INFINITESIMAL COLLINEATIONS. 295 
THEOREM 19. Every real elliptic collineation of type I can be 
generated from either two or an unlimited number of real infinitesi- 
mal collineations. 
349. Type II. The group RG,'(AA’l) contains two real 
parameters, k and t, each of which assumes in turn all real 
values. The group contains ~! one-parameter subgroups for 
which k and ¢ satisfy the relation k =a‘, Art. 299, where a is 
a constant. There is one such subgroup for each positive 
value of a. 
In order to study the distribution of the transformations in 
RG,'(AA’1) into subgroups and their generation from infini- 
tesimal transformations, we resort to the same device as in 
type I, and make k and ¢ the rectangular coordinates of a 
point ina plane. The family of curves y=a* represents the 
system of one-parameter subgroups of RG,(AA’l). For 
positive values of a these curves lie in the first and second 
quadrants and completely fill the upper half of the plane. 
There are no continuous curves for negative values of a, and 
hence continuous subgroups of RG,(AA‘l) exist only for 
positive values of a. 
Two particular curves of the family deserve special atten- 
tion. Fora very large value of a, the curve y=a’ differs 
but little from the axis x = 0; hence in the limit when a = 
the line «=0 is acurve of the family; on the other hand 
when a = 1 the curve reduces to the line y=1. In the first 
case « = 0 is the only curve of the family that penetrates into 
the lower half of the plane, and consequently the correspond- 
ing group is the only continuous subgroup of RG,(A A’) con- 
taining collineations with negative values of k. The collinea- 
tions of the group corresponding to a= are of type IV. 
The collineations of the group corresponding to a= 17 are of 
type V. 
Each one-parameter subgroup of RG,(AA’l) contains two 
infinitesimal collineations, one positive and the other nega- 
tive. Every collineation in a subgroup of RG,(AA‘l) may 
be generated from one of its infinitesimal collineations, except 
