14 TYPES AND NORMAL ForMS. 
20. Characteristic Equation in Normal Form.—Let T be 
given in the normal form 
|x i 0) 
Aron &a| 
heh Jar x Ra’| _ (kA!=A) @+AA! (1K) | 
|e OY eer (1) 
Ales 1| 
VAG ot, Fel 
The characteristic equation then becomes 
|kA’—A—p AA!(1—k) | 
Waites A Aape| ae 
Developing this we get as the characteristic equation of T in 
the normal form 
fi Ss (JES 1) ON al yA (0) (25) 
where A is the determinant Ni ‘|: 
The roots of this equation are evidently — A and —kA. 
The characteristic equation of 7’ in the normal form is 
readily found to be 
pe —29+1=0. (25’) 
21. Resultant of T and T, in Normal Form.—Let us next 
consider the resultant of two transformations JT and T,, both 
of type I, given in their explicit normal forms in homogene- 
ous coordinates. Let the equations of T, T,, and T., be as 
follows: 
x y 0 | @ y 0| 
E02 — Aa Bae Alle pyi=|A BB; 
Al BA! Al OB! OkB®! 
v1 Yi 0 v1 Yi 0 
LS Og —|-Arm * UBIO eAn| inves Aree re eens 
Aj! By kyAy’ | Aj! By ki By 
a7 Yy 0 x y 0 
Te Ose |As'| (Be “AS aris 1) Asie Bagh oB2)p 
Ail BY keAd/ Ay BY keBy 
