TYPES AND NORMAL FORMS. 15 
We get the equations of T, by substituting x, and y, from T 
in) thus, 
De AS eer (Se BRAINS 
Ter Ppt, 340) Ee AD, cosy,—= | Ar Bi Bil: 
Ay By kiAy! Ay By kBy 
where A, and A, are the determinants in 7. These last equa- 
tions readily become 
an 0 0 “sy 0 0 0 
AL TBE Av Bi. 40 AL Bia ADS OB A. 0 
T,: pit, =| A’ Bl kA’ kB’ 0}. pp,Y.=|A' BY kA’ kB 0 . (26) 
060A BR A 00 A BR Bl 
0 0 AY BY kAy 0 0 Ay By kBy 
Comparing coefficients of x and y in the two forms of T, we 
get the following equations I to IV: 
BeerA B 0 
(1) \" Ag | Bi kA! KB Oo 
| Bo! KexAglllnn \"0 An Br VAN 
0 Ay By kiAj 
A A B 0 
Az As A! kA' kB’ Oo 
(IT) = 
Ad keAd! OP ATP Ep Biy IAG 
0 Ay By kAy 
ig) A 1B 0 
Be By |B’ kA!’ kB’ 0 
(IIL) = ; 
Br. keBy | } QO 2h, tan ehh 
| 0 Ay By kBy 
Al NA B 0 | 
As Bo | A!’ kA’ kB’ 0 
(IV) | lea I 
Ay keBy/ | 0 Ai Bi Ai 
