ANALYTIC THEORY. 69 
of points on g is projectively related to the range on g, and 
the cross-ratio of any four points on g is the same as that of 
the four corresponding points on g,. So also for the two pen- 
cils of lines through P and P,, corresponding points of the 
transformation; the cross-ratio of four lines of the pencil 
through P is the same as that of their four corresponding 
lines through P,. 
THEOREM 3. The cross-ratio of any four collinear points or 
concurrent lines of the plane is unaltered by a collineation. 
82. The Line at Infinity. The collineation expressed by 
equations (1) transforms a point (x, y) into (x,, y,). If (a, y) 
be a point on the line a”’x+ b’y+c’=0, then (a, y,) isa 
point at infinity; for a’«+b’y-+c” is the common denom- 
inator of the two fractions in equations (1) and vanishes for 
all points (x,y) which lie on the line a’x%+b’y+c"”=0. 
Every point on this line is transformed into a point at infinity; 
hence the line, a’a-+b6’y-+c”’=0, is transformed into the 
line at infinity in the plane.* 
The line ax+by+c=0 is transformed into the axis 
“,=0, and the line a/x+6’y+c’=0 is transformed into 
the axis y,=0. Hence, the triangle formed by the three 
lines, aw + by+c=0, a/x+b/y+c'=0, a’4+b"y+ec"=0, 
is transformed into the triangle formed by the coordinate 
axes and the line at infinity. That the first three lines actu- 
ally form a triangle is secured by the condition A + 0. 
83. Invariant Points of a Collineation. If the collinea- 
tion expressed by equations (1) transforms any point (a, y) 
into itself, then x, and y, become «and y. The coordinates 
of such a point may be found by solving the equations 
ax+by+e aa+bly+e! 
Clearing of fractions these become 
ax? + (c’ —a)x —ce+(b’a —b)y=0), 
and by? + (a’«+c” — b')y—(a’«+c') =0. 
*For further discussion of the line at infinity see art. 88, 
