1. The notion of a perfect operator was introduced in [1], mainly for the purpose of widening the Laplace-transform interpretation'of HEAVISIDE'S symbolic calculus. A problem mentioned in [1], and related to the classical problem of characterizing Laplace transforms, is that of finding a necessary and sufficient condition for an analytic function to be the Laplace transform of a perfect operator. We give here a solution of this problem, in a way that enables us to show (in answer to a question raised in [2]) that all perfect operators are obtainable by repeated "differentiation" of continuous functions. We also discuss some implications of these results. The notation and terminology are those of E2], supplemented where necessary.