It has been established by Furth, Killeen & Rosenbluth (1963), and by Johnson, Greene & Coppi (1963), that a hydromagnetic equilibrium which is stable on a theory in which electrical resistance is ignored, may yet be unstable through finite conductivity effects. These authors have isolated and categorized several types of such instabilities which, they show, originate from the critical layer in which the perturbation wavefront is perpendicular to the equilibrium magnetic field. In this paper, the asymptotic properties of the critical layer equations, for large values of the critical layer coordinate, are obtained in a number of cases of interest, using the sheet pinch model with uniform resistivity. The mathematical approach is a novel variant of the Laplace integral representation, which allows results of greater generality to be obtained than those given by previous authors. The technique is applied first to the slow interchange mode, and the restricted (but most significant) class of solutions found by Johnson et al . is recovered. It is also shown that modes entirely localized within the critical layer do not occur. Such modes do exist for the more rapid interchange modes, and a new discussion of these is presented. Finally, the oscillatory resistive modes, which arise when the perturbation wavefront is not perpendicular to the equilibrium magnetic field, are studied by a similar mathematical method, and a class of eigenvalues is obtained.