Hotelling and Pabst [1] showed that the rank correlation coefficient had a limiting normal distribution under the equally likely permutations of the hypothesis of independence. Wald and Wolfowitz [2] developed a general theorem of this type, and Noether [3] and Hoeffding [4] have relaxed the conditions used therein. In this paper a vector form of the theorem is proved along the lines used in an example by Wald and Wolfowitz [1] but taking account of the singular cases in which the correlations approach one.