The propagation of high-frequency electrostatic waves is considered in a plasma in which there is a zero-order temperature gradient perpendicular to the uniform magnetic field. The frequency range is such that the ions do not respond to the perturbed fields (this condition is satisfied if $\frac{\ensuremath{\alpha}}{{\ensuremath{\rho}}_{i}}\ensuremath{\ll}{k}_{\ensuremath{\perp}}d$ resulting in $\ensuremath{\omega}\ensuremath{\gg}{\ensuremath{\omega}}_{\mathrm{ci}}$, $\ensuremath{\omega}\ensuremath{\gg}{\ensuremath{\omega}}_{\mathrm{pi}}$, where ${\ensuremath{\omega}}_{\mathrm{ci}}$ and ${\ensuremath{\omega}}_{\mathrm{pi}}$ are the ion-cyclotron and ion-plasma frequencies and $\ensuremath{\alpha}$, ${\ensuremath{\rho}}_{i}$, ${k}_{\ensuremath{\perp}}$, and $d$ are the scale length of the temperature gradient, the ion Larmor radius, the perpendicular component of the wave vector, and the electron Debye length, respectively). For $\ensuremath{\omega}\ensuremath{\ll}{\ensuremath{\omega}}_{\mathrm{ce}}$, ${\ensuremath{\rho}}_{e}\ensuremath{\ll}d\ensuremath{\ll}{\ensuremath{\lambda}}_{\ensuremath{\perp}}$ and a specific form of the temperature gradient the differential equation for $\ensuremath{\phi}$ is reduced to an elementary form where $\ensuremath{\omega}$, ${\ensuremath{\omega}}_{\mathrm{ce}}$ are the wave-and electron-cyclotron frequencies and ${\ensuremath{\lambda}}_{\ensuremath{\perp}}$ and ${\ensuremath{\rho}}_{e}$ the wavelength perpendicular to the uniform magnetic field and the electron Larmor radius, respectively. $\ensuremath{\phi}$ is the electrostatic potential. For ${\ensuremath{\lambda}}_{\ensuremath{\perp}}\ensuremath{\ll}\ensuremath{\alpha}$ the exact solution is very close to the local solution of Mikhailovskii and Pashitskii which neglects the effects of the boundaries. However for ${\ensuremath{\lambda}}_{\ensuremath{\perp}}\ensuremath{\gtrsim}\ensuremath{\alpha}$ the plasma is unstable to shorter axial wavelengths than predicted by the local theory. It is shown that the instability is due to the interaction of a positive energy wave with a negative energy wave. When the phase velocities of the two waves are different the plasma is stable. However, when the nonuniform plasma is adjacent to a cold resistive plasma, instability may again result. This is analogous to the resistive wall amplifier of Birdsall, Brewer and Haeff. The relevance of these results to the stability of low frequency waves in a nonuniform plasma is pointed out.