Systematic sampling methods have been used in a variety of fields, notably in forestry and ecological studies, because of their convenience in the field and their advantages over a randomized scheme if the survey is also to be used for mapping. Their use was justified by work such as that of Hasel (1938) and Osborne (1942), who examined the sampling errors of random, stratified random and systematic surveys, by subjecting detailed forestry survey data to analysis by these three methods. These analyses showed that the systematic designs were generally the most efficient. They left unsolved the problem of estimating the error of the systematic sample, although Osborne's paper gave a method based on computing the mean square serial correlation which gave reasonable agreement with experimental results. More recently, Finney (1948), also using data from forest surveys, has compared the variances of systematic samples (computed by the overlapping group method to obtain reasonable precision), random samples and stratified random samples with one and two members per stratum. In these cases Finney showed that the variance of the systematic sample differed little from the variance of a stratified random sample of the same size with one observation per stratum, but was appreciably smaller than the variance obtained with half the number of strata and two observations per stratum, which appeared to be the most efficient system giving an unbiased estimate of the variance without supplementary information. Further investigation of the variance of systematic sampling with various models seems likely to be useful. Yates (1948) examined the sampling variance for a number of particular cases and gave a form (p. 362) which, with supplementary information, provided an estimate of the sampling variance of the means of a systematic sample, or, alternatively, subject to certain assumptions, provided an upper limit to the error. Jowett (1952) gave a method for the determination of the variance of a systematic sample, on the assumption that the observations were derived from a stationary process. This was derived from earlier work by Cochran (1946) and Quenouille (1949) in the same field. In this paper we shall develop a method similar to Jowett's, but involving rather weaker assumptions; we shall also derive a form for the variance of samples with spacings less than or equal to that of the observed data (which is not covered by Jowett's method); this includes the case where the variance is derived from a single sample. This does not conflict with the often-stated principle that a systematic sample cannot by itself provide an estimate of error, since the assumptions which are made about the population are equivalent to supplementary information. Many alternative assumptions could have been made, and the justification of the particular ones chosen must be that they appear likely to apply to a wide variety of data and in the cases considered below yield results which agree with