Jackson and Bradley (1959, 1961a, 1961b) developed and studied a sequential (multivariate) $T^2$ test of hypotheses on a vector of means, and an analogous $\chi^2$ test for known covariance structure. The present paper presents the results of Monte Carlo sampling on the operating characteristics and average sample numbers (ASN) of these tests. Consideration is restricted to the behavior of these tests at specific null and alternate hypotheses $(H_0$ and $H_1)$ with nominal $\alpha$ and $\beta$ errors of .05. The empirical $\alpha$ and $\beta$ errors are, in general, less than .05 and appear to decrease as the number of variables increases. The empirical ASN are appreciably smaller than the corresponding fixed sample sizes, and approximate the ASN that Jackson and Bradley obtained using Bhate's conjecture. The estimation of the fixed sample sizes were based on the nominal $\alpha$ and $\beta$ errors of .05 while the sequential test ASN were, of course, associated with the resulting smaller error probabilities. Thus the true advantage of the sequential test is understated by the above sample size comparisons. This study investigates the behavior of the sequential test at $H_0$ and $H_1$ only. A similar study involving points between $H_0$ and $H_1$ would be of definite values for (1) ascertaining whether the advantages of the sequential procedure hold under situations other than $H_0$ and $H_1$, and (2) suggesting methods to overcome the conservatism of the test as it now stands.