The linearized equations of a gravity driven, resistive instability of a plasma supported by a planar, sheared magnetic field are shown to be characterized by a variational principle from which an eigenvalue corresponding to a periodic growth is obtained. The effect of a finite ion Larmor radius is included through off‐diagonal terms in the pressure tensor in the momentum fluid equation.When the Larmor radius is zero a numerical solution for the growth rate is found using a one parameter trial function which indicates that the correct function maximizes this rate. When the Larmor radius becomes significant, viz., with decreasing wavelength, the instability becomes an overstability with a reduced growth rate becoming proportional to the square root of the ratio of the resistivity to the wavelength. Whilst numerical results are given beyond this transition, the Larmor radius has become comparable with the thickness of an internal resistive layer and the equations cease to be reliable.