NORMAL FORMS. 115 
order around the triangle. Along one side, e. g. BC, of the 
invariant triangle we have k,=(CBXX,). If k,=1 and B 
and C do not coincide, then X and X, must coincide and every 
point on the line BC is an invariant point, and every line 
through A is an invariant line of the collineation. Thus when 
one of the cross-ratios as k, of type I becomes unity without 
B and C coinciding, T degenerates into S, a transformation 
of type IV. 
142. Cross-ratio the Same on all Lines Through A. Since 
k,=1, we have k, = = ; thus the characteristic cross-ratio 
along CA is the reciprocal of that along AB. Interchanging 
C and A in the formula for k, we get the reciprocal of k, ; 
hence the eross-ratio along AC reckoned from A to C is equal 
to that along AB reckoned from A to B, 7. e., (CAYY,) = 
(BAZZ,)=k,. The cross-ratio of the pencil through C is 
k,=C(BAPP,). But every line through A is now an inva- 
riant line and all the lines through A cut the pencil through 
C in the same cross-ratio k,. Thus we see again that the col- 
lineation S produces one-dimensional transformations along 
each of the invariant lines through A and these one-dimen- 
sional transformations are all characterized by the same 
cross-ratio k. 
143. Type IV a Special Case of Type I. A transformation 
T’ of type II is characterized by a loxodromic one-dimensional 
transformation along its invariant line AB and a parabolic 
one-dimensional transformation along its invariant line Al. 
If t, the characteristic constant of the parabolic transforma- 
tion along A/, be equal to zero, the transformation along 
Al is the identical transformation and every point on Al is 
an invariant point. The invariant figure is now the point B, 
all lines through B, and all points on/. For 7’ it was proved 
that the characteristic cross-ratios along AB and through 
