The estimators derived by Jolly and by Seber for the parameters of an open mobile animal population are maximum likelihood estimators if every live animal has the same probability of surviving until the next sample and the same probability of being observed at that time. The logical structure of the estimatinig equations is emphasised here, and is found to throw light on two classes of problem: what biases will occur if the Seber-Jolly estimates are used in situations in which the standard assumptions do not hold, and what new estimates are reasonable in these cases? The logical argument is applied to various situations including those in which survival is heterogeneous, or correlated with age, and others in which catchability is heterogeneous or affected, permanently or temporarily, by marking.