On account of the unique property of samples from a normal population that the ratio n+1 (x-Fb)/V(n + 1)/s (where jt is the population mean, x = x1/(n + 1) and ns2 =Z(Xi= 1 is the ratio of a normal deviate to a stochastically independent estimate of its variance, Student's t-test is a suitable test of significance for the mean of a normal population. However, in a variety of cases, it is necessary to test for the mean of a population which does not follow the Gaussian law. Efforts have, therefore, been made to see how far Student's distribution may be used for the purpose in non-normal populations. Due, mainly, to the analytical difficulties of the problem, no extensive theoretical discussion has yet been given. Thus, Pearson & Adyanthaya (1929), Rietz (1939) and Nair (1941) have given experimental treatments, while the theoretical discussions of some others (Rider, 1929; Perlo, 1933; Laderman, 1939) have dealt only with trivially small sample sizes. The papers by Bartlett (1935) and Geary (1936, 1947) give results true for any sample size, though they are based on certain assumptions and approximations. The present paper deals with the population considered by Geary in his 1936 paper, subject to the same approximations. The second contribution by Geary (in which is derived the t-distribution in samples from a population which departs more from normality than that considered in the 1936 paper) came to my notice too late to be made use of in the present work; but it is proposed to consider it later on. Geary (1936) has obtained the distribution of the ratio (x -It) (n + 1)/s in the case of an asymmetrical population, whose fourth and higher cumulants are zero, by neglecting squares and higher powers of the third cumulant. We know from this how far the probability of an error of the first kind (i.e. the probability of rejecting the null hypothesis when it is true) in such a population differs from that for a normal distribution, provided we may neglect the square of the standardized third cumulant yl. Here again, on account of analytical difficulties, it is not possible, except for very small sample sizes, to consider the effect of terms containing higher powers of yl. However, we can assume the result derived by Geary to be correct for very small values of yl, as also for large sample sizes-but in such cases the deviation from values of the normal theory is practically negligible. Even then, it is of interest to know whether, in using the usual tables of the t-test (based on the normal distribution), we are committing the greater error in the probability of an error of the first kind or in that of an error of the second kind. In the present paper are derived the values of the probability of an error of the second kind (and hence, of the power of the test) when the usual t-tables are used to define the critical region. It may be mentioned here that this problem is only a special case of a general investigation, on which the writer is engaged, into the effect, on statistical tests, of differences between the actual and the assumed distribution laws of the universe sampled. The solution of these problems is hampered by analytical difficulties in the derivation of the probability laws (and particularly of power functions), and the present case is one of the few in which a mathematical, though only approximate, solution has been found possible.