168 THEORY OF COLLINEATIONS. 
also invariant points and their join is another invariant line. 
The invariant figure consists therefore of a triangle (AA’A”) 
and a conic K having two of the sides as tangents and the 
third side as chord of contact. When K and / are both inva- 
riant and / does not touch K, then the two points of intersec- 
tion of | and K are also invariant ; hence the tangents at these 
two points are also invariant and their intersection is another 
invariant point. Thus we have the same figure as before and 
there is but one such variety of group, G,(AA’A’’K). 
When A is on K, the tangent / at A is also invariant; 
this case determines a two-parameter group G,(A/K). 
THEOREM 12. ‘The General Projective Group Gs has at least 
eleven varieties of subgroups of type I; these are G; (A), Gs (1), 
G;(Al), G,(AA’), G,(W), G,(AW), G:(AA'l), | Ge (AAA”); 
Gs; (K), Ge (ALK), G, (AA’A”K). 
B. GROUPS DEFINED BY LINEAR AND QUADRATIC RELATIONS. 
We shall now return to the analytic point of view and apply 
the results reached in Theorems 9 and 11 to the solution of the 
problem of finding all varieties of subgroups of G, of type I 
defined by linear and quadratic relations on the elements of 
the matrix M. 
205. Linear Functions of the Elements of the Rows. Let 
the complete family of automorphic forms be linear functions 
in the elements of the rows of M. Our set of relations FR are 
of the form 
la,t+mb\+ncq=l, 
R, -  las+mbs+neo=™m, (22) 
la,+mb3;+7c;=Nn, 
and these define the group given in the illustrative example of 
Art. 191. The form of these relations shows us at once that 
the ratios of three numbers /, m, 7, are absolutely invariant 
under all the transformations of the group. 
The geometric invariant of the group is also evident from 
the form of equations (22). These show that the point whose 
coordinates are proportioned to 1, m, 7, is invariant under all 
