*63.* This statement is equivalent to the following:

_Given three pairs of corresponding rays in two projective pencils, it is
possible to find a ray of one which corresponds to any ray of the other._




*64.* We proceed, then, to the solution of the fundamental

PROBLEM: _Given three pairs of rays, __aa’__, __bb’__, and __cc’__, of two
protective pencils, __S__ and __S’__, to find the ray __d’__ of __S’__
which corresponds to any ray __d__ of __S__._

[Figure 12]

FIG. 12


Call _A_ the intersection of _aa’_, _B_ the intersection of _bb’_, and _C_
the intersection of _cc’_ (Fig. 12). Join _AB_ by the line _u_, and _AC_
by the line _u’_. Consider _u_ as a point-row perspective to _S_, and _u’_
as a point-row perspective to _S’_. _u_ and _u’_ are projectively related
to each other, since _S_ and _S’_ are, by hypothesis, so related. But
their point of intersection _A_ is a self-corresponding point, since _a_
and _a’_ were supposed to be corresponding rays. It follows (§ 52) that