98 THEORY OF COLLINEATIONS. 
$4. Normal Forms of Equations of 
the Five Types. 
126. In $2 of chapter I it was shown that there are two dis- 
tinct types of projective transformations on a line, and that 
the analytic expressions for these two types can be put into 
elegant determinant forms in which the constants have defi- 
nite geometric meanings. It has just been shown that there 
are five distinct types of plane collineations, and our next 
task is to find normal forms of the equations of these five 
types. We shall find forms strictly analogous to those 
already found for one-dimensional transformations, article 17. 
We shall first determine the fundamental geometric prop- 
erty of a collineation of type I with reference to its invariant 
triangle. This geometric property is then expressed in 
analytic form in terms of the coordinates of a pair of corre- 
sponding points and the coordinates of the invariant points of 
the collineation. We shall first reach an implicit normal 
form and then pass to the explicit normal form by solving a 
set of linear equations. The reduction of the equations of a 
collineation to their explicit normal form, 7. e., the expression 
of a collineation in terms of its natural parameters by means 
of an elegant determinant formula, is an analytic result of 
prime importance. 
From the normal form of a collineation of type I we pass 
readily to the normal forms of collineations of the remaining 
types. 
127. Three Cross-ratios Whose Product is Unity. We shall 
now consider in detail the most general case of a collineation 
whose invariant figure is a triangle (type 1). Let the ver- 
tices of the triangle be represented by A, B, C; and the oppo- 
site sides by a, b, c, respectively. By means of a collineation 
T the line @ is transformed into itself in such a way that the 
points B and C on it are invariant points of the transforma- 
tion. Now we know that the one-dimensional transformation — 
