The proof of a theorem in the above-mentioned paper (Smith, W. L. 1955 Proc. Roy. Soc. A, 232, 6) assumes more than it should and a new proof is given. For certain (unusual) kinds of Semi-Markor process the definition given in the aforementioned paper is inadequate to define the process for all values of the time parameter, and this weakness is removed. Furthermore, a `regularity' condition for an S.M.-process to have step-function realizations (almost certainly) was given incorrectly; this error is now corrected, and certain other `regularity' conditions are discussed.