246 THEORY OF COLLINEATIONS. 
which do not belong to the same net N. A given conic K of 
the pencil S is cut by each conic of S’ in two points other than 
A. When these two points of intersection of K and K’ co- 
incide, the conics K and K’ are in contact. Each conic of S 
has a second contact with one, and only one, conic of S’. The 
locus of the points of contact of corresponding conics of S and 
S’is a line l’ through A. Therefore, we see that the two 
three-parameter groups G,(AlS) and G,(AIlS’) contain the 
same two-parameter group @,(AA/l’),_.. 
On the other hand, if the two pencils S and S’ belong to 
the same net N of conics having contact of the second order 
at A, the conics from S and S’ cannot have another contact, 
and hence two such groups G,(AlS) and G,(AIS’) have no 
collineations in common. 
In a certain net N there are ~’ pencils S, S’, S”, etce.; 
each of these pencils is the invariant pencil of a group 
G,(AlS). ©’ such groups, no two of which contain a col- 
lineation of type I in common, include all collineations of this 
type in G,(Al),_.. It is easy to see that the «’ sub- 
groups G,(AA’A”),_, inG,(Al),_. are all included in the 
co! groups G,(AlS) of the net N. If it be true that all 
eollineations of type I in G,(Al),-. are included in the 
o' groups G,(AlS) of the net N, then all collineations of 
type I ina group G,(AIS’), where S’ is a pencil of conics 
not included in N, are to be found in the ~'’ groups G,(AI/S) 
of thenet N. In fact, if we take G,( A/S’) in turn with each 
of the groups G,(AUS) of the net, we see that G,(AIS’) 
has a two-parameter subgroup G,(ABl’),_. in common with 
each group G,(AlS) of the net; the common subgroup 
G,(ABl’),_, is different for each of the groups G,(AIS) 
of the net. In this way it can be shown without difficulty 
that every collineation of type I in G,(AJ/S’) is also to be 
found in the net of groups G,(AJ/S). 
THEOREM 55. The group G@,(A/),-2 contains o? subgroups 
G,;(ALS); two subgroups G;( A/S) and G;( A/S’) contain no col- 
lineation of type Lin common when the two peneils are S and S’ be- 
