CANONICAL FORMS. 121 
149. Canonical Forms of Type II. The Cartesian normal 
form of a collineation of type II is 
a i w mH i @ 
ANB it eA PAR Baan 
A! B' 1 kA! A’ B! 1 kB! 
CoC OmtAl tae Cece OmtB- co! 
Le WN my Hh © UTES Q iF © » (15) 
AD Be aes Al Bier 1h 21 | 
AUS BUS Tawi AW Is} i Hip) 
CO C8 OD & ¢ Go | 
Let the invariant point (A, B) be taken for the origin and 
let the axes of x and y be respectively the lines AB and / of 
Bical we SeuuIno ALO hi 0 by — 0 ¢— (0) and ¢o— 1. 
equations (15) reduce to 
eed eet Sa MY ae aA j (23) 
1+ a a+ty 1+ Al x+ty 
This is a convenient form of type II when the invariant 
figure is in the finite part of the plane. If (A’,B’) be the 
point at infinity on the x-axis, then A’= — in (23). Making 
this reduction we get 
kx y 
1+ty’ Chi 1+ty~ 
‘a (24) 
Another and somewhat better canonical form is obtained 
when the point (A’, B’) is at infinity on the y-axis and (A, B) 
at infinity on the z-axis. In homogeneous form (24) may be 
written 
bi) — kan 0; —Y, 02,—2-1-b Y. (25) 
Changing « to y, y to z and z to x, this becomes 
eY,=ky, 02,=2, Ol —£- te (26) 
Passing back to Cartesian coordinates, this becomes 
Clo (21) 
(A, B) is now the point at the extremity of the w-axis and 
(A’, B’) at the extremity of the y-axis. 
150. Canonical Forms of Type III. The invariant figure 
of a collineation of type III is a line / and a point A on/. We 
