248 THEORY OF COLLINEATIONS. 
parameters / and t satisfy the relation k =a‘ for a constant 
value of a. Groups of the first class can all be built up out 
of the * two-parameter group G,'(AA’l) in the plane; groups 
of the second class can all be built up out of one-parameter 
groups G,’(AA’l),. 
300. Groups of the First Class. Our list of groups of the 
first class is already known; it consists of the six groups 
named in Theorem 22, viz.: G,/(Al’), G,(Al), G/(A’l/), 
G,’(AA’), G,’(U’), G,’(AA‘L). We wish, however, to verify 
the correctness of this list by means of the normal form of 
type II. Since each of these groups is reducible, this is an 
easy task. In this way we shall also find the & and t-relations 
that enable us to determine the list of groups of the second 
class. 
We proved in Theorem 21 that a necessary and sufficient 
condition for a group of type IJ is that all collineations in.the 
system shall have in common the same invariant lineal ele- 
ment ; and we drew the conclusion that there are three dis- 
tinct varieties of groups of type II with invariant lineal ele- 
ment, one for each lineal element in the figure (AA’l). We 
shall verify this conclusion by means of the normal form of 
type II. ; 
301. The Group G,/(Al). Let T’ and T,’ be so chosen 
that their invariant figures have the point A and the line / in 
common; and let A be the origin and / the y-axis. The nor- 
mal form of 7’ reduces to 
a 
Th oe Yea Fi Rez, 
and y, = 
Gi BR poe Be. 
1+ty+ S Spe 1+ty+ -—t) 
ADT a BEE Al 
Writing 7,’ in the same form as T’ and eliminating x, and y,, 
the resultant is found to be of the same form with the follow- 
ing conditional equations: 
(ice — ele = (2), t. = t= t;.. 
B,! B By! 
= (CaS Th) SS SF = 5) —— (en nd) EC 
(3) ris Pai re Cae) 
tv, = 
(76 ) 
