  A cubic surface in $P^3$ is known to contain 27 lines, out of which one can
form 36 Schlafli double - sixes i.e., collections $l_1,...,l_6, l'_1,...,
l'_6\}$ of 12 lines such that each $l_i$ meets only $l'_j, j\neq i$ and does
not meet $l_j, j\neq i$. In 1881 F. Schur proved that any double - six gives
rise to a certain quadric $Q$ , called Schur quadric which is characterized as
follows: for any $i$ the lines $l_i$ and $l'_i$ are orthogonal with respect to
(the quadratic form defining) $Q$. The aim of the paper is to relate Schur's
construction to the theory of vector bundles on $P^2$ and to generalize this
construction along the lines of the said theory.
