RESULTANTS. Si, 
diminishing the number of independent parameters. Many 
examples of subgroups obtained by both of the above men- 
tioned methods will be given in the following sections of this 
chapter. 
170. Invariant Figures and their Groups. We wish now 
to prove a theorem of great importance in the determination 
of continuous groups of collineations. Let 7 and T, be two 
collineations each of which leaves invariant a certain figure 
F’; since T leaves F invariant and T, also leaves F’ invariant, 
their resultant 7, must also leave F' invariant. Thus the 
entire system of collineations leaving a certain figure F’ inva- 
riant has the first group property. Since T leaves F invariant, 
it transforms points of F' into the same or other points of F’; 
hence T~ the inverse of T also transforms the points of F’ 
into the points of F, 7. e., it leaves F invariant. Thus we see 
that the system also has the second group property. Such a 
system is therefore a group. Thus we see that the invariance 
of a plane figure under a certain system of collineations is a 
sufficient condition that they form a group. It does not fol- 
low that this is a necessary condition. 
THEOREM 1. The system composed of all plane collineations 
which leave a certain figure invariant forms a group. 
$2. Resultant of Two Collineations. 
The determination of the resultant of two collineations in 
the most general form is our immediate problem. When the 
two component collineations T and T, are taken in the most 
general form and no restrictions laid upon the values of their 
parameters, their resultant T, in either order is assumed to 
be likewise in the most general form. Any other assumption 
concerning the form of T, is equivalent to some restriction on 
the values of the parameters of the components. If no re- 
strictions are laid upon the values of the parameters, it is 
clear that the collineation is of type I. The results of $$ 
