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.