GROUPS OF TYPE I. 245 
. 4s +4 AU eva 1 a 
If in addition to the condition ;, =~, we also put 7, - Al 
: AU 2a 1 a : 
(or, more generally, if we put %,+=%, and p,=4,+C) in 
the three equations of condition of G,(AlS),_, these three 
will reduce to two, viz.: k, = kk, and = z= = ap (he al) 
We thus obtain a two-parameter subgroup OF 1Ga(AUS \e=.. 
which we shall call G,(ALK),_,. The equation of the conic 
K is obtained by eliminating A’ from the two conditions 
ay th =e and 7,=4-+¢. We find that the point (A”, B”) 
always lies on the axis a? + 2hay+(h?—4aC)y? = say, 
where h, a and C are fixed numbers. The group G,(AIS) 
contains ~/ subgroups G,(AlK), one for each conic in S. 
296. Collineations Common to G,(ALK) and G,(AIK’). 
Let K and K’ be two conics of the system touching / at A. 
We wish to determine whether the two groups G,(AIK) and 
G,(AIK’) have any collineations in common. If K and K’ 
belong to the same pencil S, the two groups have in common 
the subgroup G,’’ (AIS) of type III. If the econies K and K’ 
belong to different pencils S and S’ of the same net N, they 
have three points in common at A and intersect in only one 
other common point. In this case the groups have no col- 
lineations in common. _ If the conics K and K’ do not belong 
to the same net, they may have double contact at A and A”, 
and then the two groups have the subgroup G,(AA’A’’K) in 
common since K and K’ both belong to the same pencil K,,. 
If the conics K and K’ intersect in two points other than A, 
then the groups have no collineations in common. 
297. Collineutions Common to G,(AlLS) and G,(AIS’). 
Kach pencil S is the invariant figure of a group G,(AIS), 
hence, there are ~* such groups all contained within the group 
G,(Al),_... But since G,(A1),_, contains ~‘ collineations and 
co* subgroups G,(AlS), it follows that two such subgroups 
as G,(AlS) and G,(AIlS’) must contain certain collineations 
incommon. Let us consider two pencils of conics S and S’ 
