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Phonon dispersion relations along the principal symmetry directions and heat capacities of copper, silver and gold have been calculated on the basis of a non central force model developed by us recently. The computed results are found to be in close agreement with the experimental ones.

Let Vbeafinitedirnensional vector space over thefield F and $(x) E F[x]. Let a: V--t Vbe a linear operator. Let S, be the set consisting of the vectors whose minimal polynomial 1s 4(x), together withthezero ~ector. Wegive anecessary and sufficient condition for S, to be a subspace.

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