The ordinary Wilcoxon test is used to test the null hypothesis that two populations are identical. This paper presents a technique for utilizing the Wilcoxon statistic to test a null hypothesis of a broader type, such as is encountered in the Behrens-Fisher problem, where it is required to test the equality of the means of two normal populations having (possibly) unequal variances. The discussion in this paper is on a rather technical level; for a less technical discussion, see Research Bulletin 62- 27.