex boundary is discretized
with a highly accurate spectral approach. This operator has two unstable and
two stable eigenvalues complemented by a continuous spectrum of
neutrally-stable eigenvalues. By considering a family of suitably regularized
(smoothed) eigenvalue problems solved with a range of numerical resolutions we
demonstrate that the corresponding eigenfunctions are in fact singular objects
in the form of infinitely sharp peaks localized at the front and rear
stagnation points. These findings thus refine the results of the classical
analysis by Moffatt & Moore (1978).
