Abstract. The iterated Cauchy problem under consideration is l~nk = 1 (did t -Ak) u(t) = 0 (t >1 0). (*)Here {A 1 ..... An} are unbounded linear operators on-a Banach space. The initial value problem for (*) is governed by a semigroup of some sort. When each A k is a (Co) semigroup generator, this semigroup is of class (Co) and was studied by J. T. Sandefur [26]. This result is extended to the case when each A k generates a C-regularized semigroup (with C independent of k). This means one can solve u' = Au, u (0) = fe C (Dora (A)) and get u(t)~ 0 whenever C-l f--* 0; here C is bounded and injective. When the A k are commuting generator with A k -Aj injective for k #j, then the Goldstein-Sandefur d'Alembert formula [19] is extended, viz. solutions of (*) (with suitable restrictions on the initial data) are of the form u = ~= 1 ui where ui is a solution of u; = A i ui. Examples and applications are given. Included among the examples is the establishment of a form of equipartition of energy for the Laplace equation; equipartition of energy is weUknown for the wave equation. A final section of the paper deals with the absence of necessary conditions for equipartition of energy.