1465 
are obtained by leaving a column out of the matrix of (4). As 
each matrix arising from two columns becomes zero for three points, 
the number of solutions of the system is sur. 
On @&° lie therefore sir straight lines gr, which are component 
parts of degenerated figures 9°; they form a sertuple. 
The number of the v* composed of a straight line and a conic 
may also be determined by using the invariant of the six linear 
forms of the matrix *) 
| " 
firs 
' 
Pee Bc. “Pp 
U " 
Cr CE CRT 
|= 0, where px = aa, + pbz, etc. 
We find that the invariant disappears for six values of « : 8, so 
that there will be six degenerated figures 0°. 
§ 3. Let g,° be the conic forming with the straight line g, a 
figure 0°, f, the straight line, which ®* has moreover in common 
with the plane 6, of o,°. As, with the exception of C, any point of 
®* bears only one curve 9’, the five straight lines gz (‘= 2 to 6) 
will rest on /,; the line f, will moreover be intersected twice by 
any 9°, consequently be a singular bisecant. On the other hand a 
singular bisecant can only be situated in the plane of a conic 97,. 
The sir singular bisecants fy form evidently a bisertuple with the 
siz straight lines 7. 
The remaining 15 straight lines of ®*, which we may indicate 
by Ag, ave intersected by all 9°; on each conic ¢z’ rest ten straight 
lines h,, (p and q Ek). The lines h might be called singular secants; 
for an arbitrary line intersects only three curves ¢’. 
§ 4. The system (1) may be replaced by 
Sy! aar Bb, 
3 (aaz+ Bbx) aa', + Bb'; = aa", + po", 
Tie, = c's ik Cle 
3 
So the bisecants of the curves g° are indicated hy 
4 (aax+ bz) + A (aa + Bb'r) + A" (aa";+pb"7) = 0 
Were Nea AC 0 
If X and Y are two points of a bisecant, the system 
OPL NON Nb — 058 > cr — 0} 
aia, + BLAb,=—0, Zie, =0. 
is satisfied. 
1) Cf. M. Stuyvaerr ‘Invariantologie de la cubique gauche” (Bull. de l'Acad, 
Royale de Belgique 1907, p. 515), 
