Abstract
The incitement, or provocation, to assemble the following pages was the recognition in a recent paper (5) of a geometrical property characterizing those pairs, in a pencil of quadric surfaces, such that the quadrics of a pair are both outpolar and inpolar to each other: symmetrically apolar let it be said. That there are three such pairs was known (8), but their identification by means of the principal chords of their common curve had apparently not been remarked upon. The question naturally arose as to the number and identification of such pairs of quadrics of a pencil in n-dimensional projective space [n]; it is easy to show that the number is ½n(n–1), but geometrical identification seems quite another matter and no success whatever has yet been achieved even when n is 4.
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