51.  Projective point-rows having a self-corresponding point in common. Consider now two projective point-rows lying on different lines in the same plane.  Their common point may or may not be a self-corresponding point.  If the two point-rows are perspectively related, then their common point is evidently a self-corresponding point.  The converse is also true, and we have the very important theorem: 

52. If in two protective point-rows, the point of intersection corresponds to itself, then the point-rows are in perspective position.

[Figure 11]

FIG. 11

Let the two point-rows be u and u’ (Fig. 11).  Let A and A’, B and B’, be corresponding points, and let also the point M of intersection of u and u’ correspond to itself.  Let AA’ and BB’ meet in the point S.  Take S as the center of two pencils, one perspective to u and the other perspective to u’.  In these two pencils SA coincides with its corresponding ray SA’, SB with its corresponding ray SB’, and SM with its corresponding ray SM’.  The two pencils are thus identical, by the preceding theorem, and any ray SD must coincide with its corresponding ray SD’.  Corresponding points of u and u’, therefore, all lie on lines through the point S.

53. An entirely similar discussion shows that

If in two projective pencils the line joining their centers is a self-corresponding ray, then the two pencils are perspectively related.

54. A similar theorem may be stated for two axial pencils of which the axes intersect.  Very frequent use will be made of these fundamental theorems.

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.