16 TYPES AND NORMAL FORMS. 
Since the determinant of T, is equal to the product of the 
determinants of T and T,, (art. 10), we have A,=AA, or 
using the values of the determinants (art. 18). 
Gon vamemeereial lio. alk 
ANVAsi) Be! | A’ B\ lay By|? 
The system of equations I—IV are not independent, but a 
system of three independent equations may be obtained from 
them by dividing any three of them by the fourth. Equation 
V is not independent of I—IV and may be deduced from 
them. These equations enable us to determine the natural 
parameters of T, in terms of those of T and T,. If T, or T,, 
or both T and T, are of type II, the same process of elimina- 
tion enables us to determine 7’,. 
22. Resultant of T’ and T,’ in Normal Form. The result- 
ant of two transformations of type II is usually of type I, as 
may readily be shown. We wish to determine the conditions 
that must be satisfied in order that the resultant of two 
transformations of type II shall also be of type II. Let the 
equations of 7’, T,’, and T,’ be as follows: 
ay 1 0 aay ih 10) x LO 
| A 1 A A; 1 Aj Ao 1 As 
if 0 tA’+1 | 1 0 tA, +1 yi 0 t.Ao +1 
0 = 5 I 9 din = = == 
itp 1 0 Hr Th (0) x iL (0) 
Ay Tt of aly Gh il As 1 1 
1 0 ¢ 1 0 ti df O te 
The resultant of 7’ and T,/ may also be written in the form 
lo 20 D0 |e a 0 0| 
Aa A oO ee ah A | 
i | i ® tA € @ kt O (Asn @ @ | ; (ZT) 
0 0 A, 1 Ai lo 0 Al i il 
\® @O Wi 0 tAi+1| lo o 1 0 t 
