308 THEORY OF COLLINEATIONS. 
for it was shown in Art. 361 that for thistype R=1. All 
collineations of type V leaving the line at infinity invariant 
form a two-parameter group H,'(/.), Art. 201, and this is 
therefore a subgroup of G;(l.)a. 
THEOREM 26. The group of invariant areas G;(/o)a consists of 
collineations of types I, III, and V; it contains o* one-parameter 
subgroups G;(A6C),-_;; ©% one-parameter subgroups G1//(AlS~) ; 
and o/ one-parameter subgroups 4/;/ (Alo) . 
368. The Group of Similarity, G,(oo’). We shall now 
take up the detailed study of the real four-parameter group 
G,(ow’) whose invariant figure is the line at infinity and the 
two circular points at infinity o and wo’. The group G,( ow’) 
evidently contains «* two-parameter subgroups G,(Aow’), 
where A is any point in the plane forming with w and w’ the 
triangle (Aww’). Each of these two-parameter groups 
G.(Aow’) contains 1 one-parameter subgroups G,(Aww’),. 
Since two of the vertices of the invariant triangle (Awu’) 
are conjugate imaginary points it follows that all the collinea- 
tions of type I in G,(ow’) are elliptic. There can be no real 
collineations of type II in G,(ww’) because for such collinea- 
tions the invariant points A and A’ are always real points and 
hence can not be made to coincide with w and w’. There can 
be no real collineations of type III in G,(ow’) for similar rea- 
sons. 
G,(ww’) contains real collineations of types IV and V; for 
if the line of invariant points of a collineation of type IV or 
V be the line at infinity, the imaginary points w and o’ as well 
as all real points on the line /., are invariant. 
369. Angular Magnitudes are Invariant. <A collineation 
T (ow’) which transforms any point P into P, necessarily trans- 
forms the lines joining P to w and w’ into the lines joining P, 
tow and w’. Thus isotropic* lines are transformed into iso- 
tropic lines; the system of isotropic lines in the plane is in- 
variant under the group G,(ww’). 
Two lines p and p’ meeting in P are transformed by T(ww’) 
*C. A. Scott, Modern Analytic Geometry, Art. 113. 
