The discussion on the correspondence between unstable particles and poles of the S matrix, started in a previous paper, is continued with particular attention to the decay law. It is shown that the previous formalism can be modified to obtain an S matrix, satisfying all the usual requirements of analyticity, unitarity, and asymptotic behavior, which exhibits an isolated resonance with exponential decay without an accompanying pole in the unphysical sheet. The existence of a local potential which yields the S matrix considered is also proved. © 1968 The American Physical Society.