THE GENERAL LINEAR GROUP. 305 
/. But the line PP, passes through A and is parallel to /. 
Hence the triangles PQR and P,QR are equal in area. 
. R=1, 1. e@., areas are unaltered by S. 
THEOREM 24. The value of 2, the ratio of areas for a colline- 
ation in the group @;(/), isas follows: For type I, R=kk’; for 
type I], R= in the first case, and A= inthe second case: for 
type II], #=7; for tvpe IV, R= #2 in the first case, and 2 =k in 
the second case; for type V, &=/ in both cases. 
362. Five-parameter Subgroups of G,(l.). Every six- 
parameter group G,(l) leaving invariant any line / of the 
plane, breaks up into ~! five-parameter groups of the variety 
G,;(A1) leaving a lineal element Al invariant; it also con- 
tains, Art. 209. one five-parameter subgroup of another type, 
viz.: G;(/),--,, composed of one-parameter subgroups whose 
path-curves are conics having / for common chord of contact. 
Having proved the existence of such a unique subgroup for 
every line 1, we wish to study in particular this subgroup 
when / is the line at infinity. It will be shown that this 
special subgroup of G,(/.) transforms areas into equal areas. 
It is therefore called the group of Invariant Areas. 
363. The Special Linear Group. The general linear group 
is given by the equation 
%,=anr-by--c, y,=a'x+b'y+e’. 
ne 
OOD 
A second collineation of the same group has the determinant 
D,= |G. os . The value of the determinant of the resultant 
of these two collineations is, by Art. 172, D,=DD,. If now 
Dand D, are both equal to unity then D, is also equal to unity. 
The resultant of any two collineations of the general linear 
group whose determinants are both unity is a collineation of 
the same group with determinant also unity. Hence all col- 
lineations of the general linear group whose determinants are 
equal to unity form a subgroup of the general linear group. 
—20 
The determinant of this collineation is D = 
a’ b/|° 
