A multilayer consisting of parallel sided, elastically isotropic layers with thicknesses much smaller than their dimensions in the plane of the layers may develop large coherency stresses when the layers are thin enough. General formulae are given for the stress tensors in a flat periodic fully coherent multilayer which may contain any number of layers, each with different lattice parameters, elastic constants and thicknesses. As the thicknesses of the layers increase there is an increasing tendency for the interfaces to become incoherent thereby relaxing the elastic stresses, at the expense of creating higher energy interfaces between the layers. This tendency is particularly marked for unusually thick or unusually rigid layers or for layers with very large or small lattice parameters. Unlike the case of thin films on massive substrates, there is no single thickness at which coherency is lost. In this paper, the loss of coherency is explored by searching for energy minima in irregular multilayers.