10 TYPES AND NORMAL FORMS. 
mation, not identical, belongs to one or the other of these 
types. 
13. Implicit Normal Form of Type I. A transformation 
T of type I, whose invariant points are A and A’, may be 
written in the form: 
Sa. (14) 
where the constant k is teenie in the terms of the coeffi- 
cients, a, b, c, d, as follows: 
(a+d—V(a+d)?—4(ad—be) )” 
li ers : (15) 
To show this, solve equation (14) for x,; this gives us 
(A—k A!) x—AA!(1—k) 
(1—k)u—(A’—kA) ” (16) 
-2= 
which is of the same form as 
ax +b 
1 Geta 
Comparing the coefficients of these forms, we have 
b Al—kA d 
A—kA! a 
EAA =.= on Oe 
1—k Oy Gide Se aD @? 
solving for A, A’ and k, we find 
A= a—d+ Vv (a+d)?—4 (ad—be) , 
2c 
vaya a—d—Nv(a+d)?—4(ad—be) ; 
2c 
(a+d— Vv (a+d)2—4(ad—be) ex. 
k= 4 (ad—be) (17) 
age a+d— VN (a+d)?—4 (ad—be) 
= a+d+ w(a+d)?—4 (ad—bc) } 
1+k)? d)? 
or (1+k) ns (a+d) 
k ad—be * 
The values of A and A’ thus obtained are the same as the 
roots of equation (3). Equation (14) is called the implicit 
normal form of type I. 
