278 THEORY OF COLLINEATIONS. 
these two-parameter subgroups contains ~/ singular transfor- 
mations of type HI, it follows that the group G,(l”’),__, 
contains * such singular transformations. 
The group G,(A),-_,, dualistic to G,(l’’),__,, also con- 
tains ©’ singular transformations of type II. 
THEOREM 5. The following groups, and no others, contain sin- 
cular transformations of type Il: G@,(AA‘l’),, G@s;(AA’),, 
G;(ll’),, and G,( Al), (when 7 is rational); G;(AdS), G;(A,l”) 
perio GRU ssa thiil Gel Areas 
$3. Mixed Groups. 
327. In this section the following problems will be investi- 
gated: To find (1) all collineations in the plane that leave 
fixed one vertex of a triangle and interchange the other two 
vertices; (2) that interchange a pair of points; (3) that inter- 
change a pair of lines; (4) that permute the vertices of a tri- 
angle. 
These four problems lead us to the consideration of certain 
mixed groups of collineations. A mixed group is defined asa 
system of collineations which has both group properties and 
which is composed of a continuous group and certain discon- 
tinuous groups, which interchange certain parts of the in- 
variant figure of the continuous group. For example, all 
collineations, leaving a triangle invariant as a whole, form a 
mixed group mG(AA’A"’), which consists of the continuous 
group @,(AA’A’’) and all other collineations interchanging a 
pair of its vertices and also those permuting the three vertices. 
In order to determine all the mixed groups of collineations 
in the plane, we must first examine all varieties of invariant 
figures of reducible groups of collineations. There are only 
eight varieties of such figures (Fig. 24), viz.: A point (A), a 
line (/), a lineal element (Al), a pair of points (AA’), a pair 
of lines (/l’), a point and line not incident (A,/), two points 
and two lines (AA’/’), and a triangle (AA’A”). Of these 
