points in two protective point-rows has at most two rays in common with
any pencil in the plane._ For that reason the system of rays is called _a
pencil of rays of the second order._




*58.* In the case of two perspective point-rows this system of rays
degenerates into two pencils of rays of the first order, one of which has
its center at the center of perspectivity of the two point-rows, and the
other at the intersection of the two point-rows, any ray through which may
be considered as joining two corresponding points of the two point-rows.




*59. Cone of the second order.* The corresponding theorems in space may
easily be obtained by joining the points and lines considered in the plane
theorems to a point _S_ in space. Two projective pencils give rise to two
projective axial pencils with axes intersecting. Corresponding planes meet
in lines which all pass through _S_ and through the points on a point-row
of the second order generated by the two pencils of rays. They are thus
generating lines of a _cone of the second order_, or _quadric cone_, so
called because every plane in space not passing through _S_ cuts it in a
point-row of the second order, and every line also cuts it in at most two
points. If, again, we project two point-rows to a point _S_ in space, we
obtain two pencils of rays with a common center but lying in different
planes. Corresponding lines of these pencils determine planes which are
the projections to _S_ of the lines which join the corresponding points of
the two point-rows. At most two such planes may pass through any ray