  We derive a set of equations that describe the shape and behaviour of a
single perturbed vortex line in a Bose-Einstein condensate. Through the use of
a matched asymptotic expansion and a unique coordinate transform a relation for
a vortex's velocity, anywhere along the line, is found in terms of the
trapping, rotation, and distortion of the line at that location. This relation
is then used to find a set of differential equations that give the line's
specific shape and motion. This work corrects a previous similar derivation by
Anatoly A. Svidzinsky and Alexander L. Fetter [Phys. Rev. A \textbf{62}, 063617
(2000)], and enables a comparison with recent numerical results.
