  In 1988, Brown and Ellis published [3] a generalised Hopf formula for the
higher homology of a group. Although substantially correct, their result lacks
one necessary condition. We give here a counterexample to the result without
that condition. The main aim of this paper is, however, to generalise this
corrected result to derive formulae of Hopf type for the n-fold Cech derived
functors of the lower central series functors Z_k. The paper ends with an
application to algebraic K-theory.
