THE GENERAL LINEAR GROUP. 299 
triangle A,B,C, sufficiently small. Hence we have in the 
limit 
A A 
i ae CONS. (18a) 
THEOREM 25. A collineation of the group G,(Z2) alters all 
areas by a constant ratio &. 
355. Analytic Proof of Same Property. Let (a, y), (x, y’) 
and (a’’, y’’) be the vertices of any triangle A. The corre- 
sponding vertices of A, are given by 
vw, =ex +by +c, yy =aa +by+c'; 
xy! = aa! +by!4-¢, yi! =a/x! +bly! +¢'; (19) 
¢)""=aa!l-+-by"!--c , yy! = ala! + bly!’ + c!. 
Forming the determinant which is twice the area of A, we 
have 
|v, Ya 1 ax +by +e ax+bly+e 1 CD @ fe ay) al 
| any! yy’ 1) = \aa’+by/+e a/x’+bd/y’/+c! 1) =| a’ db’ c!|. |a/ y! 1 (20) 
| a!” yi! 1 axl +by!+e a/a!+-bly+e! 1| 00 1 tell cypll il 
Ree Are DAr. 
The ratios of the two areas is therefore D which is the deter- 
minant of the collineation 7. 
356. Types of Collineations in G,(l.). It may be shown 
that the group G,(/.) contains all five types of plane collinea- 
tions. When a collineation T of type I occurs in this group, 
one side of the invariant triangle of T must coincide with the 
line at infinity, /.. A collineation T’ of type II may oc- 
cur in G,(/.) in two different ways; the line / or the line 1’ 
may coincide with /.. Collineations of type III occur in 
G,(l.) when the invariant line of 7” coincides with/l.. A 
perspective collineation S of type IV may occur in G,(l..) in 
either of two ways; the axis / of the collineation may coincide 
with /.., or the vertex A may be on the line at infinity and the 
axis pass through finite space. A collineation S’ of type V 
may occur in either of two ways in G,(/.); the axis may 
coincide with /., or the vertex A may be on /. and some 
line of the pencil through A coincide with 1... 
