114 THEORY OF COLLINEATIONS. 
In the special case that the point O lies on 1, the collineation 
of type IV reduces to one of type V. Collineations of these 
two types are perspective collineations. Fixing our attention 
on the fundamental invariant figure we see that every inva- 
riant line in the plane except / has on it two invariant points, 
O and its point of intersection with /; also every pencil hav- 
ing its vertex A on/ is an invariant pencil in the plane, and 
has in it two invariant rays, /andtheline AO. The effect of 
a perspective collineation of type IV is to move a point P along 
the invariant line OPP, to P,. Thus we have on each of the 
invariant lines through O a one-dimensional transformation 
with two invariant points O and A. Likewise in each of the 
invariant pencils with vertex on / we have a one-dimensional 
transformation of the same kind. A one-dimensional trans- 
formation is characterized by the constant cross-ratio of the 
invariant elements and every pair of corresponding elements. 
Thus along the line AO we have (AOPP,)=k. Let A, be 
another point of 1; in the pencil with vertex at A, we have 
the cross-ratio A,(AOPP,)=k. Since every line through O 
cuts this pencil in a range having the same cross-ratio k, it 
follows that the one-dimensional transformations on all lines 
through O are characterized by the same constant k; also 
the one-dimensional transformations in all pencils with ver- 
tices on / are characterized by the same value of /. 
THEOREM 27. A perspective collineation S of type IV is com- 
pletely characterized by its fundamental invariant figure and a char- 
acteristic cross-ratio #. The one-dimensional transformations along 
all invariant lines except / and in all invariant pencils except O of S 
are characterized by the same cross-ratio /. 
141. Type IV a Special Case of Type I. A perspective col- 
lineation S of type IV may be regarded as a special case of 
typeI. We proved for type I that k,k,k, = 1, where these 
quantities are the characteristic cross-ratios taken in the same 
