*83. Determination of the pencil.* We now show that _it is possible to
assign arbitrarily three lines, __a__, __b__, and __c__, of __ the system
(besides the lines __u__ and __u’__); but if these three lines are chosen,
the system is completely determined._

This statement is equivalent to the following:

_Given three pairs of corresponding points in two projective point-rows,
it is possible to find a point in one which corresponds to any point of
the other._

We proceed, then, to the solution of the fundamental

PROBLEM. _Given three pairs of points, __AA’__, __BB’__, and __CC’__, of
two projective point-rows __u__ and __u’__, to find the point __D’__ of
__u’__ which corresponds to any given point __D__ of __u__._

[Figure 20]

FIG. 20


On the line _a_, joining _A_ and _A’_, take two points, _S_ and _S’_, as
centers of pencils perspective to _u_ and _u’_ respectively (Fig. 20). The
figure will be much simplified if we take _S_ on _BB’_ and _S’_ on _CC’_.
_SA_ and _S’A’_ are corresponding rays of _S_ and _S’_, and the two