FUNDAMENTAL GROUPS. 209 
where k and k’ are the cross-ratios of the one-dimensional 
transformation along the sides AA’ and AA”’ respectively. 
These equations contain two independent parameters k and k’. 
Giving to k and k’ all possible values, we get a double infinity 
of collineations all leaving the same triangle, 4A’A”’, inva- 
riant. We wish to show that these ~* collineations form a 
group. 
Let us take any two collineations 7 and T, from this set 
and form their resultant 7T,. It is evident, geometrically, 
that T, belongs to the same set; for if T and T, each leave 
the triangle AA’A” invariant, their resultant T, must also 
leave it invariant and hence must belong to the same set. 
This may also be shown analytically. Let the equations of 7 
and T’,, be respectively: 
$a =ka $1 %2 =k, 2, 
7 , 
LP: m="y, and Ts swe=kim, (52’) 
Eliminating w,, y,, 2, we get T,. Thus 
So%e = koa, 
/ hege S242 = Kay, wherevie— ic, and ik kh! (52!) 
Since 7, is of the same form as T and T,, it belongs to the 
same set and the first group property is established. 
The inverse of T is 
Pla=kin, L. 
1 bie PPy=k*y, (52 ) 
Cae zie 
This collineation is also in the set and hence the second group 
property is established. Thus the set forms a two-parameter 
group, which will be designated by G,(AA’‘A”). 
THEOREM 32. The fundamental group of type I consists of all 
collineations of type I having the same invariant triangle; it isa 
two-parameter group whose parameters are the cross-ratios & and 4’. 
249. One-parameter Subgroups of G.(AA'A”). We shall 
now show that the group G,(AA’A’’) contains ~! one-para- 
meter subgroups. Let k’ be replaced by ky. The collineation 
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