( 752 ) 
involution triangle. If we assume a point G of the conic y, as a 
vertex of an involution triangle, then one of the other vertices must 
coincide with G , so G is a double point of the involution; y„ the 
locus of these double points, is the double curve of the involution. 
Each line l whose pole with respect to y, is no singular point of 
the involution is a side of only one involution triangle, namely of 
that triangle having the pole of l as vertex. On the other hand 
each line whose pole is a singular point is a side of an infinite 
number of involution triangles all having that point as vertex. From 
this ensues that also the lines of V form a cubic involution {i\) of 
the first rank; the polar lines of the singular points of (*,) are the 
singular lines of the tangents of y 2 are its double lines and y* 
is its double curve. Both involutions are with respect to y, polarly 
related. 
The involution triangles of y, are all polar triangles of a selfsame j 
conic y„ which is at the same time the double curve of {i 9 ). The lines 
of V f(yrm an involution (£',) which is with respect to y, the polar 
figure of («,). Each polar triangle of y, having a singular point of 
the involution as vertex is at the same time an involution triangle , || 
3. We make a point describe a line a x and we ask after the locus 
of its conjugate points. If we draw through A x , the pole of a 1 with 
respect to y,, an arbitrary line p x , then P x , the pole of p x , lies on 
a x , whilst the two points conjugate to P x lie on p x ; these two points 
lie also on the locus under discussion. Moreover A x itself is conjugated J 
to two points of a x , so that A x is a double point of this curve and 
each line through A x cuts this curve in a double point and two 
points more. Hence we find: 
If one of the vertices of an involution triangle describes a linea lt 
then the two others describe a curve a* of order four with a node 
in A lt the pole of a x with respect to y,. As a x cuts all singular 
lines , all singular points lie on « 4 . 
A few properties of this curve a* may still be given here: 
1. Let A t and A t be the points conjugated to A x , then the polar 
line of A t with respect to y, — that is the line A x A t — must 
cut a 4 in A x and in the points forming with A , an involution triangle. 
These two points are A x and ^4 S . So will a* be touched in A x by 
the lines A x A t and A X A, ; A 2 and A t are points of intersection of 
a x and a*. 
2. Besides in A , and A s the curve a* will be intersected in two 
points more by a x ; these points are at the same time the points of 
intersection of a x and y 2 . 
