Mathematics. — “A Congruence of Conics’. By Prof. Jan px 
Vers. 
(Communicated at the meeting of January 31, 1920). 
1. We shall suppose, that a trilinear correspondence’) exists 
between the ranges of points (A,), (A), (A,) lying on the crossing 
straight lines a,, a,, «,. Through each triplet of corresponding points 
A,, A,, A, let a conic 2? be passed which intersects the fixed conic 
8? twice. The congruence [27], arising in this way, will be examined 
more closely; it passes into a congruence of circles, if 8? becomes 
the imaginary circle at infinity. 
2. Pairs or LINES. In four different ways 2? can degenerate into 
a pair of straight lines. 
1. One of the lines, g, rests on a,, d,, a, the other, h, lies in 
the plane 8 of 8°. 
If we keep the point A, fixed, A, and A, describe projective 
ranges, so that g,,—=A,A, describes a quadratic scroll. There are 
therefore two lines g,, resting on a,; the two supporting points 4’, 
will be associated to A,. Each point A’, belongs to one point A, ; for 
the transversal through A’, of a, and a, determines two points A,,4A,, 
hence one point A,. Three times A, coincides with A’,; there are 
therefore three lines g,,,, each containing a group A,, A,, A,. Each 
line /,,, in B, intersecting g,,,, forms together with this line a pair 
of lines belonging to the congruence. To group 1 belong accordingly 
three systems, each consisting of a fixed straight line and a ray of 
a plane pencil. 
2. One of the lines, g,,, rests on a, and a,, the other lies in @. 
To the intersection A,* of a, with B a seroll (g,,*) is associated, 
which intersects 8 in a conic y,,”. Hach ray of the plane pencil 
(A,*, 8) intersects on y,,’ two lines g,,, and forms with each of them 
a pair of lines. Group 2 contains therefore three systems, each con- 
sisting of a ray of a plane pencil and a straight line of a quadratic 
seroll. 
3. Let us denote the point of a, associated to A,*, A,*, by A,**. 
Sach line through A,** resting on A,* A,*, forms with the latter a 
') B, Srurm, Die Lehre von den geometrischen Verwandtschaften, 1, 320, 
