1134 
group of S*; these groups of four points form a surface 1. If we 
take for the line / a line that touches the surface a’ passing through 
P in P, one of the points of the associated group will lie in P. 
The surface 77 passes therefore through P and touches there at the 
surface a’ passing through P, because the tangents of WZ in P are 
also the tangents of @ in P. The surface 11 has therefore a single 
point in P. Any line / passing through P has therefore with 7 _ 
five points in common. This surface is therefore of order five. It 
is easy to see that it is the polar surface of P with regard to the 
pencil (a’). ¢ 
If the line 7 passes through a point Q of @°, two of the surfaces 
a’, which touch at /, will coincide into the surface that touches at 
lin the point Q; the associated intersections of / and JI conse- 
quently coincide also. The surface 11 therefore passes through 9° 
and touches along this curve at the cone which projects 9’ out of P. 
The lines /, for which one of the points of the group of four 
points lying on it lies in P, are the tangents in P of the surface a’ 
passing through P. The locus of the remaining points of these groups 
is obviously the intersection of the surface // with the tangent plane 
in P, so a curve of order jive, which has a node in P. 
§ 3. If a line / intersects the curve @° in a point P, the two 
surfaces a*, which touch at /, will coincide into the surface that 
touches at / in P. If we therefore cause the line / to rotate round 
P, 2 points of the group lying on / will lie in P, so that the line 
/ intersects the surface ZI’ belonging to P only in two points outside 
P. This surface I’ has consequently in P a triple point. 
The points of 9° 
them belongs to oo* groups, while an arbitrary point belongs to 
oo* groups. 
Let us now take for the point P the-conical point of one of the 
32 nodal surfaces a*. For any of the lines / passing through P, 
this surface belongs to the surfaces a’, which touch at /, so that 
one of the points of the group lying on / lies in P. This point P 
therefore is also a singular point of S*. Any straight line passing 
through P intersects the surface 11° belonging to P in three points 
lying outside P; this surface therefore has a conical point in P. 
§ 4. We shall now consider the coincidences of S*. If two of the 
surfaces a> which touch at a line /, coincide, two of the coincidences 
of the involution that is determined by the pencil (a*) on the line / 
will coincide, This may 1 take place on account of three associated 
“ry \ 
are therefore singular points of S*; for each of | 
