*50.* It follows also that two projective pencils which have the same
center may have no more than two self-corresponding rays, unless the
pencils are identical. For if we cut across them by a line, we obtain two
projective point-rows superposed on the same straight line, which may have
no more than two self-corresponding points. The same considerations apply
to two projective axial pencils which have the same axis.




*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