292 THEORY OF COLLINEATIONS. 
RG, (ALS) in such a way that each collineation of type III 
in RG,'’(Al) belongs to one and only one such subgroup. 
Hence each real collineation of type III in the plane belongs 
to one and only one one-parameter group, RG,/’(AlS). The 
group RG,’ (ALS) has exactly the same structure as the group 
pG,(A); consequently we may state our theorem at once. 
THEOREM 17. Each real collineation of type III in the plane 
can be generated from one and only one real infinitesimal collineation. 
346. Type Tl. There are two distinct kinds of real collinea- 
tions of type Iin the plane, viz.: hyperbolic and elliptic, 
article 310. These fall into ©’ two-parameter subgroups of 
RG, :viz., hG,(AA'A”) and eG,(AA’A’’) in such a way 
that each real collineation of type I belongs to one and only 
one of these subgroups. The two cases must be treated sep- 
arately and we take up first the hyperbolic group hG,(AA’A”’). 
347. The Hyperbolic Case. The two-parameter group 
hG,(AA'A”’) has for parameters k and k’ both of which as- 
sume in turn all real values. 
The two-parameter group )G,(AA’A’’) contains an infinite 
number of one-parameter subgroups, and we proceed to de- 
termine these. All transformations inhG,(AA’A’”’) for which 
the two parameters satisfy a relation of the form k’=k’, 
where 7 is a constant, form a one-parameter subgroup ; and 
conversely, in all one-parameter subgroups, & and k’ satisfy a 
relation of this form. There are different subgroups for dif- 
ferent values of 7. Geometrically, article 252, 7 is interpreted 
as the constant cross-ratio of certain four points on the tan- 
gent to a path curve of hG,(AA’A”’),: viz., the point of tan- 
gency T and the points of intersection of the tangent with 
the sides of the invariant triangle. These four points are all 
real for a real hyperbolic group and hence ¢ is also real. 
In order to study the distribution of the ~* collineations of 
hG.,(AA’'A’’) into one-parameter subgroups we resort to a 
geometrical device as follows: Let k and k’ be the rectangu- 
lar coordinates of a point in a plane (not to be confused with 
