GEOMETRIC METHOD. 163 
$4. Groups of Type I Determined by Linear 
and Quadratic Relatiolns. 
A. SUBGROUPS OF G,; GEOMETRIC METHOD. 
We shall now attack the problem of the determination of 
all varieties of subgroups of G, of type I which are defined 
by a linear or quadratic set of relations R. Before taking 
up the general analytic method of solving this problem it is 
desirable to consider it synthetically and give, as it were, a 
geometrical forecast of the principal results. The geomet- 
rical method is lacking in rigor but is valuable for the light 
which it throws on the problem. 
197. Number of Collineations of Type I. We have seen 
that every collineation of type I leaves a triangle invariant 
and is further characterized by two independent cross-ratios 
k and k’. Every collineation of type I depends therefore upon 
eight constants, viz., the six coordinates of the three vertices 
of the invariant triangle and these two independent cross- 
ratios. Since each of these constants may assume o’ differ- 
ent values, we see that there are ~‘ collineations of type I in 
the plane. We are also enabled to distinguish two distinct 
kinds of variable parameters, viz., coordinates of invariant 
points and characteristic cross-ratios. 
If we suppose the vertices of the invariant triangle to re- 
main the same but let the cross-ratios k and k’ vary through 
all possible values, we get a system of ~? different collinea- 
tions all having the same invariant triangle. By Theorem 1 
of this chapter these form a group which may be designated 
by G,(AA’A”’), the two parameters being k and k’. 
Since there are ’ triangles in the plane, it follows that 
there are ~* such two-parameter groups as G,(AA’A’’) in the 
plane. Hence, the group of all collineations in the plane G, 
contains ~* two-parameter subgroups G,(AA’A’’). Notwo 
of these two-parameter groups can have a collineation of 
