A theorem on quadratic forms and its application in the calculus of variations

Principal result.Let P{z), Qi(z), ■ ■ ■ , Qr(z) be quadratic forms in the real variables Zi, ■ ■ ■ , zm with real coefficients having the following properties: the form P(z) is positive at each point (z)^(0) at which the forms Qi(z)> ' " ' > Qr(z) vanish simultaneously;for every set of constants 51; • ■ ■ ,Sr, not all zero, the quadratic form SiQi(z)] is indefinite; for every linear subspace L on which the quadratic forms Qi, ■ ■ ■ , Qr do not vanish simultaneously, save at (z) = (0), there is a linear combination SiQi(z) of these forms which is positive definite on L. The principal result given in the present note is given in the following: Theorem i.Under the above hypotheses there exists a set of constants Si, ■ • • , Sr such that the quadratic form P(z, S) = P(z) + S.Qfz) (i=l,---,r) is positive definite in the variables zx, ■ ■ ■ , zm.The last two hypotheses may be dropped when r = \.This theorem was proposed in a somewhat different form by Bliss in 1937 in a seminar on the calculus of variations.It is useful in sufficiency proofs for multiple integral problems.Proofs of the theorem for the case r = 1 were given in the seminar by Albert,% Reid, § McShane and Hestenes, each using a different method.The last two of these proofs have not been published.The proof of Theorem 1 here given is due to McShane and is an immediate extension of the one given by McShane for the case r = \.In the next section Theorem 1 will be applied to the case in which (z) = (xi, ■ ■ ■ , xn, y1; • • • , yn) and the forms Qi, ■ ■ ■ , QT are the two-rowed minors of the wX2-dimensional matrix (xk yk).If n = 2 then r = 1 and Theorem 2 below is an immediate consequence of Theorem 1.If n>2 the further result described in Theorem 3 below is * Presented separately to the Society, by McShane on April 8, 1939, and by M. R.