GROUPS OF TyPES II, III. 247 
long to the same net V; they havea subgroup G:(AA/‘/),-. in 
common when S and 8’ belong to different nets. Kach group 
G,( ALS) contains o/ subgroups G.( Al A), one for each conic in 8. 
298. The Group G,(K). Among the groups of the third 
class must be included the irreducible group G,(K). It con- 
tains, Art. 204, the subgroups G,( AIK) and G,(AA‘A’k), 
both of type I and third class. These subgroups have both 
been found again by other methods. 
We have thus found eleven varieties of groups of the third 
class. Five of these eleven groups, viz.: G,(AAA’),, 
G,(AA'l),, G,(AA’),, G,(U), and G,(Al),, where r= —1, 2 
or 4, are only special cases of groups of the second class and 
not sufficiently peculiar to warrant listing them separately. 
On the other hand the six groups G,( AIK), G,(K), G,( AIS), 
G,(Al),__,, G,(A),--, and G,(l),__, are essentially distinct 
from the groups listed in the second class. 
THEOREM 56. There are six distinct varieties of groups of tpye 
li, Ula GSS, wine (n(4WiO), “Cal O\n Ea(ZUS)), Ga(OMW)eer 
G;(A),=-1 and G;(7),—-1. 
B. GROUPS OF TYPES II AND III. 
We pass now to the problem of determining a complete list 
of the varieties of groups of type II. We have already found 
in $5 a complete list of the different varieties of groups of 
type II defined by sets of linear relations on the elements of 
M with the additional condition D=0. In § 7 we discussed 
the fundamental group of type II, G.’(AA’l), and its sub- 
groups G,/(AA‘l), and found, Art. 211, that no collineation 
of type II can leave a conic invariant; hence there are no 
groups of type II defined by quadratic relations on the ele- 
ments of WM. 
299. Two Classes of Groups of Type IT. Weshall find two 
distinct classes of groups of type II, viz.: (1) those in which 
the two parameters k and ¢ are independent of each other ; 
and (2) groups made up of collineations in each of which the 
