  The C-spectral sequence was introduced by Vinogradov in the late Seventies as
a fundamental tool for the study of algebro-geometric properties of jet spaces
and differential equations. A spectral sequence arise from the contact
filtration of the modules of forms on jet spaces of a fibring (or on a
differential equation). In order to avoid serious technical difficulties, the
order of the jet space is not fixed, i. e., computations are performed on
spaces containing forms on jet spaces of any order.
  In this paper we show that there exists a formulation of Vinogradov's
C-spectral sequence in the case of finite order jet spaces of a fibred
manifold. We compute all cohomology groups of the finite order C-spectral
sequence. We obtain a finite order variational sequence which is shown to be
naturally isomorphic with Krupka's finite order variational sequence.
