Explicit formulas are derived for the sum rules for the frequency-dependent hyperpolarizability-diagonal-components. These are the counterparts to the Cauchy moments for the dynamic polarizabilities. The formulas allow for the frequency dependence of any nonlinear optical process to be expressed as a single general expansion up to terms which are of fourth power in the optical frequencies, X(alpha,alpha,...,alpha)(n)(-omega(sigma);omega(1),...,omega(n)) = X(alpha,alpha,...,alpha)(n)(0) + AW(2) + BW22 + B’W-4, where omega(sigma) = Sigma(i) omega(i), W-2 = omega(sigma)(2) + omega(1)(2) + ... omega(n)(2), and W-4 = omega(sigma)(4) + omega(1)(4) + ... omega(n)(4) (in conventional notation X(1) = alpha, X(2) = beta, X(3) = gamma, etc.). The advantages of determining the frequency dependence of all NLO processes, for a given species, in a single calculation are stressed. We focus mainly on the sum rules (A, B, and B’) for X(3) and X(5). These are applicable to both atoms and molecules (with the exception of X(5) for noncentrosymmetric molecules) and we evaluate them, using near-exact wave functions, for H and He. It is apparent that B’ is generally smaller than B and this accounts for the reasonable success of the Shelton-Bishop dispersion formula which is often used to fit experimentally-derived dynamic hyperpolarizabilities.