244 THEORY OF COLLINEATIONS. 
shall go on to show that G,(Al),_, also contains other varie- 
ties of of subgroups. 
Al 2 
294. The Growp G,(AIS),_.. If we set =, a and 
A." (or in more general f pe aS 
By! Ay ’ O ig al torm B! mau as A /? 
2 are : Vigie NE 
in the conditional equations of G,(Al),_., we have 7 = 45 
(or more generally ay the a) and IV reduces to III. We 
thus have left the equations of a three-parameter subgroup of 
G,(Al),_.. It is not difficult to see that this three-parameter 
group leaves invariant a pencil of conics S touching / at A and 
should therefore be designated by G,(AJUS),_,.. The equa- 
tions of such a system of conics may be written 
x? + 2hey+ Cy? =40y, (75) 
where C is the parameter of the pencil. The polar of any 
point A’ on / with respect to this pencil of conics is 
AlexthA'y=2ay or paca : 
Hence the condition a +h= = , implies that the point (A’B’”) 
is on the polar of the point (A’, O) with respect to every conic 
of the pencil given by equation (52). 
It is easy to show by direct substitution that the transforma- 
tion T which satisfies the above conditions leaves invariant 
the pencil of conics 
e+ 2heytCy=4ay. 
The group G,(Al) contains therefore ©’ subgroups 
G,(AIS),_., one for each pencil S touching / at A. 
295. The Group G,(ALK). Thepencil Scontains ~! con- 
ics and is invariant under ’ collineations of the group 
G,(AlS),_.. If we choose from the pencil S a single conic 
K, and from the group those collineations which leave K alone 
invariant, we shall have a subgroup of G,(A/S),_.. 
