GROUPS OF TYPE I. 241 
lowing varieties of subgroups, G,(AA’),-.,  G,(Il’),-2, 
G,(AA'l’),_... We must examine the three remaining con- 
figurations, viz.: (AlK), (ALS) and (ALN), and see if these 
are invariant under certain groups. 
291. The Group G,( AIK). Let us take any conic K of the 
system touching / at A. We can construct ~’ triangles, Fig. 
29, having one vertex at A, another at A’ on l, and a third 
Fic. 29. 
vertex A” on the conic, such that AA’ and A’ A” are tangents 
to K and AA” is the chord of contact. Belonging to each of 
these triangles is a one-parameter group whose path-curves 
are conics all touching AA’ at A. Each of these groups has 
K among its invariant path-curves. Hence these ~* collinea- 
tions leaving invariant the configuration (A/K) form a group 
of two parameters, G,(AlK), the two parameters being the 
cross-ratios k along | and the position of the point A” on K. 
The group G,( A/),-. contains ~* such subgroups, one for each 
conic in the system touching / at A. 
THEOREM 55. All collineations leaving invariant the configura- 
tion consisting of a conic, one of its tangents, and the point of con- 
tact, form a two-parameter group G, (Al). 
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