  The study of MHD waves is important both for understanding heating in the
solar atmosphere and for solar atmospheric seismology. The analytical
investigation of wave mode properties in a cylinder is of particular interest
in this domain, as many atmospheric structures can be modeled as such in a
first approximation. We use linearized ideal MHD to study quasimodes (global
modes that are damped through resonant absorption) with a frequency in the cusp
continuum, in a straight cylinder with a circular base and an inhomogeneous
layer at its boundary which separates two homogeneous plasma regions inside and
outside. We are in particular interested in the damping of these modes, and
shall hence try to determine their frequency as a function of background
parameters. After linearizing the ideal MHD equations, we find solutions to the
second-order differential equation for the perturbed total pressure in the
inhomogeneous layer in the form of Frobenius series around the regular singular
points that are the Alfv\'en and cusp resonant positions, as well as power
series around regular points. By connecting these solutions appropriately
through the inhomogeneous layer and with the solutions of the homogeneous
regions inside and outside the cylinder, we derive a dispersion relation for
the frequency of the eigenmodes of the system. From the dispersion relation, it
is also possible to find the frequency of quasimodes even though they are not
eigenmodes. As an example, we find the frequency of the slow surface sausage
quasimode as a function of the inhomogeneous layer's width, for values of the
longitudinal wavenumber relevant for photospheric conditions. The results were
found to match well the results found in another paper which studied the
resistive slow surface sausage eigenmode. We also discuss the perturbation
profiles of the quasimode and the eigenfunctions of continuum modes.
