At an elliptic magnetic stagnation line, the combined effect of viscosity ν and resistivity η stabilizes the ideally unstable kinks with short axial wavelengths λ. For small ν and η the critical wavelength below which all modes are stable is O(ν1/4 η1/4). If only one of the two parameters ν and η is nonzero (purely resistive or purely viscous plasma), the modes remain unstable, but the maximum growth rate is O(λ2/η) or O(λ2/ν), as opposed to the ideal growth rate (ν = η = 0) which is O(1). This result provides one possible mechanism to explain the absence of these instabilities in Z-pinch experiments such as EXTRAP. It is derived for the simple but generic case of the incompressible motion about circularly cylindrical equilibria with purely azimuthal magnetic fields and arbitrary axial current density profiles. For the equilibrium with constant current density the entire short wavelength spectrum is explicitly given. The eigenvalues (but not the eigenfunctions) are invariant upon interchanging ν and η. The eigenvalues pertaining to different radial mode numbers have spacing O(λ) and are on curves in the complex frequency plane which depend only on the azimuthal mode number m and on the two parameters λ2/ν and λ2/η.