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._