*Project Gutenberg*. Public domain.

*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 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 through *S*. It is called *a pencil
of planes of the second order*.

## PROBLEMS

*1. * A man *A* moves along a straight road
*u*, and another man *B* moves along the
same road and walks so as always to keep sight of *A*
in a small mirror *M* at the side of the road.
How many times will they come together, *A* moving
always in the same direction along the road?

2. How many times would the two men in the first
problem see each other in two mirrors *M* and
*N* as they walk along the road as before? (The
planes of the two mirrors are not necessarily parallel
to *u*.)

3. As A moves along *u*, trace the path
of B so that the two men may always see each other
in the two mirrors.