4 ONE-DIMENSIONAL PROJECTIVE TRANSFORMATIONS. 
(a—p)x+ by =0, 4 
cL (ad =) y= 0: ©) 
If these equations are simultaneous, their resultant vanishes; 
thus 
a ‘a | = 
Developing the determinant we get the quadratic equation, 
p. —(G-@)io 1 (a0) —0e)i— 0) (5) 
which is called the characteristic equation of 7; its roots 
may be equal or unequal. 
In the first case, suppose that equation (5) has two distinct 
roots, p, and ,. If one of these roots as p, be substituted for 
p in (4), these become simultaneous and may be solved for 
the ratio «:y. The value of this ratio x:y gives the coordi- 
nates of the invariant point corresponding to p, If p, the 
other root of the characteristic equation, be substituted for p 
in (4), these again become simultaneous and their common 
solution gives the coordinates of the invariant point corre- 
sponding to p.. 
In the second case, suppose the characteristic equation (5) 
has a pair of equal roots. Then there is only one value of p 
which, when substituted in equation (4), makes them simul- 
taneous. It follows in this case that T has only one invariant 
point, or as we may say, two coincident invariant points. 
6. Pseudo-transformations.—If the determinant of T van- 
ishes, the transformation is called a pseudo-transformation. 
In defining the transformation it was expressly stated that 
the determinant must not be zero. This condition excludes 
just these transformations called pseudo-transformations. 
The equation of the transformation is written 
ax+6. 
1. = 
i! ca +d? 
if the determinant ad — be = 0, thend = ae Substituting this 
value of d in the equation, we have 
