NORMAL FORMS. 
132. Determinant of Normal Form. 
of the normal form of T may be found as follows: 
A= —| 
| A B 
= [kA kB 
| ki A" k/ BY” 
BTA 
BI I GAY 
B’ 1k A" 
fe} HL 133 
BIST IB! 
B’ 1 kB! 
B 
B 
B! 
Alen 
= i; k/| A’ 
| All Bl 
[A 1 A 
| A’ 1 kA’ 
| A” 1 kA” 
A 1B 
AU ISK B! 
All of k! B” 
AQ 1 
Aik 
Avi kt 
, g/ 
| 
gall pit 
1 
1 
1 
1 
t/  |4 7 
1 
1 
1 
1 
105 
The determinant A 
ABA 
A! Bl kA! 
A” BU AN 
A BRB 
| A’ Bl kBY 
| A”! B"’ k/B” 
ANB A 
A Bek 
All B' ki 
B' 
Bl 
Ay See 
B’| |» 
A B 
B’ 
whose a, 3’, etc., are the minors of A, B’, ete., in 
A! 
All 
THEOREM 22. 
A Beal 
Jey Gil 
EB 
Sa elie! 
A 
A! 
All 
Bo is 
jess Bll 
BY 1 
The determinant of the normal form of the col- 
lineation 7 is equal to the product of the cross-ratios & and k/ into 
the cube of twice the area of the invariant triangle. 
133. Characteristic Equation of the Normal Form of T. 
The characteristic equation of the normal form of T is readily 
written down as follows, compare Art. 20: 
Ba eA 
iB ih a |) = (0) 
Bl" 1 kt A” | 
B ih 3 
BI SL kB! 
Br it kB" 
B 1 1 
B 1 k 
Be tig 
A 
A! 
All 
Al 
| 
fal | 
k A’ 
ki A” 
B 
kB’ 
kt! BY 
il Z 
k 
A B A 
At Bl kA! 
All BY ki A” 
JA Jes 183 
A’ Bl kB = (). 
All B" k! Bi | 
ara Geol nit 
A’ B k |—o | 
Alt Br kl! | 
