TYPES AND NORMAL FORMS. 9 
11. Commutative Transformations. Two transformations 
T and T, are said to be commutative when the resultants, 
taken in either order, are equal; 7. e., when 7, and 77’ of the 
last article give the same transformation. We may find the 
conditions that must be satisfied in order that T and T’, are 
commutative by equating corresponding coefficients in equa- 
tions (12) and (12’). We thus get 
ee ESR 
ai—d =e by *) C1 gi 
as the necessary conditions of commutativity. 
The invariant points of T and T, are given by the roots of 
the respective equations, art. 4, 
cx? — (a —d)x —b=0 and ca’ — (a,—d,)x —b,=0. 
The conditions of commutativity show that these two equa- 
tions have the same roots. It will be shown later that these 
necessary conditions are also sufficient. Hence 
THEOREM 7. Two projective transformations T and Ti are com- 
mutative when and only when they have the same invariant points. 
$2. Types and Normal Forms of Projective 
Transformations. 
12. Two Types of Projective Transformations. The inva- 
riant points of a transformation T are given by the roots of 
the quadratic equation (3). The roots of this equation are: 
(A, A’) miosd= VN (a+d)?— 4 (ad — be) 
2c 
(13) 
These two roots are distinct or coincident, according as 
(a+d) —4(ad — bc) 40, 
or = 0. 
Thus there are two distinct types of transformation. The 
first type is characterized by the fact that it has two invariant 
points, while the second type has only one. Every transfor- 
