351] 
ON CUBIC CONES AND CURVES. 
413 
The two forms x 3 + y 3 + z 3 + Olxyz = 0, (X + Y + Z) :i + GkX YZ = 0 ; Enumeration of the 
Cones comprised therein. Nos. 30 and 31. 
30. I form the following Table: 
64 S 3 - T 2 
T 2 
1 
k 
s 
T 
1 ~6TS 3 ’ 
— GO 
- 00 
+ oo 
— 00 
+ 00 
0 1 
|(i+ V3) 
-3-V3 
f(3 + 2\/3) 
0 
81 (45 + 26 V3) 
1 
>- complex, 
- 
- 
+ 
+ 
+ 
+ > 
1 
2 
_ 
9 
ÏF 
2 7 
8 
0 
0 
acnodal, 
- 
- 
+ 
+ 
- 
_ ' 
0 
-4 
0 
1 
- 1 
± 00 
i 
4 
9 
“T 
0 ¡5 
2 5T 
351 
5 12 
7 2 9 
5 12 
5 1 2 
FTF 
- simplex trilateral, 
i(V3-i) 
-3 +V3 
- f (- 3 + 2 V3) 
0 
- 81 (- 45 + 26 V3) 
1 
+ 
- 
- 
- 
- 
+ > 
1 
0 
0 
-27 
-729 
00 
simplex neutral, 
+ 
+ 
+ 
- 
- 
i- simplex quadrilateral. 
00 
00 
00 
- 00 
00 
And I further describe as follows the nature of the cones which correspond to 
the several values of k and l. 
31. I between — oo and — or k between — oo and — f. 
The cone is complex. In the series, viz. corresponding to l = — (1 + V.3) or k = — 3 — V3, 
there is a special form which may be called the complex harmonic, viz. the four 
tangents from any line of the cone form a harmonic system: but observe, qua complex 
cone, the tangents are all real or all imaginary. I — — ^ (form fails), k = — •§, the cone 
is acnodal. I between — \ and 1, or k between — f and 0; the cone is simplex 
trilateral. In the series, viz. corresponding to 1 = 0 or k = — 4, there is a special 
form which might be called the quasi-neutral, the speciality having however reference 
to the imaginary inflexions, viz. corresponding to each real inflexion we have two 
imaginary inflexions such that the three tangents meet in a line. 
There is also corresponding to l = j, or k = — f, a form which seems to be a special 
one, though I have not ascertained wherein that speciality consists.