296 THEORY OF COLLINEATIONS. 
the collineations with negative k in the perspective subgroup 
a=o, The collineations properly of type I] in RG,(AA'l) 
for which k is negative do not belong to its continuous sub- 
groups and can not be generated from infinitesimal collinea- 
tions of the group. 
THEOREM 20. Every real collineation of type Il, for which 
k is positive, belongs to one and only one one-parameter group 
RG,( AA’l), and can be generated from one and only one real in- 
finitesimal collineation: no real collineation of type I, for which 
k is negative, can belong toa one-parameter group nor can it be 
generated from a real infinitesimal collineation. 
$5. The General Linear Group, G, (1. )- 
350. In chapter IV we determined all varieties of sub- 
groups of the general projective group G,. Some of these 
groups are of special importance when the invariant figure of 
the group is especially related to the Absolute of the Kuclid- 
ian plane, 7. e., to the line at infinity and the two circular 
points. In the present section we shall study in detail the 
general linear group, G,(/.) and its subgroups: the special 
linear group, G,(/.), or group of Invariant Areas, the group 
of Similarity, G,(...'), and the group of all Motions of a 
rigid body in the Euclidian plane, G, (.,.,’),--:. 
351. Invariant Line at Infinity. The six-parameter 
group of collineations whose equations are in the linear form 
G=ar+by+¢, y,=—a'x bye’, (15) 
is called the general linear group. In the general linear frac- 
tional transformation the line represented by the common 
denominator is transformed into the line at infinity (Art. 
82). In the above form the denominator is a constant and 
represents the line at infinity, which is thus transformed into 
itself. 
Equations (15) contain six constants or parameters. The 
first group property is shown at once by eliminating «, and y, 
