The background of this problem is developed in an earlier paper.' We here complete the reduction referred to on page 518 as the third problem. That is, we here determine conditions on the coefficients of a positive quaternary quadratic form which define a reduced form. This reduced form has the property that in every class of equivalent forms there is one and only one reduced form. In other words, every form is equivalent to one and only one reduced form. These conditions are in the form of linear inequalities on the coefficients and fall into two groups. First, there are inequalities which are completely symmetric, that is, are independent of the numbering of the variables of the form. These conditions are, except for those mentioned in the previous paper, of a distinctly new type and possess many obvious advantages one of which is the brevity with which the conditions may be stated. Second, there are the usual inequalities which control permutations of the variables. In the reduction, no distinction is made between proper and improper equivalence. That is, transformations of determinant -1 are used as freely as those of determinant +1. However if one desires a set of reduced forms of given determinant having the property that every form of that determinant is properly equivalent to one and only one form of the set, the set may be found as follows. First, find the set of reduced forms of the given determinant by the inequalities of this paper. Then for each reduced form which has no automorph of determinant -1 we will have another form obtained from it by, for instance, changing the signs of the coefficients of xIx2, x1x3 and X1X4 in the form. Hence the number of forms in what we might call the properly reduced set will be the number of forms of the reduced set plus the number of forms in that set without automorphs of determinant -1. In the table of automorphs at the close of the paper, the automorphs of determinant -1 are starred. S. B. Townes, has, in an unpublished Chicago dissertation (1936), carried through the Eisenstein reduction for positive quaternary quadratic forms. Though it has the advantage of giving a leading coefficient which is the minimum of the form, etc., the proof is very much longer and the conditions on the form are more complex. The comparative shortness of this proof is due in large part to the symmetric conditions above referred to. Furthermore, the minimum of any reduced form of this paper can be obtained quickly.