*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 projective to the point-row _B_.
Project the point-row _K_ from _M_ and we get a point-row _D_ on _AC_
again, which is projective to the point-row _B_. For every point _B_ we
have thus one and only one point _D_, and conversely. In other words, we
have set up a one-to-one correspondence between the points of a single
point-row, which is also a projective correspondence because four harmonic
points _B_ correspond to four harmonic points _D_. We may note also that
the correspondence is here characterized by a feature which does not
always appear in projective correspondences: namely, the same process that
carries one from _B_ to _D_ will carry one back from _D_ to _B_ again.
This special property will receive further study in the chapter on
Involution.




*38.* It is seen that as _B_ approaches _A_, _D_ also approaches _A_. As
_B_ moves from _A_ toward _C_, _D_ moves from _A_ in the opposite
direction, passing through the point at infinity on the line _AC_, and
returns on the other side to meet _B_ at _C_ again. In other words, as _B_
traverses _AC_, _D_ traverses the rest of the line from _A_ to _C_ through
infinity. In all positions of _B_, except at _A_ or _C_, _B_ and _D_ are