A composite hypothesis asserting some kind of equality about two distributions may be tested by use of a two-sample statistic. It is often true that if one sample size is allowed to become infinite in this statistic then it is interpretable as one for testing a simple hypothesis about the distribution furnishing the finite sample. (The simple hypothesis is furnished by the “infinite sample.”) Wilcoxon's two-sample test studied in this way becomes a useful tool for certain kinds of problems. A rank test of Lehmann's reduces to either Fisher's test for combining independent tests of significance or to one once proposed by Karl Pearson for the same purpose. Two natural one-sample limits of two-sample median tests are also presented.