*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 separated from each other by _A_ and _C_. *39. Harmonic conjugate of the point at infinity.* It is natural to inquire what position of _B_ corresponds to the infinitely distant position of _D_. We have proved (§ 27) that the particular quadrangle _K_, _L_, _M_, _N_ employed is of no consequence. We shall therefore avail ourselves of one that lends itself most readily to the solution of the problem. We choose the point _L_ so that the triangle _ALC_ is isosceles (Fig. 7). Since _D_ is supposed to be at infinity, the line _KM_ is parallel to _AC_. Therefore the triangles _KAC_ and _MAC_ are equal, and the triangle _ANC_ is also isosceles. The triangles _CNL_ and _ANL_ are therefore equal, and the line _LB_ bisects the angle _ALC_. _B_ is therefore the middle point of _AC_, and we have the theorem _The harmonic conjugate of the middle point of __AC__ is at infinity._ [Figure 7] FIG. 7