QUADRATIC RELATIONS. AEE 
The set of relations of the kind R, secures the invariance of 
the line 
(Ul! + pm! + qn’) «+ (pl! + mm! +rn') y+ (ql’/+rm'/+nn')z=0. 
But this line is no other than the polar of the point (1’, m’, n’) 
with respect to the conic, 
la? + my? + nz? + 2pxy + 2quze+ 2ryz = 0. 
In like manner the two sets R, and R, imply a linear set of 
the kind R, which secures the invariance of the pole of the 
line l’ with respect to K. Hence the two groups G,(AK) 
and G,(lK) are of the same variety. The geometric inva- 
riant of the group is the triangle (A A’A’’) and the conic K, 
related to it as shown in Art. 204; the group is designated 
by G,(AA/A"K). 
If the two sets of relations R, and RF, are so related to 
each other that the following condition exists, 
Ul/2 + mm/2+nn/2+ 2pl’/m!'+ 2ql'’n'’+2rm'n'=0, 
then there are two independent parameters and the group 
defined is a two-parameter group. The invariant point is on 
the conic K and the invariant line touches K at the invariant 
point. The group is designated by G,( AIK). 
213. Groups Defined by Other Sets of Relations. When 
we examine the complete cubic family of automorphic forms 
we see at once that the group defined by such a family is not 
a continuous group; for a set of relations R derived from a 
cubic family imposes nine conditions on eight independent 
elements of M and these conditions can be satisfied by only a 
finite number of sets of values of these elements. Hence in our 
study of continuous groups of plane collineations we can not 
make use of complete families of automorphic forms of degree 
higher than two. 
We can not impose simultaneously two independent sets of 
quadratic relations upon the elements of M and thereby ob- 
tain a continuous group; for in such a case the number of 
conditions (ten) again exceeds the number of independent 
elements (eight) of M, and if a group is thus obtained it 
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