class of "best" estimators, it is shown in this paper that supeRn(O, m) -) lLas n -4 oo if and only if m/n - O and m-+ oo as n oo; that minm supoRn(O, m) 1 + Cnt* as n oo; and that the minimax sample size is m Cn1 as n - oo. 1. Introduction. This investigation treats the problem of estimating the common mean , of two populations using a fixed number n of observations. If the population variances were known, the most efficient procedure would be to take all n observations from that population with the smaller variance. When prior information about the variances is lacking or is too vague to be quaintified, it is natural to consider the procedure which consists of taking a preliminary sample of size m from each population, computing estimates of the variances, and then taking the remaining n - 2m observations from that population with the apparently smaller variance. Since, if m is chosen too large or too small, the advantage of the two-stage sampling scheme over the procedure of simply taking n/2 observations from each population will be lost, the problem arises of determining for some good estimator an optimum choice of m as a function of n, not dependent on the unknown variances. As an example, we may suppose that we have available two devices for measuring a physical constant, that each measurement is expensive or time consuming so that their total number is limited, and that we wish to estimate the constalnt as accurately as possible. For related work on two-stage experiments with a fixed total sample size, reference should be made to Ghurye and Robbins [2], where it is shown that the ratio of the variance of a certain two-stage estimator for the difference of two means to the minimum variance tends to uniity as the sample size increases, and