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.

A slow steady motion of incompressible viscous liquid in the space between two fixed concentric spherical boundaries is considered. The motion arises from two point-sources of strengths ±2m at arbitrary points A, B on the outer sphere r = a. The velocity is calculated as the vector sum of the velocities in two simpler motions in each of which there is an axis of symmetry so that a stream function can be used. The force exerted by the liquid on the inner boundary r = b is similarly the resultant of two forces, each passing through the common centre of the spheres; it can be simply expressed in terms of a, b, m and the vector .

If f(z) is a function of the complex variable z, regular in a neighbourhood of z = 0, with f (0) = 0 and f' (0) ≠ 0, then the equation w = f(z) admits a unique solution, regular in some neighbourhood of w = 0, given by

where C is an appropriate contour encircling z = 0. These formulae are well-known, being stages in the proof of the classical reversion † formula

due to Lagrange.

A graph is said to be k-chromatic if its vertices can be split into k classes so that two vertices of the same class are not connected (by an edge) and such a splitting is not possible for k−1 classes. Tutte was the first to show that for every k there is a k-chromatic graph which contains no triangle [1].