240 THEORY OF COLLINEATIONS. 
We shall denote the configuration, see Fig. 26, by Sen A"K) 
and the group is G,(AA‘A’K). 
289. Subgroups of G,(A, 1”), G,(A), G,(l) when r = — 1. 
The three groups G,(A), G,(1) and G,(A,1) have the single 
k-relation kk,’ =kk'k,k,’, but not the double k-relations. 
Hence these groups do not break up into groups of the second 
class for all values of ry. But for r = 1 we do find a subgroup 
in each of the above groups of the first class. Let k’=k" 
and k,'’=k,; in the relation k,k,'!=kk,k'k,’, then, we have 
keeles’ — (hic, "> 4 TE mows — sl) othenwk. kan oes Ene 
two cross-ratios in the resultant have the same relation to 
each other as in the components. Hence for r = — 1 we find 
the three groups G,(A,l’’),__,, G,;(A),-_,, and G,(l),__,. 
THEOREM 52. The eroups G,(4,/”), G.(4) and G,_(2) each 
contain one and only one subgroup for a constant value of 7, viz.: 
WALL MN 7) p—lig Gisl (CAD) her) Gis (dy) pene 
290. The Structure of G,(Al)r=2. The group G,(Al) 
y = 2 breaks up into subgroups in several different ways, and 
accordingly its structure is peculiar and worthy of attention. 
There are °* distinct conics touching /at A. This system 
of conics is composed of 1 nets N, such that each net con- 
tains ©” conics having contact of the second order at A, and 
hence all conics of a net N have a common circle of curvature 
at A. Each of the circles touching / at A is the circle of cur- 
vature of such a net. 
Each net N is composed of 1 pencils S such that the conics 
of each pencil have contact of the third order at A. The 
polar of the point at infinity on / with respect to the conics of 
the pencil S is a line l’ through A, the line of centers of S. 
Each line through A is the line of centers of one of the pen- 
cils of the net N. Since there are /’ nets N, each line 
through A is the line of centers of ~’ pencils S. The system 
of conics touching / at A is, therefore, composed of * 
pencils S. 
Now we know that the group G,(Al),_, contains the fol- 
