The reciprocation of one quadric into another

Abstract

It is a familiar fact that if two quadric forms, in (n + 1) homogeneous variables, be each expressible as a sum of squares, of (n + 1) independent linear functions of the variables, then they are polar reciprocals of one another, in regard to any one of 2n quadrics. The question arises whether this is true for any two quadrics. Segre states that this is an unsettled question. A solution is given, however, by Terracini for any two non-degenerate quadrics, supposed to have been reduced to the Weierstrass canonical form. The present note has the purpose of pointing out that a solution is derivable from a remark made by Frobenius; this requires a knowledge of the roots of the equation satisfied by the matrix of the two quadrics.