THE GENERAL LINEAR GROUP. 297 
from two transformations of the group, T and 7,, whose 
equations are (15) and (15,): 
e=4,0,+by,+¢,35 Yy.=a,/u,+b/y,+¢,. (15,) 
The resultant is also a linear transformation. The second 
group property is shown by solving equations (1) for x, and 
y,. The inverse of T is also linear and the second group 
property is established. 
THEOREM 21. All linear transformations in two variables form 
a Six-parameter group G,;(/.) whose invariant figure is the line at 
infinity. 
392. Purallel Lines are Transformed into Parallel Lines. 
Since the line at infinity is an invariant line of the group 
G,(l..) every point at infinity remains at infinity and hence 
parallel lines are transformed into parallel lines by every 
transformation of the group. The common direction of the 
system of parallel lines may be altered but the fact of paral- 
lelism is preserved. 
This conclusion may be shown analytically as follows: Let 
the equation of a system of parallel lines be written 
Au, + By, + C=0, (16) 
where A and B are constants and Ca variable parameter. 
Substitute for «, and y, their values in the equations 
v,=ar+byt+e, y,=a'«+b'y+c’, and we get 
x(Aa-+ Ba')+y(Ab+ Bb')+ Ac+ Be’ +C=0. (16) 
Since the coefficients of « and y are constants and only C 
varies we have again a system of parallel lines. 
353. Parabolas are Transformed into Parabolas. All 
conies touching the invariant line at infinity are transformed 
into conics touching the same invariant line; hence the trans- 
formations of the group G,(/.) transform the system of ©‘ 
parabolas of the plane into the same system of parabolas. 
The general equation of a parabola is 
(ax, + Gy,)?+ 29x, + 2fy,+d =0. (9) 
