306 THEORY OF COLLINEATIONS. 
This subgroup is called the Special Linear Group and is given 
by the equations 
v=ax+byte, y=au«+b'y+c', 
with the condition |%, 4 == il 
364. Areas are Transformed Into Equal Areas. It was 
_ shown in Art. 355 that a collineation of the general linear 
group alters all areas of the plane in the same ratio. Thus it 
was shown that A, = DA, when A is the original area, A, the 
transformed area, and D the determinant of the collineation. 
In the special linear group D= 1; hence A,= A, 7. e., every 
area is transformed into an equal area. 
THEOREM 25. The Special Linear Group transforms every area 
into an equal area; 2. e., it is the group of Invariant Areas G;(J.)A. 
365. Collineations of Type Tin G,(lxo2)a. It was shown 
in Art. 357 that for a collineation of type I in G,(/. ) the con- 
stant ratio of areasis R=kk'’=k'+r. If this ratio is equal 
to unity, then we must have r=—i1. Butif r= —1 then 
the path-curves of the one-parameter group G,(ABC),__, are 
conics, Art. 254, having AA’ and A A” for common tangents 
and A’A” for chord of contact. The line A’A” is now the line 
at infinity and hence the conics having double contact at A’ 
and A” are concentric conics having the same asymptotes. 
When A’ and A” are real points, the path-curves are concen- 
tric hyperbolas having the same asymptotes. When A’ and 
A” are conjugate imaginary points the path-curves of a one- 
parameter group are similar and concentric ellipses. 
In the case of a hyperbolic one-parameter subgroup of 
G,(l~2) (when A’ and A” are real), we can readily see that 
the areas are transformed into equal areas, for all tangents 
to a hyperbola, Fig. 40, form with the asymptotes triangles of 
equal areas. Likewise all segments of one hyperbola cut off 
by tangents to a similar and concentric hyperbola have equal 
areas (Salmon 396). In the case of an elliptic subgroup, 
where the path-curves are similar and concentric ellipses, 
areas are evidently transformed into equal areas. 
