250 THEORY OF COLLINEATIONS. 
303. The Group G,/(A‘l). Let T’ and T,’ be so chosen 
that their invariant figures have the point A and the line l’ in 
common, and let A be the origin and /’ the x-axis. The nor- 
mal form of T’ reduces to 
c 
ku + —-(i—k)y 
c! 
t, = | v, z (80) 
k—-1 We ce (1—k)} 21 (Same denominator.) 
1+ = 45 + —— . 
A’ (acl el) Lae 
Writing T,’ in the same form and eliminating, the resultant is 
also of the same form with the following conditional equa- 
tions: 
Qik — Kier 
ease Se, Kyat 
= 
A, A! Ay! 
(3): ese Shee Gee (81) 
C2! c! cy! 
te ete ics Be ath, 12% 
Cys US es 8a ee 
c./ co! A)! c! c;! ce! A! 
ne Gem 
= c G ky | Hy. Oye (| ) 
c! A)! (a A,’ 
Hence the existence of the group G,(Al’) is verified. 
304. The Group G,( AA’). If we make A,=A in the 
conditional equations of the group G,/(A’l’) we find the con- 
ditional equations of the group G,’ (4AA’) as follows: 
(1) io (C4) ea Se 
i 82 
(3) 3 -) = mk + e-D (ee) 
We also obtain the same equation by making A,’ = A’ in the 
conditional equations of the group G,/( Al) and writing ¢ for 
= . Hence the existence of the group G,’(AA’) is proved, and 
also that it is a subgroup of both G,/(A/l’) and G,/(Al). 
305. The Group G,/(ll’). If we make B’ = 0 in the normal 
form of the group G,’( Al), we obtain the normal form of a 
group G,’ (Il’) as follows : 
ka Yy 
’ Th = 
rena see oe Tap aang 
y+ AV’ x y A’ x 
to 
(86 ) 
