120 THEORY OF COLLINEATIONS. 
148. Canonical Forms of Type I. Let a collineation T of 
type I be given in the homogeneous normal form 
o/h co Ok | 12 ) 2 
A Binh AS Al PAMAY 98s Gn cB 
A Bo kA |> PYr=\ar B CO kB 
All Br Cc" kk! A” All Bl Cc" k/ B’ 
x y 2 0 
ALOR aCIC 
A’ BC! ik 
A!’ Bl Cc" k'C” 
If the invariant triangle be taken as the triangle of refer- 
ence these equations are greatly reduced. Let the coordi- 
nates of the vertices of the invariant triangle be (0,0, C), 
(A’, 0,0), (0,B”,0). Substituting these values in equations 
(12) we get 
ot, — AB’ Ca. py — it AUB Cy. 02, — Alb Cz. 
Setting p =p’ A’B’’C where p’ is a new proportionality fac- 
tor we have 
? 
px, = 
pz, = (12) 
pla = kx, 
( ply = ky, (21) 
Pa =z. 
Equations (21) constitute the homogeneous canonical form 
of the collineation T, the triangle of reference being the in- 
variant triangle. 
We can pass from homogeneous coordinates to Cartesian 
coordinates by dividing the first and second equations of (12) 
by the third, and then making the z’s and C’sall unity. Also 
we can get the canonical form of the collineation in Cartesian 
coordinates by making the same changes in equations (21). 
Thus we get 
( ; z ey: (22) 
in which the invariant triangle is made up of the coordinate 
axes and the line at infinity. The constant cross-ratios along 
the a- and y-axes are respectively k and k’. 
THEOREM 30. The homogeneous and Cartesian canonical forms 
of a collineation of type I are respectively 
Pm =k, ce 
(Fina and ea? 
PA =2, 
