GEOMETRIC THEORY. 45 
CeO Ci — a1 Hee eal 
a2) ° @=) wet TRS 2 
Simplifying, this reduces to the’ bilinear form 
TX, — pu + sxu,—q=0; 
or solving for x,, we have 
et 
This equation represents the relation between # and «,, any 
two corresponding points in a projective transformation. 
This equation therefore represents a projective transforma- 
tion, and all properties of the transformation may be deduced 
analytically from the equation. 
THEROEM 30. A projective transformation of the points ona 
line is represented analytically by a linear fractional equation in one 
variable. Or otherwise expressed,a linear fractional transforma- 
tion in one variable is a projective transformation. 
55. Summary. In $1 we defined a projective transforma- 
tion analytically by the linear fractional equation, 
_ +6 ( 1 ) 
ce+d?’ 
and proceeded to deduce the properties of projective trans- 
formations from this definition. In the present section we have 
defined a projective transformation geometrically, and have 
shown that its analytical representation is a linear fractional 
transformation of the form of equation(1). This proves that 
the transformation as defined in two entirely different ways 
is one and the same, and that the definitions are in harmony. 
$7. Geometric Theory of Projective 
Transformations. 
56. The instrument described in $6 for constructing a pro- 
jective transformation of the points on a line / by means of a 
conic K touching two lines / and /’ can be used to establish 
many important properties of such transformations. While 
