GROUPS OF TYPES. Der 
$9. Groups of Types I, II and III. 
In the last $ we found the complete list of varieties of sub- 
groups of G, that contain only collineations of types IV and 
V. In the present $ we take up the problem of finding the 
complete list for types I, II and III. In § 4 we found all va- 
rieties of subgroups of G, that are defined by linear and quad- 
ratic relations on the elements of M; and it was there 
expressly assumed that the elements of M were not subject 
to any of the restrictions that would make the characteristic 
equations of M have multiple roots. Hence the groups of the 
lists given in Theorem 12 are all of type I. In $5 we found 
all varieties of subgroups of types II and III that are defined 
by linear relations in the elements of M. But it does not fol- 
low that these are the complete lists of groups of these types, 
for each of these groups may have subgroups defined by some 
additional condition and still of their respective types. 
For example we found in § 7 that the group G,( AA’ A’’) has 
«o/’ subgroups of type I each characterized by a constant value 
of r. We shall find that each of the groups listed in Theorem 
12, except G,(AA’A” K), has one or more subgroups of type 
I; and that each group listed in Theorem 22 has «1 subgroups 
of type II. 
A. GROUPS OF TYPE I. 
285. Three Classes of Groups of Type I. We wish to 
make a rational classification of the groups of type I. In so 
doing we shall find three distinct classes of such groups, viz.: 
(1) Those containing collineations in which the cross-ratio 
parameters k and k’ are independent of each other; groups 
of this kind are made up of two-parameter groups of the kind 
G,(AA’A”); (2) another class containing collineations in 
which the cross-ratio parameters satisfy the relation hk’ = k’ 
for a constant value of 7; the groups of this class are made 
up of one-parameter groups of the kind G,(AA’A’’),; (3) a 
class of groups for which r = — 1, 2, 1/2, these are made up 
