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.