92 THEORY OF COLLINEATIONS. 
all simultaneously zero, then the collineation has an invariant 
figure unlike any of the above cases. The single root p, gives 
us a single invariant point (or line). The double root p, gives 
coo! invariant points which lie on a line and o¢ invariant lines 
which pass through a point. The single invariant point given 
by p, does not lie on the line of invariant points given by ., 
since the first minors of (9) do not all vanish when 9, is sub- 
stituted for p in (8). Hence, when the above conditions are 
satisfied, the invariant figure consists of a line of invariant 
points and a single invariant point not on this line; the col- 
lineation is of type IV. 
117. Type V. If A(o)=0has a triple root for which all 
the first minors of (9) are simultaneously zero, there are «+ 
invariant points which lie on a line and ~’ invariant lines 
which pass through a point. The intersection of these in- 
variant lines is an invariant point which must be one of the 
points on the line of invariant points. Hence, when the 
above conditions are satisfied the collineation is of type V. 
118. Geometric Construction of Types of Collineation ; 
Type I. The method of constructing a collineation given in 
article 98 also shows in an elegant manner the five types of 
plane collineations. When the conics K and K’ of Fig. 12 
are not in contact, the invariant figure is a triangle. Hence, 
the collineation determined by two conics, which do not 
touch each other, is of type I. 
For special positions of the conics K and K’ the invariant 
figure will be different. Thus special cases arise when the 
conics touch one another, or touch the line / at the same point, 
or have contact of the second or third order, etc. These 
special cases give rise to the other four types of collineations. 
119. Type II. Let us next consider the case of a collinea- 
tion where the two conics touch one another. The invariant 
figure in this case consists of b, the common tangent to the 
two conics, A their point of contact, and another common 
tangent a intersecting b at B, Fig. 15a. 
