ANALYTIC THEORY. 71 
line. It is to be understood that not every point on c is 
transformed into itself; this is true only of A and B. But 
every point onc, except A and B, is transformed into some 
other point alsoonc. Thus the points on ¢ undergo a one- 
dimensional projective transformation, A and B being the 
invariant points of the transformation. The same holds 
for the other sides, a and 0, of the invariant triangle. In like 
manner the pencils through the invariant points A, B, C un- 
dergo one-dimensional projective transformations. 
THEOREM 5. A plane collineation of the most general kind 
leaves a triangle invariant, and produces a one-dimensional project- 
ive transformation along each of the invariant lines and through 
each of the invariant points. 
85. The Identical Collineation. The question at once pre- 
sents itself whether there exist collineations in the plane 
which leave invariant more than three points or more than 
three lines. Suppose we have a collineation T leaving inva- 
riant the triangle ABC and a fourth point D of the plane, 
such that no three of the four points are in a line (Fig. 10). 
Fic. 10. 
The lines AD, BD, and CD, are invariant lines of the trans- 
formation 7, since they are the joins of two invariant 
points. The pencil of lines through one of these invariant 
points, for example A, undergoes a one-dimensional trans- 
formation, which leaves three of its lines invariant: such a 
one-dimensional transformation is an identical transformation 
and leaves all lines through A invariant (Chapter I, Theorem 
