*66. Lines joining four points of the locus to a fifth.* Suppose that the points _S_, _S’_, _B_, _C_, and _D_ are fixed, and that four points, _A_, _A__1_, _A__2_, and _A__3_, are taken on the locus at the intersection with it of any four harmonic rays through _B_. These four harmonic rays give four harmonic points, _L_, _L__1_ etc., on the fixed ray _SD_. These, in turn, project through the fixed point _M_ into four harmonic points, _N_, _N__1_ etc., on the fixed line _DS’_. These last four harmonic points give four harmonic rays _CA_, _CA__1_, _CA__2_, _CA__3_. Therefore the four points _A_ which project to _B_ in four harmonic rays also project to _C_ in four harmonic rays. But _C_ may be any point on the locus, and so we have the very important theorem, _Four points which are on the locus, and which project to a fifth point of the locus in four harmonic rays, project to any point of the locus in four harmonic rays._ *67.* The theorem may also be stated thus: _The locus of points from which, four given points are seen along four harmonic rays is a point-row of the second order through them._ *68.* A further theorem of prime importance also follows: _Any two points on the locus may be taken as the centers of two projective pencils which will generate the locus._ *69. Pascal’s theorem.* The points _A_, _B_, _C_, _D_, _S_, and _S’_ may thus be considered as chosen arbitrarily on the locus, and the following remarkable theorem follows at once. _Given six points, 1, 2, 3, 4, 5, 6, on the point-row of the second order, if we call_ _L the intersection of 12 with 45,_ _M the intersection of 23 with 56,_ _N the intersection of 34 with 61,_ _then __L__, __M__, and __N__ are on a straight line._ [Figure 13] FIG. 13