497 
and as now the plane through ¢ and the chord a through P is deter- 
mined unequivocally, and as in this plane only two chords 6 lie, 
point P counts only once. 
3. Through &* pass four quadratic cones whose vertices we shall 
call 7’, … 7. These vertices too behave themselves somewhat irregularly 
with respect to the question put originally, for an arbitrary plane 
e.g. through the line « passing through 7’, contains always two chords 
hb, so that also the four vertices of the cones regarded by themselves 
satisfy the given question an infinite number of times; nevertheless 
these points are for 2° only single points. 
This can be proved most easily with the aid of the edges of the 
tetrahedron 7’,....7,. Let us consider e.g. 7,7, and let us regard 4* 
as the intersection of the two cones having 7’, and 7’, as vertices. All 
points P of 7,7, have with respect to the first cone only one polar 
plane a,, viz. the plane 7,7,7’,, and likewise with respect to the 
second cone only one polar plane z,, viz. 7,7,7, ; the line of inter- 
section 7,7, is therefore the line s for all points P of 7,7’, or in other 
words the planes Ps (or b,6,) for all points of 7,7, form a pencil 
of planes around the edge 7,7. The question is to find the points 
P of TT, for which the chord a of 4° passing through P lies in 
the plane Ps and to this end we have but to intersect each plane Ps 
by 4°, by means of which we find in each suchlike plane three 
chords a forming altogether a scroll 2* of order four with £* as 
a nodal curve and s as asingle directrix. For, through a point of s only 
one chord a passes, whilst in a plane through s three of suchlike 
chords are lying, and through a point of 4° evidently two chords a 
pass intersecting s. Now this scroll * intersects 7’, 7’, in four points, 
but to these 7, and 7’, themselves do not belong, because no reason 
whatever can be given why of the three chords « in the plane 7, 7,7’, 
e.g. just one should pass through 7’,; so we find on 7,7’, four points 
of intersection besides the two vertices of the cones, and as the latter 
of course likewise belong to the surface they count once on 7,7, 
and therefore likewise in general. 
If we determine the points of intersection of 2° with the chord a 
through 7’, then we find that the two points which this chord has 
outside 4 in common with the surface ($ 1) coincide with 7), which 
with a view to the preceding means that a touches the surface in 7, 
We endeavour also to acquire on this specia! chord a the (2,2) 
correspondence of § 1, which is easily done and where we have but 
this to remark, that in the plane 5,5, as well as in the plane b,*b,* 
the four points of £* lie two by two on two lines through 7). If 
33 
Proceedings Rayal Acad. Amsterdam. Vol. XY. 
