CANONICAL FORMS. 123 
151. Canonical Forms of Type IV. The canonical forms 
of a collineation of type IV may be most easily obtained from 
those of type I or of type I]. The homogeneous normal form 
of type IV, when the invariant figure is in the finite part of 
the plane, is gotten by making k= 1, or k’=1, or k’=kin 
equations (21). Thus making k’ = 1, we get 
90) = kay 
@ =4, (30) 
If we make k’=7 in (22) we get the Cartesian canonical 
form of type IV where the axis of the collineation is the 
y-axis and the vertex is at infinity on the x-axis. Thus 
(are (31) 
If k’=k in (22), the collineation is of type IV with the ver- 
tex at the origin and the axis of the collineation at infinity. 
Thus : 
(saa (32) 
yi=ky. 
If we make t=0 in equations (23), the resulting collinea- 
tion is of type IV with the axis along the y-axis and the ver- 
tex on the w-axis at a distance A’ from the origin. We thus 
get 
=F eile (aus (33) 
152. Canonical Forms of Type V. The canonical forms of 
type V are readily obtained from those of type II, by making 
k=1; for when k=1 in the equations of type II the one- 
dimensional transformation along the line joining (A, B) and 
(A’, B’) is an identical transformation, and type II reduces to 
type V. Making k= 17 in (23) we get 
tat ty ? Uae ¥ (32) 
The vertex is now at the origin and the axis of the elation 
is the y-axis. Making k=1 in (27) we get 
t,=a+t, ¥,=Y. (35) 
