NORMAL FORMS. 99 
of the points on a line, which leaves two points of the line in- 
variant, is characterized by the constant cross-ratio of the in- 
variant points and any pair of corresponding points (Art. 15). 
Let k, be the characteristic cross-ratio of the one-dimen- 
sional transformation along the line a. In like manner we 
have transformations of one dimension along each of the in- 
variant lines 6 and c. We shall call their characteristic cross- 
ratios k, and k, respectively. In reckoning these cross-ratios 
the points will be taken always in the same order around the 
triangle. Thus we see that every collineation of type I in the 
plane determines three characteristic cross-ratios along the 
three invariant lines. It is also evident that the pencil of 
lines through the vertex A of the invariant triangle is trans- 
formed into itself in such a way that the rays AB and AC are 
invariant rays of the transformation. Also the cross-ratio of 
the invariant rays and any pair of corresponding rays of the 
pencil is constant for all pairs of corresponding rays; this 
cross-ratio is equal to k,, the characteristic cross-ratio along 
the side a opposite A. Similar considerations apply to the 
pencils of rays through the invariant points Band C. We 
shall now proceed to show that these three cross-ratios are not 
independent, but are connected by a very simple relation. 
nies aye 
