1135 
The intersections of 9* with a tangent plane of G are the images 
of four circles passing through a given point and touching at C,, 
C, (§ 3). 
The circles touching at a given straight line are represented by 
a cylindrical surface that envelops G and of which the straight 
lines are perpendicular to the given straight line consequently 
parallel to the plane XOY. 
Mathematics. — “A Quadruply Infinite’ System of Point Groups 
in Space’. By Dr. Cus. H. van Os. (Communicated by Prof. 
JAN DE VRrES). 
(Communicated in the meeting of January 27, 1917). 
Let a pencil (a°) be given, consisting of cubic surfaces a®. An 
arbitrary straight line / is touched by four surfaces a° of the pencil. 
As the space contains oo* lines /, there are oo“ groups of four points 
of contact. We shall indicate this system of groups of four points by S*. 
§ 1. If we take for the line / a line g lying on one of the sur- 
faces a°, the four surfaces mentioned coincide with this surface a’, 
while the points of contact become indefinite. These straight lines 
g are therefore singular lines of S*. They form a ruled surface 2, 
of which we shall determine the order. 
A line g intersects a second surface a* in three points lying on 
the base-curve 9° of the pencil (a*); the lines g are therefore trise- 
cants of the curve @°. If on the other hand we consider a trisecant 
of 9’, the surface a’, which passes through an arbitrary point of 
this trisecant will have four, consequently an infinitely great number 
of points in common with it, so that the trisecant is a straight line y. 
Through an arbitrary point pass 18 bisecants of 9° *), the genus 
of o° amounts consequently to 4 X 8 x 7—18 = 10. If we therefore 
project the curve o’ out of one of its points, we get as projection 
a curve of order eight with 5 X 7 X 6-10 = 11 nodes. Through 
the said point pass therefore 11 trisecants of 0’, so that the surface 
R has the curve o° as 11-fold curve. 
A surface «@ intersects the surface R along the curve ¢" and 
according to the 27 straight lines g lying on a’, the order of R 
amounts to 42. 
§ 2. Any line / passing through a given point P contains one 
1) Cf. e.g. ZeuTHEN, Lehrbuch der abzählenden Geometrie, page 46. 
