In the past fifteen years there have been many papers on the analysis of tagging or mark-recapture experiments to estimate the parameters of natural populations. While early treatments were linmited to populations assumed to be 'closed'-i.e., with no mortality or recruitment, emigration or immigration-more recent papers treat the more realistic cases in which at least some of these processes are considered. In particular, reference may be made to Darroch [1959], Seber [1962], Paulik [1963], and Parker [1963]. Seber treats in full detail the problem of estimating mortality rates and population sizes from a series of tagging experiments at appropriate intervals of time. Basically the idea here, which in fact had been given earlier by Ricker [1958], is that each new tagging experiment gives information on the tags surviving to that time from earlier tagging experiments. Thus mortality rates, and hence population sizes, are estimable, whence recruitments are easily deduced. However if only a single tagging experiment is conducted, this method is not available and a different approach is necessary. This is so for the data that Parker analyzes. His data refer to a series of experiments he conducted in which tagging was conducted essentially at a single point in time followed by recoveries of tags over a number of intervals. The situation is such that neither mortality nor recruitment can be neglected. As he notes, the treatments of recruitment have often assumed that recruitment is proportional to population size, which is not a realistic assumption for a fish population. As an alternative assumption he proposes to consider recruitment as being constant over the summer season, but of course not necessarily so between seasons. In an earlier paper [1955] Parker had also considered the problem of estimating the population size without making any assumptions as to the recruitment pattern. It is the aim of the present paper to reconsider the Parker problem with no assumptions as to the nature of recruitment, both for the case