CONTINUOUS GROUPS. 133 
erty; for k,=kk, and when |k|<1 and k,|< 1, then also is 
k, <1and the first group property is established. The in- 
verse of #,—ke is c=—k“a,; when |k| <1, then is |k~| >. 
No transformation in the selected system has its inverse also in 
the system ; hence the system is not a group according to the 
definition. 
163. Parameters of a Group. When a system of collinea- 
tions having 7 parameters is a group, the parameters of the 
system, as defined in Art. 160, become the parameters of the 
group, which is called an r-parameter group. For example 
the system of collineations given by the canonical form of 
type III, Art. 150, 
%,=x2-+2at (2) 
¥,=y+ta+at?+ht, 
depends upon three parameters, a, h, and t. It will be proved 
later that this system forms a group. Giving to a, h, and 
t, all possible values we have ° collineations which form a 
three-parameter continuous group; continuous, because of 
the continuous variation of its parameters. Two consecutive 
collineations in a continuous group differ only by infinitesimal 
values of one or more of its parameters. 
164. Classification of Groups. Continuous groups of col- 
lineations may be classified in several ways ; according to the 
number of their parameters, according to the figures which 
they leave invariant, or according to the types of collineations 
composing them. The best plan of classification is one that 
arranges them according to the types of collineations con- 
tained in them. We shall therefore speak of groups of type 
I, type III, ete.; the group mentioned in the previous article 
is of type III, because the collineations which go to form the 
group are mostly of type III. 
165. Group Notation. In chapter II we made use of the 
notation T, T’, T’, S, S’, as suitable designations for the 
five types of collineations. Groups of the five types will be 
designated respectively by G, G’, G’, H, H’. The number 
