FIG. 9


*47. Superposed fundamental forms. Self-corresponding elements.* We have
seen (§ 37) that two projective point-rows may be superposed upon the same
straight line. This happens, for example, when two pencils which are
projective to each other are cut across by a straight line. It is also
possible for two projective pencils to have the same center. This happens,
for example, when two projective point-rows are projected to the same
point. Similarly, two projective axial pencils may have the same axis. We
examine now the possibility of two forms related in this way, having an
element or elements that correspond to themselves. We have seen, indeed,
that if _B_ and _D_ are harmonic conjugates with respect to _A_ and _C_,
then the point-row described by _B_ is projective to the point-row
described by _D_, and that _A_ and _C_ are self-corresponding points.
Consider more generally the case of two pencils perspective to each other
with axis of perspectivity _u’_ (Fig. 9). Cut across them by a line _u_.
We get thus two projective point-rows superposed on the same line _u_, and
a moment’s reflection serves to show that the point _N_ of intersection
_u_ and _u’_ corresponds to itself in the two point-rows. Also, the point
_M_, where _u_ intersects the line joining the centers of the two pencils,
is seen to correspond to itself. It is thus possible for two projective
point-rows, superposed upon the same line, to have two self-corresponding
points. Clearly _M_ and _N_ may fall together if the line joining the
centers of the pencils happens to pass through the point of intersection
of the lines _u_ and _u’_.

[Figure 10]