A new type of axisymmetric magnetohydrodynamic equilibrium is presented. It is characterized by a region of pressure and safety factor variation with a short scale length imposed as a perturbation. The equilibrium consistent with these profile variations can be calculated by means of an asymptotic expansion. The flexibility obtained by generating such equilibria allows for a close examination of the mechanisms that are relevant to ballooning instabilities – ideal-MHD modes with large toroidal mode number. The so-called first and second regions of stability against these modes are seen well within the limits of validity of the asymptotic expansion. It appears that the modes must be localized in regions with small values of the local shear of the magnetic field. The second region of stability occurs where the local shear is large throughout the range where the magnetic-field-line curvature is destabilizing.