TYPES AND NORMAL ForRMS. 13 
18. Determinants of Normal Forms. The determinant of 
the explicit normal form of type I is found as follows: 
x 1 0 
Aiea 
_|A’ 1 KA’ _ (kA A)n +AA!(1—k) 
Gee i ~ (k=1)"+(A'—kA) 
Pla anc 
At ae 
|kA'—A AA!(1—k) 
AS ek (23) 
k= A! ll 
The determinant of the a normal form of type II is 
|1+tA —tA2 
= | =]. (24) 
t 1—tA 
19. Type II as the Limiting Form of Type I. It is evident 
that type II is the limiting form of type I when the two inva- 
riant points coincide. From equation (14) we see that k = 1 
; 1—k ; . 
when A=A’. The fraction Te i becomes indeterminate 
when A=A’. Putting for A, A’ and k their values from 
(17), we have: 
lim i1-k a BC 
A'=A A-A!~- atd° 
2¢ : lim 1-6 _ 
a bp MENCES ig og Ee 
By means of this relation the normal form of type II can 
be deduced directly from that of type I. Dividing both 
numerator and denominator of (16) by A—A’, we get: 
(A=kA') , _ AAMU—*) 
je AE a EA) 
nape C= AeA, 3 
(A=A)) “(A= A/) 
Putting A’=A and , ie et a i =t, this reduces to (19) . 
In the explicit eae form of type I, (22), subtract the 
second row from the last in each determinant, divide through 
by A’—A, and pass to the limit. In this way we get the 
explicit normal form of type II. 
