The generalised product moment distribution in a normal system

In a previous paper ( 1 ) the authors used the theory of the moment generating function to deduce the known random sampling distributions of the estimated variance and co-variance in a system of variables following the normal law of frequency. In this paper we shall go much further. By means of the same general method the simultaneous distribution of the ½ p ( p + 1) second order moment statistics in a normal system of p mutually correlated variables will be deduced. It will be shown to be completely independent of that of the p sample means, and incidentally the method of proof to be developed will be found, in the special cases p = 1 and 2, to be an improvement on that given in the earlier paper, being independent of previous results reached by other authors.