DO 
lying in «; then we can also find in 3 a curve of order 6 which 
must lie on the surface. Finally in a third plane y we have now 
of course to assume arbitrarily only 6 points and then the surface 
is determined; for every arbitrary fourth plane cuts the three curves 
lying in a, 8, y together in 18 single points, and 4” in three points 
which must be double points, by which the section of the surface 
to be constructed is determined. Besides 4° we therefore want 
18 + 12 + 6 = 86 points to determine the surface; so the condition 
that 4% isa nodal curve is equivalent to 83 — 36 = 47 single conditions. 
If &* is-to lie on the surface of order six, then we have to take care 
that it must have twenty-five points in common with the surface ; so 
k® as a double curve and #* as a single curve absorb 47 + 25 = 72 
single conditions, so that but 88—72—11 conditions are left. Now 
a common chord of 4* and &* bas with every surface of order six 
passing twice through /* and once through /* in its points of inter- 
section with both curves exactly six points in common with this; 
thus by distributing the eleven points which are left among eleven 
of the twenty common chords, we can be sure that also these 
eleven chords will come to lie on the surface. However, we know 
that on our surface 2° all the twenty common chords lie; so we 
can state the following theorem: the twenty common chords of k* 
and k* lie on a surface 2° of order 6 passing twice through k* and 
once through k*; it is the locus of all the points of space for which 
the triplet of chords a + 26 ts complanar. 
6. The first polar surface of an arbitrary point © of space 
with respect to 2° is a surface #,° of order five passing once 
through #°; the complete section with 2°, which must be of order 
thirty, breaks up into #® counted twice and a residual section 7?‘ 
of order twenty-four, from which ensues immediately that the apparent 
circuit of 2° out of an arbitrary point of space on an arbitrary plane 
is a curve of order twenty-four. 
The curve 7** has as is easy to see twelve points in common 
with #°. The second polar surface of O, viz. a surface I,‘ of order 
four, does not contain #*, so it intersects it in twelve points; these 
are the points which #* and 7? have in common. If namely we connect 
Q with an arbitrary point P of 77‘, then OP is a tangent in P of 
2°; now if P lies on 4 then OP touches in P one of the sheets 
of 2° passing through /*, but in consequence of this on the line 
OP lie united in P three points of 2°, and therefore two of J7,, and 
one of 7/,. Each of these twelve points counts for three coinciding points 
of intersection of 2° with its two polar surfaces; for, if we intersect 
