*61. Tangent line.* We have shown in the last chapter (§ 55) that the
locus of intersection of corresponding rays of two projective pencils is a
point-row of the second order; that is, it has at most two points in
common with any line in the plane. It is clear, first of all, that the
centers of the pencils are points of the locus; for to the line _SS’_,
considered as a ray of _S_, must correspond some ray of _S’_ which meets
it in _S’_. _S’_, and by the same argument _S_, is then a point where
corresponding rays meet. Any ray through _S_ will meet it in one point
besides _S_, namely, the point _P_ where it meets its corresponding ray.
Now, by choosing the ray through _S_ sufficiently close to the ray _SS’_,
the point _P_ may be made to approach arbitrarily close to _S’_, and the
ray _S’P_ may be made to differ in position from the tangent line at _S’_
by as little as we please. We have, then, the important theorem

_The ray at __S’__ which corresponds to the common ray __SS’__ is tangent
to the locus at __S’__._

In the same manner the tangent at _S_ may be constructed.




*62. Determination of the locus.* We now show that _it is possible to
assign arbitrarily the position of three points, __A__, __B__, and __C__,
on the locus (besides the points __S__ and __S’__); but, these three
points being chosen, the locus is completely determined._