_If two projective point-rows, superposed upon the same straight line, have more than two self-corresponding points, they must have an infinite number, and every point corresponds to itself; that is, the two point-rows are not essentially distinct._ If three points, _A_, _B_, and _C_, are self-corresponding, then the harmonic conjugate _D_ of _B_ with respect to _A_ and _C_ must also correspond to itself. For four harmonic points must always correspond to four harmonic points. In the same way the harmonic conjugate of _D_ with respect to _B_ and _C_ must correspond to itself. Combining new points with old in this way, we may obtain as many self-corresponding points as we wish. We show further that every point on the line is the limiting point of a finite or infinite sequence of self-corresponding points. Thus, let a point _P_ lie between _A_ and _B_. Construct now _D_, the fourth harmonic of _C_ with respect to _A_ and _B_. _D_ may coincide with _P_, in which case the sequence is closed; otherwise _P_ lies in the stretch _AD_ or in the stretch _DB_. If it lies in the stretch _DB_, construct the fourth harmonic of _C_ with respect to _D_ and _B_. This point _D’_ may coincide with _P_, in which case, as before, the sequence is closed. If _P_ lies in the stretch _DD’_, we construct the fourth harmonic of _C_ with respect to _DD’_, etc. In each step the region in which _P_ lies is diminished, and the process may be continued until two self-corresponding points are obtained on either side of _P_, and at distances from it arbitrarily small. We now assume, explicitly, the fundamental postulate that the correspondence is _continuous_, that is, that _the distance between two points in one point-row may be made arbitrarily small by sufficiently diminishing the distance between the corresponding points in the other._ Suppose now that _P_ is not a self-corresponding point, but corresponds to a point _P’_ at a fixed distance _d_ from _P_. As noted above, we can find self-corresponding points arbitrarily close to _P_, and it appears, then, that we can take a point _D_ as close to _P_ as we wish, and yet the distance between the corresponding points _D’_ and _P’_ approaches _d_ as a limit, and not zero, which contradicts the postulate of continuity. *50.* It follows also that two projective pencils which have the same center may have no more than two self-corresponding rays, unless the pencils are identical. For if we cut across them by a line, we obtain two projective point-rows superposed on the same straight line, which may have no more than two self-corresponding points. The same considerations apply to two projective axial pencils which have the same axis.