FUNDAMENTAL GROUPS. 205 
242. Reduced Normal Form of Other Subgroups of G,. 
Let T and T, have in common the invariant point A and the 
invariant line 1; the group is then G,(Al), and T, will also 
leave A and / invariant. Making A=B=B’=0and A,=B,= 
B,’=0 in equations I-IX, we find A,= B,= B,/=0. Equa- 
tions V, IX, and X reduce respectively to 
Ke CAI. Co ACB C3kAy Bees. Ve 
ALB CL AUBIN CRA Be Cr: IX’. 
kel Al Bees — ik AV BC kek APB C,. X’. 
Dividing V’ by IX’ we get k,’=k’k,’; dividing X’ by V’ we 
get k, =kk,. 
In like manner we can find the reduced normal forms of 
each of the remaining groups in the list of Art. 209 and 216, 
and verify in this way their k-relations as given in Art. 234, 
ele: 
We have thus verified, by means of the reduced normal 
forms of the reducible subgroups of G,, the theory of the k- 
relations stated in Theorems 27 and 28. The group G,(K) 
being irreducible the theory of its k-relations can not be veri- 
fied in this way; it will, however, be amply confirmed later. 
$Z. Fundamental Groups, One-Parameter 
Groups and their Path-Curves. 
243. We shall now consider for each type of collineation a 
certain group which we shall call the fundamental group of 
that type. We have shown, Chapter II, Theorem 18, that 
each type of collineation has its own characteristic invariant 
figure. We have also shown that all collineations of a given 
type which leave invariant this characteristic figure form a 
group, the so-called fundamental group for that type. The 
fundamental group of type I is the group G,(AA’A”’), Art. 
197, whose invariant figure is the triangle (AA’A’’). The 
fundamental group of type II is G,’(AA‘l), Art. 226; that of 
