A long-sought method of factor analysis—involving the minimization of off-diagonal residuals of the correlation matrix—is developed in this paper. Now that it is available, it might well replace the principal-factor and the maximum-likelihood solutions as the preferred initial factorisation of a correlation matrix. The mathematical method found to be most effective for attack on this problem is the Gauss-Seidel process. A program has been written in FORTRAN for the minres solution. A special feature provides for the control of Heywood cases, which have plagued some other methods of analysis. Minres results are illustrated for several examples, and comparisons made with principal-factor and maximum-likelihood solutions.