Growth rates of axisymmetric modes of elongated tokamak plasmas, in the presence of a conducting shell, are computed, and the effects of plasma-wall distance, current profile and elongation are analysed. For highly elongated plasmas, it is found that the side walls are more important than the top and bottom walls for stabilizing these modes. Furthermore, it is shown that decreasing the width of the radial current profile has a much greater effect on vertical stability than increasing the width of the conducting shell by the same amount. Finally, the conditions for axisymmetric stability during the evolution from a circular plasma to a highly elongated racetrack are investigated.