270 THEORY OF COLLINEATIONS. 
A,’ are not collinear with either A or A’. Let k,= —; 
k’=k and k,,=k~". The resultant of the two one-dimen- 
sional transformations along the line AA’ is identical since 
both invariant points are common and the cross-ratios have 
reciprocal values. The resultants of the one-dimensional 
transformations of the pencils through both A and A’ are 
parabolic, since they each have only one invariant line in 
common and reciprocal values of the cross-ratios. Hence the 
resultant of T and T, is an elation S’(al) where « is some 
point on/. It is easy to show that all elations in H,’(/) are 
contained in G,(AA’),. 
Analytic Method. The equations of the normal form of 
G,(AA’), are ; 
ket Ue — ky in 
Ea ee yee) rN (2) 
ap th y+ Same denom. 
A! A! B" 
boa fi se | Fav! 
Let k=1, B”=0 and lim. = (= = -) =t, then 
k—1 /ANk (ke! —1) ikea Tg 
lim. —— ( ai; ) 
tions (6) reduce to 
i 28 BL : Pate 6’) 
v= 1+ty ’ Y, = ty ( 
These are the equations of the group H,’(/). 
In like manner the structural formule of the other group 
of this class may be verified. 
321. Third Class. The list of groups of the third class 
shows structural formule as follows: 
G.( ALK) = 0'G,(AA’A”)+ G// (AIS), 
G,(K) = 0'G,(AA’A”) + ~0'G,"(AlS), 
= at where x issome constant. Equa- 
GAIS) 0 G,(AA'A”)+  G,(AlS) 
ACA) Ee Sete 
G,(A,1”),..;= ©°G,(AA’A") + 01H) (AAI) +S. T., 
G.("),.., = 04G,(AA'A")+ 0! H/(Al) +58. T., 
$2 
PS 
| 
8 
+G,(AA’A") + 01H(1) +8. T. 
