Experiments by Huppert (1982) have demonstrated that a finite volume of fluid placed on an inclined plane will elongate into a thin sheet of fluid as it slides down the plane. If the fluid is initially placed uniformly across the plane, the sheet retains its two-dimensionality for some time, but when it has become sufficiently long and thin, the leading edge develops a spanwise instability. A similarity solution for this motion was derived by Huppert, without taking account of the edge regions where surface tension is important. When these regions are examined, it is found that the conditions at the edges can be satisfied, but only when the singularity associated with the moving contact line is removed. When the sheet is sufficiently elongated, the profile of the free surface shows an upward bulge near the leading edge. It is suggested that the observed instability of the shape of the leading edge is a result of the dynamics of the fluid in this bulge. The related problem of a ridge of fluid sliding down the plane is examined and found to be linearly unstable. The spanwise lengthscale of this instability is, however, dependent on the width of the channel occupied by the fluid, which is at variance with the observed nature of the instability. Preliminary numerical solutions for the nonlinear development of a small disturbance to the position of a straight leading edge show the incipient development of a finger of fluid with a width that does not depend on the channel size, in accordance with the observations.