TA~}, which all pass through the line _g_, pass
also through the four harmonic points _A_, _B_, _C_, _D_, so that {~GREEK SMALL LETTER ALPHA~} passes
through _A_, etc. Then it is clear that any plane {~GREEK SMALL LETTER PI~} through _A_, _B_, _C_,
_D_ will cut out four harmonic lines from the four planes, for they are
lines through the intersection _P_ of _g_ with the plane {~GREEK SMALL LETTER PI~}, and they pass
through the given harmonic points _A_, _B_, _C_, _D_. Any other plane {~GREEK SMALL LETTER SIGMA~}
cuts _g_ in a point _S_ and cuts {~GREEK SMALL LETTER ALPHA~}, {~GREEK SMALL LETTER BETA~}, {~GREEK SMALL LETTER GAMMA~}, {~GREEK SMALL LETTER DELTA~} in four lines that meet {~GREEK SMALL LETTER PI~} in
four points _A'_, _B'_, _C'_, _D'_ lying on _PA_, _PB_, _PC_, and _PD_
respectively, and are thus four harmonic hues. Further, any ray cuts {~GREEK SMALL LETTER ALPHA~}, {~GREEK SMALL LETTER BETA~},
{~GREEK SMALL LETTER GAMMA~}, {~GREEK SMALL LETTER DELTA~} in four harmonic points, since any plane through the ray gives four
harmonic lines of intersection.

*35.* These results may be put together as follows:

_Given any two assemblages of points, rays, or planes, perspectively
related to each other, four harmonic elements of one must correspond to
four elements of the other which are likewise harmonic._

If, now, two forms are perspectively related to a third, any four harmonic
elements of one must correspond to four harmonic elements in the other. We
take this as our definition of projective correspondence, and say:

*36. Definition of projectivity.* _Two fundamental forms are protectively
related to each other when a one-to-one correspondence exists between the
elements of the two and when four harmonic elements of one correspond to
four harmonic elements of the other._

[Figure 6]

FIG. 6

*37. Correspondence between harmonic conjugates.* Given four harmonic
points, _A_, _B_, _C_, _D_; if we fix _A_ and _C_, then _B_ and _D_ vary
together in a way that should be thoroughly understood. To get a clear
conception of their relative motion we may fix the points _L_ and _M_ of
the quadrangle _K_, _L_, _M_, _N_ (Fig. 6). Then, as _B_ describes the
point-row _AC_, the point _N_ describes the point-row _AM_ perspective to
it. Projecting _N_ again from _C_, we get a point-row _K_ on _AL_
perspective to the point-row _N_ and thus proj