THE GENERAL LINEAR GROUP. 309 
into p, and p,’ meeting in P,. Since cross-ratios are unaltered 
by a collineation, the cross-ratio of the pencil P(pp’/oo’) is 
equal to that of P,(p,p,/oo’). But the cross-ratio of the pen- 
cil formed by two intersecting lines p and p’ and the isotropic 
lines through the point of intersection measures the angle be- 
tween p and p’.* Hence the angle between p and p’ is equal 
to the angle between the two corresponding lines p, and p,’. 
This is true for all points of the plane. Therefore all angles 
in the plane are transformed into equal angles; in other words, 
all angular magnitudes are invariant for the group G,(00’). 
Since all angular magnitudes are unaltered by a collinea- 
tion T(oo’), it follows that any figure is transformed into a 
similar figure. Shape is conserved but size is not necessarily 
conserved. The group G,(ww’) is therefore called the Group 
of Similarity. The group G,(o0’) is a subgroup of G,(U.); 
hence every collineation in G,(ww’) alters all areas by a con- 
stant ratio R. But if angular magnitudes are unaltered and 
areas are altered by a constant ratio FR, it follows that all 
linear magnitudes are also altered by a constant ratio R’ 
which is given by R’= NVR. 
THEOREM 27. Every collineation of the group @,(¢’) leaving 
invariant the line at infinity and the two circular points, trans- 
forms angles into equal angles and every figureintoa similar figure 
and alters every linear magnitude by a constant ratio. 
370. The Path-curves of the Group G,(Aow’),. The path- 
curves of a one-parameter group G,(Aow’), require special 
attention because of their remarkable form. We proved in 
Art. 252 that the path-curves of the one-parameter group 
G,(ABC), have the property that the cross-ratio of the pencil 
formed by a tangent to a path-curve at P, and the three lines 
from the point of contact to the invariant points is constant 
and equal to r for all points of the plane. Since Pu and Pw’ 
are the isotropic lines through P, it follows that the angle be- 
tween AP and the tangent to the path-curve at P is constant 
for all points of the plane. The only plane curve for which 
*C. A. Scott, Modern Analytic Geometry, Art. 273, 
