ONE-DIMENSIONAL PROJECTIVE TRANSFORMATIONS. 5 
Cm een ao? (6) 
which shows that every point on the line is transformed into 
the fixed point a The inverse of the transformation T' is 
written 
=== da+b 
cxi—a ~ 
uv 
The determinant of this is also ad—be, which equated to zero 
also gives d= a Substituting this value of d in the last equa- 
tion, we have 
b(ex—a) ay 
Wy a(cxi—a) > 
t= =e} (7) 
which shows that every point on the line is transformed by 
the psuedo-transformation (7) into the fixed point — a 
The invariant points of a pseudo-transformation are also 
given by equation (38). Putting d=" in this equation, it 
breaks up into . 
(«-$) (w+) =0; (8) 
thus showing that > and — 7 are the invariant points of the 
pseudo-transformation. 
THEOREM 2. A pseudo-transformation transforms every point on’ 
the line into one or the other of its invariant points. 
7. Three Conditions Determine a Projective Transforma- 
tion.—The equation of a_projective transformation T contains 
three independent constants, viz., a:b: ¢: d. We infer, there- 
fore, that three conditions determine such a transformation. 
In particular, three points and their corresponding points de- 
termine uniquely and completely a projective transformation. 
Let x’, x’, x’ be any three points on a line, and «,', x,/’, x,/”’ 
their corresponding points, respectively. Substituting succes- 
sively in (1) the coordinates of each pair of corresponding 
points, we have three equations, viz. : 
