The three pairs of opposite sides of a hexagon inscribed in a point-row of the second order meet in three points on a line.

71.  Harmonic points on a point-row of the second order. Before proceeding to develop the consequences of this theorem, we note another result of the utmost importance for the higher developments of pure geometry, which follows from the fact that if four points on the locus project to a fifth in four harmonic rays, they will project to any point of the locus in four harmonic rays.  It is natural to speak of four such points as four harmonic points on the locus, and to use this notion to define projective correspondence between point-rows of the second order, or between a point-row of the second order and any fundamental form of the first order.  Thus, in particular, the point-row of the second order, {~GREEK SMALL LETTER SIGMA~}, is said to be perspectively related to the pencil S when every ray on S goes through the point on {~GREEK SMALL LETTER SIGMA~} which corresponds to it.

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 section is a point-row of the second order.  It will later appear that a point-row of the second order is a conic section.  In the future, therefore, we shall refer to a point-row of the second order as a conic.