134 THEORY OF COLLINEATIONS. 
of parameters in the group will be expressed by a subscript ; 
thus a two-parameter group of type I will be written G,. It 
is oftentimes desirable to express in the symbol the figure 
left invariant by all the collineations of the group. This is 
done by enclosing in parenthesis the symbol for the invariant 
figure. Thus the symbol G,’’(Al) designates the three- 
parameter group of type III, leaving invariant the lineal 
element Al. This is the group whose equations are given in 
Art. 163. 
166. Groups of the Same Variety. Two groups composed 
of the same type of collineations and having the same number 
and kind of parameters are said to be of the same variety 
when their invariant figures differ only in position, shape or 
size. For example there are ~’ lines in the plane and each 
line is invariant under the ~’ collineations of a six-parameter 
group. These ’ groups are all of the same variety. It is 
unnecessary to study more than one group of each variety. 
We shall enumerate and investigate forty-four varieties of 
groups of plane collineations. Groups of the same variety 
are also said to be equivalent, according to the definition of 
groups given in Art. 35. Thus if G operated on by T gives 
G’ by the formula G’ = T-'GT, then G and G’ are groups of 
the same variety. 
167. Determination of the Resultant. A necessary condi- 
tion that a given system of collineations forms a group is that 
the system should possess the first group property, 7. e., the 
resultant of any two collineations of the system is one of the 
same system. The process of finding the resultant of two 
collineations is one of the most important operations we shall 
make use of in the present chapter. We must therefore ex- 
amine the process in detail. 
Let a collineation T be given in homogeneous coordinates 
by the equations, 
ae »Y,2), 
es Py »Y,2), (3) 
Y,z); 
