  The ratio of the period of the fundamental mode to that of the first overtone
of kink oscillations, from here on the "period ratio", is a seismology tool
that can be used to infer information about the spatial variation of density
along solar magnetic flux tubes. The period ratio is 2 in longitudinally
homogeneous thin tubes, but it differs from 2 due to longitudinal
inhomogeneity. In this paper we investigate the period ratio in longitudinally
inhomogeneous prominence threads and explore its implications for prominence
seismology. We numerically solve the two-dimensional eigenvalue problem of kink
oscillations in a model of a prominence thread. We take into account three
nonuniform density profiles along the thread. In agreement with previous works
that used simple piecewise constant density profiles, we find that the period
ratio is larger than 2 in prominence threads. When the ratio of the central
density to that at the footpoints is fixed, the period ratio depends strongly
on the form of the density profile along the thread. The more concentrated the
dense prominence plasma near the center of the tube, the larger the period
ratio. However, the period ratio is found to be independent of the specific
density profile when the spatially averaged density in the thread is the same
for all the profiles. An empirical fit of the dependence of the period ratio on
the average density is given and its use for prominence seismology is
discussed.
