. 
Mathematics. — “On an involution among the rays of space’. By 
G. SCHAAKE. (Communicated by Prof. Jan pr VRrws). 
(Communicated at the meeting of January 25, 1919). 
In the Proceedings of this Academy of the meeting of September 
29, 1918, XXVII, p. 256 a communication of Prof. Jan pr Vrins is to be 
found, dealing with an involution of rays determined by four plane 
pencils of rays which are arbitrarily chosen in space. In the sequel 
this investigation will be completed on certain points 
1. In § 4 of the communication referred to above a system of 
o* ruled quadries is mentioned of which each regulus has one 
generator in each of the three line-pencils (4), (c) and (d). The num- 
ber of these surfaces passing through three points is there determined 
by choosing one point in each of the planes 8, y and À containing 
the pencils. Then in the first place the rays 4, ce and d, which pass 
through these points, certainly furnish a ruled surface (bed) which 
satisfies the condition. In addition to this, however, it is possible to 
construct for instance a surface of which the generators passing through 
the points chosen in the tangent planes 8 and y do not belong to the 
system of rays 6, c, d, whilst the third generator passing through 
the point in the plane 0 is one of the system (d). Hence the system 
of quadries is not linear. 
Now consider the quadries of the system which pass through an 
arbitrary point P. On each of these surfaces lies a line, passing 
through P, which meets the lines 6, c and d, by which the surface 
is defined. 
The rays passing through P determine a trilinear correspondence 
Tp between the pencils (4), (c) and (d). Each triplet furnishes a 
surface of our system of quadrics passing through P. Hence, if we 
want to know how many of the ruled quadrics pass through three 
given points P, Q and FR, we must discover how many triplets of 
rays the three trilinear correspondences Tp, Tg and 7'r have in 
common. 
Such a trilinear correspondence, however, is represented by an 
equation between the direction-parameters of 6, c and d. If we regard 
these quantities as coordinates with respect to a Cartesian system of 
axes in space, the equation represents a cubic surface, for conve- 
