TYPES OF COLLINATIONS. 91 
nates of an invariant point of the collineation. Substituting 
successively the three roots of A(p)=0 in equations (8) and 
solving each simultaneous system, we obtain the coordinates 
of three invariant points. 
If equations (2) be interpreted in line coordinates, the same 
analytic work gives us the coordinates of three invariant lines. 
These three invariant points and three invariant lines form 
the vertices and sides of an invariant triangle. Hence, when 
A(p) = 0 has three distinct roots, the collineation leaves a tri- 
angle invariant and is of type I. 
114. Type II. If the cubic, A(p) =9, has a single root o, 
and a double root p,, the invariant figure of the collineation (2) 
is no longer a triangle. If the first minors of (9) are not all 
simultaneously zero, the collineation has two invariant points 
and two invariant lines. The invariant point (or line) corre- 
sponding to the double root p, may be regarded as two coinci- 
dent invariant points (or lines). The single root p, gives us 
an ordinary invariant point (or line). The two invariant 
lines intersect in one of the invariant points (the double one) 
and the two invariant points lie on one of the invariant lines 
(the double one). Hence, when A(p) = 0 has a double root 
and the first minors of (9) are not all simultaneously zero, the 
collineation is of type II. 
115. Type II. If A(p)=Ohasa triple root and the first 
minors of (9) do not all vanish, then there is only one value of 
p that makes (8) simultaneous. Hence the collineation in 
this case leaves only one point invariant. In line coordinates 
the same conditions shew that the collineation leaves only 
one line invariant. The condition that the invariant point 
lies on the invariant line is evidently satisfied, so that the in- 
variant figure is a lineal element. Hence, when A(p) = 0 
has a triple root and the first minors of (9) are not all sim- 
ultaneously zero, the collineation is of type III. 
116. Type IV. If A(p)=0hasa double root p, such that 
when p, is substituted for p in (8) the first minors of (9) are 
