CONTINUOUS GROUPS. 131 
groups of each of the five types of collineation and their one- 
parameter groups and path-curves; and in §8 the whole 
theory of groups of perspective collineations is developed by 
the same instrument. The list of all varieties of continuous 
sub-groups of G, is completed in §9 and a table of these groups 
is given. Groups of real collineations are treated in $10. 
$1. Theory of Continuous Groups of Col- 
lineations. 
160. Systems of Collineations. In the last chapter we 
studied in detail the properties of each of the five types of 
plane collineations. We shall now consider the properties of 
certain infinite systems of these collineations. 
Let a collineation T of type I be given in the canonical 
form, Art. 148, 
T : cee (1) 
yi=k'y. 
Let k and k’ each assume in turn all possible values; we 
get thereby a system of ~° different collineations. . This sys- 
tem of collineations has the important property that each col- 
lineation of the system leaves invariant the triangle formed 
by the w- and y-axes and the line at infinity. The two quan- 
tities k and k’ are called the parameters of the system, which 
is therefore called a two-parameter system. In general a 
system, S,, of ”collineations (7 <8), which is obtained by 
varying 7 independent parameters, is called an r-parameter 
system. The r-parameter system is said to be a continuous 
system when it contains the collineations corresponding to 
every possible complex value of the v independent parameters. 
161. Component and Resultant Collineations. Let T and 
T, be any two plane collineations. 7 transforms the points 
P, P’, P”, ete. of the plane into new positions P,, P,’, P,’’, 
ete., and the lines /, /’, 1’’, etc. of the plane into new posi- 
Mons LL ete, --L,.transtormsythe points 2;,, 25 °P,”", 
