Mathematics. — “On an involution among the rays of space”. 
By Prof. Jan pe Vrins. 
(Communicated at the meeting of September 29, 1918). 
1. Among the rays of space four arbitrary plane pencils of rays 
determine an involution of pairs of rays; each pair consists of the 
two transversals ¢,t’ of four rays a,b,c,d appertaining one to each of 
the four pencils. The pencil (a) we shall also denote by (4,«); A 
is the vertex, « the plane of (a). Similarly the other pencils are 
denoted by (B,8), (Cy), (D,9). 
A straight line ¢ determines four rays a,6,c,d, which, in general, 
have still another transversal ¢’. If «,b‚c,d appertain to a quadratic 
regulus, then each line of the complementary, regulus is conjugated to 
every other line of this complementary regulus. 
If ¢ describes the pencil (Yv), ¢’ engenders a ruled surface which 
we shall denote by (¢’)", where « is the degree of this surface. By 
the rays of (77) the four pencils (7), (0), (c), (d) are rendered projective. 
2. We now suppose that in some way a projective correspondence 
is established between these pencils and consider the ruled surface 
T engendered by the transversals 4! of four corresponding rays. 
From A the pencils of rays (6), (c), (d) are projected by three 
projective pencils of planes; hence three transversals of corresponding 
rays 6,c,d exist which intersect the corresponding ray a at A, so 
that 4,B,C,D are triple points of 7. Similarly «,g,y,d are triple 
tangent planes, since they contain three lines /. 
The intersection of 7’ and « thus consists of three straight lines 
and a curve. which has a triple point at A. Since every ray a 
contains the points of transit of a pair of transversals ft’, the said 
curve is of the fifth degree. Hence T'is a ruled surface of the eighth 
degree. 
Hight tangents of the curve «°‚ which 7’ has in common with a, 
pass through A; it follows from this that 7’ contains eight rays t, 
which coincide each with its corresponding ray ¢’. 
3. If ¢ is made to describe the pencil of lines (7x), the ruled 
surface 7’ breaks up into the pencil (¢) and a ruled surface (¢’)’. 
Hence the transformation (t,t) converts a plane pencil of lines into 
a ruled surface of the seventh degree. 
