258 THEORY OF COLLINEATIONS. 
If 7, is chosen so that c,=c, then c¢, also equals ¢ and 
O.=0-+ 0,3 we, elet 0 eles 29+) )) Hence if ci be kept 
constant and # alone varies, we have a one-parameter sub- 
group eG,(AA’A”),. The two-parameter group eG,(AA’A”) 
contains ©‘ one-parameter subgroups, one for each value of ¢ 
ink=ele+?)é, 
There are two subgroups of eG,(AA’A’’) of special impor- 
tance, viz.: forc=o and c=]a—. When’¢—o. —eoand 
k’ =e-*#; in order that the equation k’ = k” should be satis- 
fied by these values of k and k’, we must put r=— 1. Hence 
the path-curves of the group eG,(AA’A”)._, area pencil of 
conics having double contact at a pair of conjugate imaginary 
points A’ and A”. The conics of such a pencil have no real 
TGs le 
points in common. The conics of such a pencil are either real 
or pure imaginary without a real point. 
If ¢ approach © and ¢ approach o at the same time, c# may 
approach a finite numbern. Insucha case k=e", a real 
number. 
The cross-ratio k’’ along the line A’A” must be of the form 
e’® , since the one-dimensional transformation along that side 
