This study is concerned with flow of liquid films down inclined plates, which carry an increasing amount of fluid in the downstream direction. It is supposed that by some mechanism, e.g. condensation, the fluid enters the film at its outer edge. The rate at which this mass-addition occurs is governed by some power law. Only those cases are discussed in which the fluid is predominantly inviscid. The no-slip condition at the wall, which is violated by the inviscid flow leads to the introduction of a viscous sublayer near the wall. A singular perturbation technique is used to develop higher approximations.