252 THEORY OF COLLINEATIONS. 
and G,’(A’l’),.. The group G,'( Al’) does not break up into 
subgroups of the second class. 
THEOREM 58. There are five varieties of groups of the second 
class of type II, viz.: G( 44/l'),, Ge ( AA’), Gs! (ll), Gy (Al), and 
Gs ( Ay ers 
308. Groups of Type III. We found in Art. 227, that 
there is only one variety of group of type III defined by linear 
relations on the elements of M and D=0 and D’=0. This 
is the group G,’’( Al) which is the fundamental group of the 
type. This fundamental group was investigated in § 7, and it 
was found to contain two varieties of subgroups, viz.: 
G.''( ALIN) and G,’’(AlS). Hence these three varieties of 
groups of type III form the complete list. 
THEOREM 59. There are three varieties of groups of type ITI, 
viz.: G,’ (Al), GJ’( AL), Gy’ (ALS). 
C. TABLE OF GROUPS OF COLLINEATIONS OF THE PLANE. 
In this table, the collineation groups of the plane are classi- 
fied according to the five types of collineations. Each group 
is designated by an appropriate symbol. The self-dualistic 
groups are enclosed in brackets, thus: | G,(AA‘A’’) | ; a pair 
of dualistic groups are bracketed together, thus: } ee! r. 
Similar tables are given by Lie, ‘‘Continuierliche Gruppen,” 
pp. 288-291, and by Meyer, ‘‘Chicago Congress Papers,” pp. 
188-190; but in these tables of Lie and Meyer the notation 
and classification is entirely different. The numbers on the 
right refer to Lie’s and Meyer’s tables respectively. 
