he 
having with 4° the line / in common. So the locus of Q’ = kl is 
a curve /* through FR and R’, as / is to have two points in common 
with it and R and Z2 correspond amongst other points with them- 
selves; the curves 4’ and 0‘ have four more points in common. 
§ 4. If / is a unisecant of e* in RF, the locus (Q) degenerates 
into the tangent r and a 4°. Any plane through / contains besides 
R three points Q; of these two must be combined with A, if the 
plane contains the tangent 7. The quadrisecants q of / become here 
trisecants; for 7 rests on each of the polar lines g of / ($2). The 
plane gr touches 9‘ in Mè and contains therefore two points Q united 
in A. In relation with the results obtained we conelude from this 
that by the correspondence (PQ) to a unisecant of v* a twisted 
curve of order jive is conjugated having a node in the point common 
to the unisecant and 9, the nodal tangents lying in the plane /r. 
So the curve is of rank ten. Through / pass six common tangential 
planes of of and 4°, 
§ 5. The vertices 7; of the four quadratrie cones containing 9% 
are singular points of the correspondence (P,Q). For 7, 
chords and the corresponding points Q lie on the conic t°, common 
to the polar plane rt, = 7,77’, of 7, and the quadratic cone with 
T, as vertex. 
To the line 7,7, as locus of points P correspond in the first place 
the two conics 1,’ and r,* and moreover the line 7,7’, counted twice. 
bears oc! 
For the points Sp in any plane through SS, form a complete 
quadrangle of which 7, and 7, are diagonal points; in the third 
diagonal point @,, and Q,, coincide, whilst of the remaining four 
points Q two lie in +r, and two in tr. So to any point of 7,7, 
correspond two points of 7,7, and inversely. 
If 7 contains the point 7, only, the six points Q lying in a plane 
) through / consist of two points in +, and ont,” and of four points 
lying on the line common to 4 and the polar plane of / with respect 
to the cone projecting g* out of 7). Then the curve (Q) breaks up 
into the conte 1,7 and a plane curve 4*. In the two tangential planes 
of the cone passing through / the two points Q lying on t,* coincide 
with two of the remaining four if a point of intersection of T° 
and 4* where the latter is touched by the edge of contact. 
§ 6. Let us now consider the surface of the points .Q corre- 
sponding to the points P of a plane M. If Se are the points common 
to WZ and 9%, the six lines SiS; form the intersection of 77 with the 
Gor 
