  For axisymmetric models for coronal loops the relationship between the
bifurcation points of magnetohydrodynamic (MHD) equilibrium sequences and the
points of linear ideal MHD instability is investigated imposing line-tied
boundary conditions. Using a well-studied example based on the Gold-Hoyle
equilibrium, it is demonstrated that if the equilibrium sequence is calculated
using the Grad-Shafranov equation, the instability corresponds to the second
bifurcation point and not the first bifurcation point because the equilibrium
boundary conditions allow for modes which are excluded from the linear ideal
stability analysis. This is shown by calculating the bifurcating equilibrium
branches and comparing the spatial structure of the solutions close to the
bifurcation point with the spatial structure of the unstable mode. If the
equilibrium sequence is calculated using Euler potentials the first bifurcation
point of the Grad-Shafranov case is not found, and the first bifurcation point
of the Euler potential description coincides with the ideal instability
threshold. An explanation of this results in terms of linear bifurcation theory
is given and the implications for the use of MHD equilibrium bifurcations to
explain eruptive phenomena is briefly discussed.
