*Project Gutenberg*. Public domain.

*6.* A line is drawn cutting the sides of a
triangle *ABC* in the points *A’*,
*B’*, *C’* the point *A’*
lying on the side *BC*, *etc*. The harmonic
conjugate of *A’* with respect to *B*
and *C* is then constructed and called *A"*.
Similarly, *B"* and *C"* are constructed.
Show that *A"B"C"* lie on a straight line.
Find other sets of three points on a line in the figure.
Find also sets of three lines through a point.

CHAPTER III — COMBINATION OF TWO PROJECTIVELY RELATED FUNDAMENTAL FORMS

[Figure 9]

FIG. 9

*47. Superposed fundamental forms. Self-corresponding
elements.* We have seen (§ 37) that two projective
point-rows may be superposed upon the same straight
line. This happens, for example, when two pencils
which are projective to each other are cut across
by a straight line. It is also possible for two
projective pencils to have the same center. This
happens, for example, when two projective point-rows
are projected to the same point. Similarly, two
projective axial pencils may have the same axis.
We examine now the possibility of two forms related
in this way, having an element or elements that correspond
to themselves. We have seen, indeed, that if
*B* and *D* are harmonic conjugates with
respect to *A* and *C*, then the point-row
described by *B* is projective to the point-row
described by *D*, and that *A* and *C*
are self-corresponding points. Consider more
generally the case of two pencils perspective to each
other with axis of perspectivity *u’* (Fig.
9). Cut across them by a line *u*.
We get thus two projective point-rows superposed on
the same line *u*, and a moment’s reflection
serves to show that the point *N* of intersection
*u* and *u’* corresponds to itself
in the two point-rows. Also, the point *M*,
where *u* intersects the line joining the centers
of the two pencils, is seen to correspond to itself.
It is thus possible for two projective point-rows,
superposed upon the same line, to have two self-corresponding
points. Clearly *M* and *N* may fall
together if the line joining the centers of the pencils
happens to pass through the point of intersection
of the lines *u* and *u’*.

[Figure 10]

FIG. 10

*48.* We may also give an illustration of a
case where two superposed projective point-rows have
no self-corresponding points at all. Thus we
may take two lines revolving about a fixed point *S*
and always making the same angle a with each other
(Fig. 10). They will cut out on any line *u*
in the plane two point-rows which are easily seen to
be projective. For, given any four rays *SP*
which are harmonic, the four corresponding rays *SP’*
must also be harmonic, since they make the same angles
with each other. Four harmonic points *P*
correspond, therefore, to four harmonic points *P’*.
It is clear, however, that no point *P* can coincide
with its corresponding point *P’*, for
in that case the lines *PS* and *P’S*
would coincide, which is impossible if the angle between
them is to be constant.