122 THEORY OF COLLINEATIONS. 
wish to find the reduced form of the equations of type III 
when the origin is taken at A and the w-axis along the line /. 
The normal form of type III is given in equations (17) Art. 139. 
If the invariant point is at the origin, then A=0 and B=0. 
If the invariant line coincides with the axis of x, then a= 1 
and (>=0; fora and @ are respectively cosine and sine of the 
angle which / makes with the axis of x. Since a’ and 3’ 
are the direction cosines of the normal to s at AB, they be- 
come 0 and 1 respectively for this position of the invariant 
figure, ‘Substitutme A — 0) (B— 0) a— 6 — 0.0 —0) Boa 
in (17) we get 
“+2 at y ae y 
Ti = 1+ta+ (at?+ht)y ’ Ui Faire ae ana (28) 
Equations (28) may also be put in the form 
v1 x 
SS 
ae (28a) 
1 1 x 2 
Baal Coane +ht). 
Equations (28) constitute the canonical form of type III 
when the invariant figure is in the finite part of the plane. 
A second canonical form is obtained by putting the inva- 
riant line at infinity and the invariant point A at the ex- 
tremity of the y-axis. To do this we make equations (28) 
homogeneous by introducing z as follows : 
pv,= “2+ 2at y, 
PY:=Y, 
p2,=2+tx+(at?+ht)y. 
Changing y into z, z into x, and « into y, these equations 
become, when z is made unity, 
v,=xt+ty+(at?+ht), y,=y+2at. (29) 
THEOREM 31. The canonical forms of a collineation of type ITI, 
when the invariant figure is in the finite or infinite part of the 
plane respectively, are 
EN fea aie eee trere ya, SE Oe 
Yl y y Y1 Y¥ 
and 
m=a+tyt+tat+ht, y=y+2at. 
