This paper may be regarded as a sequel to a previous papers(1) in these Proceedings. The vector and matrix notation of that paper used for a statistical sample is systematized somewhat further, so that while a sample S refers as before to the matrix of nm values (a sample of m observations in one variate only being a row vector), we writefor the linear regression formula between the dependent and independent variates into which a sample is supposed partitioned (in place of equation (12) of (1)). More generally, a third submatrix S0 is partitioned off, and its effect eliminated (corresponding to equation (13) of (1)), but without loss of generality we assume that S2 in equation (1) above can always stand for S2.0 if necessary.