*72. Determination of the locus.* It is now clear that five points,
arbitrarily chosen in the plane, are sufficient to determine a point-row
of the second order through them. Two of the points may be taken as
centers of two projective pencils, and the three others will determine
three pairs of corresponding rays of the pencils, and therefore all pairs.
If four points of the locus are given, together with the tangent at one of
them, the locus is likewise completely determined. For if the point at
which the tangent is given be taken as the center _S_ of one pencil, and
any other of the points for _S’_, then, besides the two pairs of
corresponding rays determined by the remaining two points, we have one
more pair, consisting of the tangent at _S_ and the ray _SS’_. Similarly,
the curve is determined by three points and the tangents at two of them.




*73. Circles and conics as point-rows of the second order.* It is not
difficult to see that a circle is a point-row of the second order. Indeed,
take any point _S_ on the circle and draw four harmonic rays through it.
They will cut the circle in four points, which will project to any other
point of the curve in four harmonic rays; for, by the theorem concerning
the angles inscribed in a circle, the angles involved in the second set of
four lines are the same as those in the first set. If, moreover, we
project the figure to any point in space, we shall get a cone, standing on
a circular base, generated by two projective axial pencils which are the
projections of the pencils at _S_ and _S’_. Cut across, now, by any plane,
and we get a conic section which is thus exhibited as the locus of
intersection of two projective pencils. It thus appears that a conic