The results given here represent an extension of previous work [1, 2] in which the author considered the oscillations of a plane current-vortex sheet in an ideal perfectly conducting fluid. In this paper we consider the effects of curvature of the sheet in a direction transverse to the velocity and magnetic field direction. This problem may be regarded as that of finding longitudinal small oscillations on a jet of fluid which moves along the lines of force of an impressed magnetic field. For oscillations, whose wavelength is small by comparison with the radius of curvature of the section of the jet, it is to be expected that the criterion for stable or unstable oscillations will be the same as for the plane case examined previously, and this is verified. When one considers the other extreme, in which the wavelength of the oscillations is large, the analysis shows that the magnetic field aligned to the jet has the effect of stabilising the jet, irrespective of the magnetic field strength. The magnetic field thus behaves for large wavelengths in the same way as a surface tension does for small wavelengths. For values of the applied magnetic field which would make the current-vortex sheet without curvature unstable, it is seen that there is a single transition from instability to stability as the wavelength increases. It is shown also that when small wavelengths are stable, in addition to large wavelengths, it does not necessarily mean that the jet will be stable for all wavelengths. Criteria are deduced to distinguish this case from another in which the jet remains unstable for a simple bounded range of intermediate wavelengths.