CANONICAL FORMS. 119 
A collineation of type V may also be regarded as a special 
case of type III when the one-dimensional parabolic trans- 
formations along Al and through A become identical. 
147. Normal Form of Type V.* The normal form of type 
V is found by making k = 7 in that of type II, or by making 
a=0 and h=0 in that of type III. Making k= 1 and the 
usual reductions in equations (15) we get 
Tey) a 10. Deny aL V0 
AID NIS I “TL. Al Al Ie} tl /8 
Cimcon OmaCr \ Cie ye 0) eis 
ec ec! 0 At+e | ec c} O Bt+e¢d | 
i , and y, = ————_—__,__ (20) 
ah Op wl 0) (3 Oil) 
Ae By Ve 87 AN BAP 3 
wk @) Gi dn OO @ 
cel KONE Chicks <6 
$5. Canonical Forms of Collineations. 
The normal forms of plane collineations given in $4 
are perfectly general. The axes of reference have no special 
relations to the invariant figure of the collineation, but the 
normal forms of the equations may often be greatly simplified 
by choosing the axes of reference in special positions with re- 
spect to the invariant figure. The equation of a collineation 
in a very simple form will be designated as a canonical form. 
Often a given collineation can be reduced to two or more very 
simple forms; in such a case we speak of two or more canon- 
ical forms of the same collineation. 
*The normal forms of the equations of the five types of plane collineations were 
first given by Prof. Gabriele Torrelli in the Rendiconti di Circolo Matematico di 
Palermo, Tome viii, pp. 41-54. They were found independently by the writer and 
published by him in the Kansas University Quarterly, vol. vill, pp. 45-66. As pre- 
sented here they differ in minor details from Torrelli’s forms and from my own pre- 
vious forms, 
