*55. Point-row of the second order.* The question naturally arises, What
is the locus of points of intersection of corresponding rays of two
projective pencils which are not in perspective position? This locus,
which will be discussed in detail in subsequent chapters, is easily seen
to have at most two points in common with any line in the plane, and on
account of this fundamental property will be called a _point-row of the
second order_. For any line _u_ in the plane of the two pencils will be
cut by them in two projective point-rows which have at most two
self-corresponding points. Such a self-corresponding point is clearly a
point of intersection of corresponding rays of the two pencils.




*56.* This locus degenerates in the case of two perspective pencils to a
pair of straight lines, one of which is the axis of perspectivity and the
other the common ray, any point of which may be considered as the point of
intersection of corresponding rays of the two pencils.




*57. Pencils of rays of the second order.* Similar investigations may be
made concerning the system of lines joining corresponding points of two
projective point-rows. If we project the point-rows to any point in the
plane, we obtain two projective pencils having the same center. At most
two pairs of self-corresponding rays may present themselves. Such a ray is
clearly a line joining two corresponding points in the two point-rows. The
result may be stated as follows: _The system of rays joining corresponding