( 627 ) 
Mathematics. — “The plane curve of order 4 with 2 or 3 cusps 
and O or 1 nodes as a projection of the twisted curve of 
order 4 and of the 1% species.” By Prof. H. pe Vrizs. 
(Communicated in the meeting of January 30 1909.) 
1. If two quadratic cones are situated arbitrarily with respect 
to each other, they intersect each other in a twisted curve 7* of 
order 4 and of the 1“ species. If we suppose the plane t to be brought 
through a point O of the nodal curve of the developable belonging 
to rf in which plane lie the two tangents of r* passing through 
QO, then this plane must intersect the two cones according to conics 
k?,, k*,, touching each other in the points of contact O,, O, of 
the two indicated tangents with rf. We shall now suppose the first 
cone to be deterinined by the base-curve 4’, and the vertex A, the 
second by 4°, and the vertex S. The plane r is a double tangential 
plane of vr‘, so it must be a tangential plane of one of the four 
quadratic cones passing through rf; i.e. in t, and on the line O,0,, 
lies the vertex H of a third double projecting cone of7*; and finally 
the vertex 7’ of the fourth cone must then lie in the common polar 
plane of H with respect to the cones [R] and [S], and this plane 
must pass through O, because the double curve of the developable 
of r* consists of four plane curves of order four situated in the faces 
of the tetrahedron RS7 H, and O, as a point of this double curve, 
must thus lie in one of those faces, namely in the polar plane RS 7’ 
of H, because the points 0, and O,, whose tangents intersect each 
other in QO, lie on a straight line through H. The cone | 7’] intersects 
t in a conic A, likewise touching in O, and OQ, the lines OO,, 
OO,; the cone [H] on the contrary has with t only the line 0,0, 
counting double in common. 
2. If we project 7‘ out of O on an arbitrary plane 2, then the 
projection is a plane curve £* with two cusps in the points of inter- 
section of this plane with OO,, OO,; it is convenient to take for 
this plane of projection the polar plane of O with respect to the 
cone [H], because then O,, O,, together with two other important 
points — of which we shall soon hear more — coincide with their 
projections; the cuspidal tangents are nothing but the traces of the 
osculating planes of 7* in O,, O, with 2. 
The plane 2 intersects the cone [H] in two generatrices ; 
one is O,0,, the other intersects r‘ in two points D,,D, coinciding 
with their central projections on zr, and in which &‘ touches the 
