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Let D be a linear partial differential operator between vector bundles on a differentiable manifold X of dimension n. Let D be the sheaf of germs of differentiable functions on X. For every h E Z a spectral sequence (hEPq) is associated to D. When D satisfies appropriate regularity conditions these spectral sequences degenerate for all sufficiently large h and hE2?o is the pth Spencer cohomology for D. One can compute hE2 as the cohomology at APT* C, Rh -p.q of a complex 0 --> Rhq * * Al T* (D0, Rh -i1q --> An T'( 0Rh n,q ? 0. When q=0 this complex coincides with the usual (first) Spencer complex for D. These results give a generalization of Spencer's theory. The principal importance of this generalization is that it greatly clarifies the role played by homological algebra in the theory of overdetermined systems of linear partial differential equations.

The attempt which WV. V. D. Hodge made in Chapter III of his beautiful book' to establish the existence of harmonic integrals with preassigned periods has not been entirely successful because the proof is partly based on a false statement (p. 136) concerning the behavior of the solution of a non-homogeneous integral equation when the spectrum parameter approaches an eigen value. In a Princeton seminar on the subject, H. F. Bohneriblust pointed out that counter examples are readily available even for linear equations with a finite number of unknowns. For instance the equation Xx + Ax = c with