*33. Four harmonic lines.* We are now able to extend the notion of harmonic elements to pencils of rays, and indeed to axial pencils. For if we define _four harmonic rays_ as four rays which pass through a point and which pass one through each of four harmonic points, we have the theorem _Four harmonic lines are cut by any transversal in four harmonic points._ *34. Four harmonic planes.* We also define _four harmonic planes_ as four planes through a line which pass one through each of four harmonic points, and we may show that _Four harmonic planes are cut by any plane not passing through their common line in four harmonic lines, and also by any line in four harmonic points._ For let the planes α, β, γ, δ, which all pass through the line _g_, pass also through the four harmonic points _A_, _B_, _C_, _D_, so that α passes through _A_, etc. Then it is clear that any plane π 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 π, and they pass through the given harmonic points _A_, _B_, _C_, _D_. Any other plane σ cuts _g_ in a point _S_ and cuts α, β, γ, δ in four lines that meet π 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 α, β, γ, δ 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