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where N(v) is the number of normal modes with frequencies less than v\ and v-l is the largest frequency of a normal mode. To apply Eqs. (1) or (2) one must either know all the frequencies of normal modes or their distribution function. In the case of complex molecules and crystalline solids composed of N (N = 0( 1023) in the latter case) particles, the procurement of this knowledge becomes equivalent to the determination of the (3N — 6) normal modes of N coupled oscillators (6 being the number of translational and rotational degrees of freedom). Mathematically this means finding the zeroes of a characteristic determinant of order (3iV — 6) or the roots of a (3N — 6)th degree equation. In highly symmetrical molecules and crystal lattices this characteristic determinant is factorable into several smaller ones; but to date N(v) has never been derived exactly for any molecular model of a real crystal. Numerical approximations have been obtained for N(v) but without any estimate of their errors.2 By taking traces of powers of the matrix of the determinant mentioned above one can obtain the moments of the distribution function dN(v). It is the purpose of this note to show how, without a knowledge of N{v) or the frequencies of the individual normal modes, one can obtain by the method of mechanical quadratures averages over the normal modes

Averages over normal modes of coupled oscillators with application to theory of specific heats