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In attempting to clear the ground for an attack on the problem of relative maxima and miniima of two quadratic or hermitian forms with aii infinite number of variables, the author found that various questions insistently arose concerning the equivalence of pairs of forms in a finite number of variables and their reduction to normal type. The methods of attack in this theory as set forth in the literature seem to lack systematic unity. Not only are different devices employed for the treatment of the various phases of the problem, but these devices vary for different cases of the same phase. In investigating the problem of maxima and minima the author uncovered fundamental principles which may serve as a new interpretation of the problem of equivalence and make possible a iiew and more unified conception of the treatment. This new foundation for the theory of bilinear forms is set forth in this memoir while the problem of relative extrema is developed in the succeeding one.t To exhibit the formal connection between the problems of relative extrema and of equivalence of pairs of forms, we note that in considering the bi linear forms i,n 1,n Atx y) a B(x, y') bx vZ, yJ _ ijt s j y. j xj jYi ii tJ

A new method in the equivalence of pairs of bilinear forms