( 753 ) 
3. Besides in these last points a* will still be cut by y, in 6 points 
more, the tangents in these 6 points to a* must pass through 
A x . From this ensues that a* is of the tenth class, by which the 
PiiticKER numbers of a 4 are entirely determined (n = 4, m — 10, 
d = 1). This holds, for it is easy to investigate that a* cannot 
possess a double point differing from A x . 
4. If a vertex of an involution triangle describes a line, on which 
lies a singular point, the curve described by the two other vertices 
degenerates into the polar line of that singular point and a curve 
which must be of order three. If a vertex of an involution triangle 
describes a singular line s, then one of the other two vertices will 
be a fixed point, namely the pole of s and the other point will 
describe s itself and as many other lines as there are singular points 
on s. As both points together must describe a curve of order four, 
three singular points will lie on s. In like manner each singular 
point is point of intersection of three singular lines. 
If now again a x is an arbitrary line and if a 4 has the same signi¬ 
fication as above, then the curve a* will cut a line b x four times; 
from this ensues that four times a point of and a point of b x are 
vertices of a selfsame involution triangle. These vertices we call 
P x , Q x , E x , S x and , Q, , E 2 , S 3i whilst the third vertices of these 
triangles may be represented by I\ , Q 3 , R 3 , S 3 ; farthermore T x is 
the point of intersection of a x and b x and T, and T 3 are the two 
points forming with T x an involution triangle. 
If now a point describes the line b x , then its conjugate points 
describe a curve ft* of order four; a 4 and ft* have 16 points of inter¬ 
section. These are: 
1. the two points T t and T 3 ; 
2. the four points P 3 , Q t ,R t ,S t ; 
3. ten points more having the property that to each of them two, 
so an infinite number of pairs of points, are conjugated and which 
are thus the singular points. Therefore: 
The involution (i 3 ) has 10 singular points; their polar lines are 
the 10 singular lines of (i' 3 ). 
These singular elements have such a position that on each of these 
lines three of these points lie and that in each of the points three of 
the lines intersect each other ; so they form a configuration (10,, 10,), 
If s lt is a singular line and S x3 its pole with respect to y,, then 
there are besides S 13 still 6 singular points not lying on s xt . If S lt 
is one of these points and * l# the polar line of S i3> then the point 
of intersection of s l3 and s l3 is at the same time the pole of S X3 S X3 * 
