( 756 ) 
must be the same as the points of intersection of « 8 and because* 
on «, no more than four singular points can lie. 
So all conics passing through S 12 and S lt and farthermore described* 
around one, hence around an infinite number of involution triangles | 
will form a pencil; the two other base points of this pencil are also j 
singular points. We determine these first: if we choose as ft the 
pair of lines S 2i and S as and as ft the pair S lA and S t6 , it is evident* 
that S lt and S 1& are the discussed base points. Therefore: If the 
10 singular points, hence also the double curve y s , of the involution^ 
are hnoum, we can generate the involution triangles in this way. * 
We can construct five different pencils of conics of which each % 
conic is described around an infinite number of polar triangles of 
which are then at the same time the involution triangles in view; i 
the base points of these pencils consist of the sets of points (S lt , SyM 
S 14 ,S lt ), (S l2 ,S„,S„,S 26 ), (S lt , S tt , S tA , S„), (S u , S 2i , S 34 , S ti ) and 
{S xi , s„, s tt , s 4i ). M 
These pencils we shall call in future respectively {Bi), (5,), (£,), 
(B 4 ) and (B t ). I 
If a x and a t are two conics, the first taken arbitrarily out of( J S 1 ),| 
the second arbitrarily out of {B % ), these two will have four points 
of intersection, viz. S lt and the vertices of an involution triangle. | 
Now it can happen in two different ways that 2 of these points of 
intersection coincide: 1. S 12 can be at the same time a vertex of the 
involution triangle 2. one of these vertices can lie on the double^ 
curve y s . In each of these two cases « x and « a will have only threej 
different points in common, but they will touch each other moreover ] 
in one of these points. 
7. Out of these 5 pencils we choose one — e.g. {Bi) — arbitra- i 
rily; an arbitrary conic ft out of {B t y is described around an infinite 
number of involution triangles whose vertices form in its circum¬ 
ference an involution of order three. The latter has four double points ? 
in the points of intersection of ft with y„ the double curve of the 
involution (*,). Inversely the conics of the pencil {Bi) determine an 
involution of order four on y 2 ; the latter has 6 double points in 
the points in which y, is touched by a conic out of. (J5 X )- I* 1 eac ^ 
of these points three points have thus coincided, forming together a 
group of (i,). 
The involution {i s ) has 6 triple points; in each of the points y,is 
touched by a conic out of each of the pencils {Bi), (B 2 ), {B t ), {B t ), 
and {Bi). 
